PARALLEL DISTRIBUTED

PROCESSING

Computational Models of Cognition and Perception

Editors

Jerome A. Feldman Patrick J. Hayes David E. Rumelhart

Parallel Distributed Processing: Explorations in the Microstructure of Cognition. Volume 1: Foundations, by David E. Rumelhart, James L. McClelland, and the PDP Research Group

Parallel Distributed Processing: Explorations in the Microstructure 0/ Cognition. Volume 2: Psychological and Biological Models, by

James L. McClelland, David E. Rumelhart, and the

PDP Research Group

Neurophilosophy: Toward a Unified Science o/the Mind-Bra;n, by

Patricia S. Churchland

Qualitative Reasoning About Physical Systems, edited by

Daniel G. Bobrow

Visual Cognition, edited by Steven Pinker

PARALLEL DISTRIBUTED

PROCESSING

Explorations in the Microstructure

of Cognition

Volume 1: Foundations

David E. Rumelhart James L. McClelland

and the PDP Research Group

Chisato Asanuma Francis H. C. Crick Jeffr<sup>e</sup>y L. Elman Geoffrey E. Hinton Michael 1. Jordan

Alan H. Kawamoto Paul W. Munro Donald A. Norman Daniel E. Rabin Terrence 1. Sejnowski

Paul Smolensky Gregory O. St<sup>o</sup>n<sup>e</sup> Ronald 1. Williams David Zipser

Institute for Cognitive Science University of California, San Diego

A Bradford Book

The MIT Press Cambridge, Massachusetts London, England

Twelfth printing, 1999

© 1986 by The Massachusetts Institute of Technology

All ri ghts reserved. No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recordin g, or information

storage and retrieval) without permission in writing from the publisher.

Printed and bound in the United States of America

Library of Congress Cataloging-in-Publication Data

Rumelhart, David E.

Parallel distributed processing.

(Computational models of cognition and perception)

Vol. I by David E. Rumelhart. James L. McClelland and the PDP Research Group.

"A Bradford book.'·

Includes bibliographies and indexes.

Contents: v.l. Foundations-v.2. Psychological

and biological models.

I. Human information processing. 2. Cognition. I. McClelland, James L.

II. University of California. San Diego. PDP Research Group. Ill. Title. IV. Series. BF 45 S. R853 1986 153 85-24073 ISBN 0-262-18120-7 (v.1 ) he ISBN 0-262-68053-X (v. I ) pb 0-262-13218-4 (v.2) 0-262-63110-5 (v.2) 0-262-18123-1 (set) 0-262-631 12-1 (set)

Preface

Acknowledgments

VOLUME 1 FOUNDATIONS

Addresses of the PDP Research Group

Part I THE PDP PERSPECTIVE

Contents

ix

xv xix

1 The Appeal of Parallel Distributed Processing 3

1. L. MCCLELLAND, D. E. RUMELHART, and G. E. HINTON

2 A General Framework for Parallel Distributed Processing 45

D. E. RUMELHART, G. E. HINTON, and 1. L. MCCLELLAND

3 Distributed Representations 77

G. E. HINTON, 1. L. MCCLELLAND, and D. E. RUMELHART

4 PDP Models and General Issues in Cognitive Science 110

D. E. RUMELHART and 1. L. MCCLELLAND

Part II BASIC MECHANISMS 147

5 Feature Discovery by Competitive Learning 151

D. E. RUMELHART and D. Z1PSER

6 Information Processing in Dynamical Systems:

Foundations of Harmony Theory 194

P. SMOLENSKY

7 Learning and Relearning in Boltzmann Machines 282

G. E. HINTON and T. 1. SEJNOWSKI

8 Learning Internal Representations by Error Propagation 318

D. E. RUMELHART, G. E. HINTON, and R J. WILLIAMS

Part III FORMAL ANALYSES 363

9 An Introduction to Linear Algebra in Parallel Distributed

Processing 365

M. I. JORDAN

10 The Logic of Activation Functions 423

R. J. WILLIAMS

11 An Analysis of the Delta Rule and the Learning of

Statistical Associations 444

G.O STONE

12 Resource Requirements of Standard and Programmable

Nets 460

J L. MCCLELLAND

13 P3: A Parallel Network Simulating System 488

D. ZIPSER and D. E. RABIN

References 507 Index 517

VOLUME 2 PSYCHOLOGICAL AND BIOLOGICAL MODELS

Preface to Volume 2

Addresses of the PDP Research Group

Part IV PSYCHOLOGICAL PROCESSES

IX xi

14 Schemata and Sequential Thought Processes in PDP Models <sup>7</sup> D. E. RUMELHART. P. SMOLENSKY, 1. L. MCCLELLAND, and G. E. HINTON

15 Interactive Processes in Speech Perception:

The TRACE Model 58

J. L. MCCLELLAND and J. L. ELMAN

16 The Programmable Blackboard Model of Reading 122

J L. MCCLELLAND

17 A Distributed Model of Human Learning and Memory 170

J. L. MCCLELLAND and D. E. RUMELHART

18 On Learning the Past Tenses of English Verbs

D. E. RUMELHART and J. L. MCCLELLAND

19 Mechanisms of Sentence Processing: Assigning Roles

216

to Constituents 272

J. L. MCCLELLAND and A. H. KAWAMOTO

Part V BIOLOGICAL MECHANISMS 327

20 Certain Aspects of the Anatomy and Physiology of the

Cerebral Cortex 333

F H c. CRICK and C. ASANUMA

<sup>21</sup>Open Questions About Computation in Cerebral Cortex <sup>372</sup>

T. 1. SEJNOWSKI

22 Neural and Conceptual Interpretation of PDP Models 390

23 Biologically Plausible Models of Place Recognition and

Goal Location 432

D. ZIPSER

24 State-Dependent Factors Influencing Neural Plasticity:

A Partial Account of the Critical Period 471

P. W. MUNRO

25 Amnesia and Distributed Memory J. L. MCCLELLAND and D. E. RUMELHART

Part VI CONCLUSION

26 Reflections on Cognition and Parallel Distributed

Processing D. A. NORMAN Future Directions

References

Index

503

529

531

547 553 581

Preface

One of the great joys of science lies in the moment of shared discovery. One person's half-baked suggestion resonates in the mind of another and suddenly takes on a definite shape. An insightful critique of one way of thinking about a problem leads to another, better understanding. An incomprehensible simulation result suddenly makes sense

as two people try to understand it together.

This book grew out of many such moments. The seeds of the book were sown in our joint work on the interactive activation model of word perception. Since then, each of us has worked with the other and with other collaborators. The results of these collaborations are reported in several of the chapters of this book. The book also contains many chapters by other colleagues whose explorations have become intertwined with ours. Each chapter has its own by-line, but each also reflects the influences of other members of the group. We hope the result reflects some of the benefits of parallel distributed processing! The idea of parallel distributed processing-the notion that intelligence emerges from the interactions of large numbers of simple processing units-has come and gone before. The idea began to seem more and more attractive to us as the contrast between our convictions about basic characteristics of human perception, memory, language, and thought and the accepted formal tools for capturing mental processes became more apparent. Symbol-processing machines, for all their Turing equivalence, had fa�}RdHtlJ}j<AiaHfnt�1 frameworks for capturing

the simple insights about the interactive nature of processing that had lead to such models as the HEARSAY model of speech understanding. More generally, they had failed to provide a framework for representing knowledge in a way that allowed it to be accessed by content and effectively combined with other knowledge to produce useful automatic syntheses that would allow intelligence to be productive. And they made no contact with the real strengths and weaknesses of the hardware in the brain. A Cray computer can perform on the order of 100 million double-precision multiplications in a second, but it does not exhibit natural intelligence. How then are we to understand the capabilities of human thought, given the time constants and noisiness inherent in neural systems? It seemed obvious that to get any processing done in real time, the slow, noisy hardware in the brain would have

to do massively parallel processing.

As our interest in parallel mechanisms developed, we began to study the work of others who shared our convictions and to build on their work. Particularly important in this regard was Hinton and 1. A. Anderson's (1981) Parallel Models oj Associative Memory. Indeed, we see our book as a descendant of their book on two accounts. First, the material presented here represents further developments on the work presented in Hinton and Anderson's book. Second, we owe a particular intellectual debt to both Hinton and Anderson. Our interest in distributed, associative memories goes back to interactions with Jim Anderson, beginning as early as 1968. Our interest in these topics began in earnest, however, during the period when we were developing the interactive activation model of word perception, in 1979, shortly after Geoffrey Hinton began a postdoctoral fellowship at UCSD. Geoffrey's crisp explanations showed us the potential power and generality of models created from connections among simple processing units, and fit together nicely with our own developing conviction that various aspects of perception, language processing, and motor control were best thought of in terms of massively parallel processing (see McClelland, 1979, and Rumelhart, 1977, for our earliest steps in this direction). The project culminating in this book formally began in December, 1981 when the two of us and Geoffrey Hinton decided to work together exploring the implications of network models and to write a book outlining our conclusions. We expected the project to take about six months. We began in January 1982 by bringing a number of our colleagues together to form a discussion group on these topics. During the first six months we met twice weekly and laid the foundation for most of the work presented in these volumes. Our first order of business was to develop a name for the class of models we were investigating. It seemed to us that the phrase parallel distributed processing (PDP

for short) best captured what we had in mind. It emphasized the parallel nature of the processing, the use of distributed representations and distributed control, and the fact that these were general processing systems, not merely memories we were studying, as the phrase associative memory suggests. Thus the PDP research group was born. Hinton and McClelland left after the first six months- Hinton to CMU and McClelland to MIT and later to CMU. The PDP research group, however, has continued regular meetings at UCSD up to the present time. The group has varied from five or six of us at times to as many as 15 or more at other times, and there is now a parallel group of about 15 or

so psychologists and computer scientists at CMU.

Shortly after leaving UCSD in 1982, Hinton began working with Terrence Sejnowski on the Boltzmann machine (Chapter 7) and decided to dt"�p from the role of organizer of the project to a con<sup>t</sup> <sup>r</sup><sup>i</sup> buto<sup>r</sup> , so he could spend more time working on the implications of the Boltzmann machine. Thus, the primary responsibility for putting the book together fell to the two of us. At first we expected to complete the book within a year after we began our work. Soon, however, it becam<sup>e</sup> clear that there was much work to be done and many directions to explore. Thus, our work continued and expanded as we and our colleagues followed the implications of the PDP approach in many dif-

ferent ways .

A good deal has happened since we began this project. Though much of the initial groundwork was laid in early 1982, most of the material described in these volumes did not take its present form until much

later.

The work has been interdisciplinary and represents what we consider a true cognitive science approach. Although the two of us have been trained as cognitive psychologists, the PDP group as a whole includes people from a wide range of backgrounds. It includes people trained in physics, mathematics, neuroscience, molecular biology, and computer sciences, as well as in psychology. We also envision an interdisciplinary audience for our book. We are cognitive psychologists and we hope, primarily , to present PDP models to the community of cognitive psychologists as alternatives to the models that have dominated cognitive psychology for the past decade or so. We also, however, see ourselves as studying architectures for computation and methods for artificial intelligence. Therefore, we hope that this book will be seen as relevant to researchers in computer science and artificial intelligence. Also, the PDP approach provides a set of tools for developing models of the neurophysiological basis of human information processing, and so we hope portions of these books will seem relevant to neuroscien-

ORGANIZATION OF THE BOOK

Our book consists of six parts, three in each of the two volumes. The overall structure is indicated in the accompanying table. Part I provides an overview. Chapter 1 presents the motivation for the approach and describes much of the early work that lead to the developments reported in later sections. Chapter 2 describes the PDP framework in more formal terms. Chapter 3 focuses on the idea of distributed representation, and Chapter 4 provides a detailed discussion of several general issues that the PDP approach has raised and explains how these

issues are addressed in the various later chapters of the book.

The remaining parts of the book present different facets of our explorations in parallel distributed processing. The chapters in Part II address central theoretical problems in the development of models of parallel distributed processing, focusing for the most part on fundamental problems in learning. The chapters in Part III describe various mathematical and computational tools that have been important in the development and analysis of PDP models. Part IV considers

> A CONDENSED TABLE OF CONTENTS VOLUME I

I. THE PDP PERSPECTIVE II. BASIC MECHANISMS

1. The Appeal of PDP 5. Competitive Learning 2. A Framework for PDP 6. Harmony Theory 3. Distributed 7. Boltzmann Machines Representations 8. Learning by 4. General Issues Error Propagation

IV. PSYCHOLOGICAL PROCESSES

14. Schemata and PDP 15. Speech Perception 16. Model of Reading 17. Learning and Memory 18. Morphology Acquisition 19. Sentence Processing

VOLUME II

V. BIOLOGICAL MECHANISMS

20. Anatomy and Physiology 21. Computation in the Brain 22. Neural and

Conceptual Levels 23. Place Recognition 24. Neural Plasticity 25. Amnesia

III. FORMAL ANALYSES

9. Linear Algebra 10. Activation Functions II. The Delta Rule 12. Resource Requirements 13. Parallel Network Simulator

VI. CONCLUSION

26. Renections Future Directions

applications and implications of PDP models to various aspects of human cognition, including perception, memory, language, and higherlevel thought processes. Part V considers the relation between parallel distributed processing models and the brain, reviews relevant aspects of the anatomy and physiology, and describes several models that apply PDP models to aspects of the neurophysiology and neuropsychology of information processing , learning , and memory. Part VI contains two short pieces: a reflection on PDP models by Don Norman and a brief

discussion of our thoughts about promising future directions .

How to read this book? It is too long to read straight through. Nor is it designed to be read this way. Chapter 1 is a good entry point for readers unfamiliar with the PDP approach, but beyond that the various parts of the book may be approached in various orders, as one might explore the different parts of a complex object or machine. The various facets of the PDP approach are interrelated, and each part informs the others; but there are few strict sequential dependencies. Though we have tried to cross - reference ideas that come up in several places, we hope that most chapters can be understood without reference to the rest of the book. Where dependencies exist they are noted in the intro-

ductory sections at the beginning of each part of the book.

This book charts the explorations we and our colleagues have made in the microstructure of cognition. There is a lot of terrain left to be explored. We hope this book serves as a guide that helps others join us

in these ongoing explorations.

December 1985 James L. McClelland

PITTSBURGH, PENNSYLVANIA

David E. Rumelhart LA JOLLA, CALIFORNIA

Acknowledgments

As we have already said, nearly all the ideas in this book were born out of interactions, and one of our most important acknowledgments is to the environment that made these interactions possible. The Institute for Cognitive Science at UCSD and the members of the Institute have

made up the core of this environment.

Don Norman, our colleague and friend, the Founder and Director of the Institute, deserves special credit for making ICS an exciting and stimulating place, for encouraging our explorations in parallel distributed processing, and for his central role in arranging much of the financial support this book has benefited from {of which more below} . The atmosphere depends as well on the faculty, visiting scholars, and graduate students in and around ICS. The members of the PDP Research Group itself, of course, have played the most central role in helping to shape the ideas found in this book. All those who contributed to the actual contents of the book are listed on the cover page; they have all contributed, as well, in many other ways. Several other participants in the group who do not have actual contributions to the book also deserve mention. Most prominent among these are Mike Mozer and Yves Chauvin, two graduate students in the Cognitive Science Lab, and Gary Cottrell, a recent addition to the grou<sup>p</sup>from the University of

Rochester.

Several other membe<sup>r</sup>s of the intellectual community in and around ICS have played very important roles in helping us to shape our thoughts. These include Liz Bates, Michael Cole, Steve Draper, Don Copyrighted Material

Gentner. Ed Hutchins, Jim Hollan, Jean Mandler, George Mandler, Jeff Miller, Guy van Orden, and many others, including the participants

in Cognitive Science 200.

There are also several colleagues at other universities who have helped us in our explorations. Indeed, the annual connectionist workshops (the first of which resulted in the Hinton and Anderson book) have been important opportunities to share our ideas and get feedback on them from others in the field. and to learn from the contributions of others. Jim Anderson, Dana Ballard, Jerry Feldman, Geoff Hinton and Terry Sejnowski all had a hand in organizing different ones of these meetings; and we have learned a great deal from discussions with them and other participants, particularly Andy Barto, Scott Fahlman, Christof von der Malsburg, John Hopfield, Dave Touretzky, and more recently Mark Fanty and Gene Charniak. McClelland's discussions at MIT (particularly with Jerry Fodor and Molly Potter) helped in the clarification of several aspects of our thinking, and various colleagues at and around CMU -particularly John Anderson, Mark Derthick, Dave Klahr, Brian MacWhinney, and Jeff Sokolov-have contributed a great deal through discussions over the last year and a half or so, as we have worked toward the completion of the book. Others one or both of us have interacted with a great deal include Bill Brewer, Neal Cohen, AI Collins, Billy Salter, Ed Smith, and Walter Schneider. All of these people have contributed more or less

directly to the development of the ideas presented in this book.

An overlapping group of colleagues deserves credit for helping us improve the book itself. Jim Anderson, Andy Barto, Larry Barsalou, Chris Reisbeck, Walter Schneider, and Mark Seidenberg all read several chapters of the book and sent useful comments and suggestions. Many other people read and commented on individual chapters, and we are sincerely grateful for their careful contributions, which we acknowledge

in the appropriate chapters.

This project owes a tremendous amount to the help of the excellent staff of the Institute for Cognitive Science. Kathy Farrelly, in particular, has played an enormous role in all aspects of the production of the book; her cheerful, thoughtful, and very careful assistance made the production of the book run much more smoothly than we have had any right to hope and allowed us to keep working on the content of some of the chapters even as the final production was rolling forward on other sections. Eileen Conway's assistance with graphics and formatting has also been invaluable and we are very grateful to her as well. Mark Wallen kept the computers running, served as chief programming consultant and debugger par excellence, and tamed troff, the phototypesetter. Without him we would never have gotten all the formatting to come out right. Karol Lightner worked very hard toward the end of the Copyrighted Material

project on final proofing and indexing, and Sondra Buffett, as the Administrative Director of ICS, held everything together and kept everything running throughout the entire four years of the project.

Our project has been supported by a number of different agencies and foundations. Primary support came from the System Development Foundation and the Office of Naval Research. The System Development Foundation has provided direct support for the PDP research group through a grant to Norman and Rumelhart, and has also supported several of the individual members of the group (Crick, Hinton, Sejnowski, and Zipser). ONR contracts that have contributed support include N00014-79-C-0323, NR 667-437; N00014-85-K-0450, NR 667-

548; and N00014-82-C-0374, NR 667-483.

The people behind both SDF and ONR deserve acknowledgment too. The entire PDP enterprise owes a particular debt of gratitude to Charlie Smith, formerly of SDF, who appreciated the appeal of para\1el distributed processing very early on, understood our need for computing resources, and helped provide the entire PDP research group with the funds and encouragement needed to complete such project. Henry Halfr, formerly of ONR, was also an early source of support, encouragement, and direction. Charlie Smith has been succeeded by Carl York, and Henry Halff has been succeeded by Susan Chipman, Michael Shafto, and Harold Hawkins. We are grateful to a\1 of these people for their commitment to the completion of this book and to the

ongoing development of the ideas.

Several other sources have contributed to the support of individual members of the group. These include the National Institute of Mental Health, through a Career Development Award-PHS-MH-00385-to McCle\1and and post-doctoral fe\1owships to Paul Smolensky and Paul Munro under Grant PHS-MH-14268 to the Center for Human Information Processing at UCSD. Smolensky received support in the form of a fellowship from the Alfred P. Sloan Foundation, and some of McClelland's work was supported by a grant from the National Science Foundation (BNS-79-24062). These and other sources of support for specific individuals or projects are acknowledged in the appropriate

chapters.

Fina\1y, we would like to thank our wives, Heidi and Marilyn. Their understanding, encouragement, and support throughout the four years of this project helped to make the process of bringing this book to life

much more rewarding than it might have been.

JLM/DER

Addresses of the PDP Research Group

Chisato Asanuma

Francis H. C. Crick

Jeffrey L. Elman

Geoffrey E. Hinton

Michael I. Jordan

Alan H. Kawamoto

Salk Institute P.O. Box 85800 San Diego, C A 92138

Salk Institute P.O. Box 85800 San Diego, CA 92138

Department of Linguistics

University of California, San Diego

La Jolla, CA 92093

Department of Computer Science Carnegie-Mellon Univer<sup>s</sup>ity Pittsburgh, PA 15213

Department of Computer and Information Science University of Massachusetts Amherst, MA 01003

Department of Psychology Carnegie-Mellon Uni versity Pittsburgh, PA 15213

James L. McClelland

Paul W. Munro

Donald A. Norman

Daniel E. Rabin

David E. Rumelhart

Terrence J. Sejnowski

Paul Smolensky

Gregory O. Stone

Ronald 1. Williams

David Zipser

Department of Psychology Carnegie-Mellon University Pittsburgh, PA 15213

Department of Information Science

University of Pittsburgh Pittsburgh, PA 15260

Institute for Cognitive Science University of California, San Diego

La Jolla, CA 92093

Intellicorp

1975 EI Camino Real West Mountain View, CA 94040

Institute for Cognitive Science University of California, San Diego

La Jolla, CA 92093

Department of Biophysics Johns Hopkins University Baltimore, MD 21218

Department of Computer Science

University of Colorado Boulder, CO 80309

Center for Adaptive Systems Department of Mathematics

Boston University Boston, MA 02215

Institute for Cognitive Science Universi�y of California, San Diego

La Jolla, CA 92093

Insitute for Cognitive Science University of California, San Diego

La Jolla, CA 92093

PART I

THE PDP PERSPECTIVE

CHAPTER 1

The Appeal of

Parallel Distributed Processing

J. L. McCLELLAND, D. E. RUMELHART, and G. E. HINTON

What makes people smarter than machines? They certainly are not quicker or more precise. Yet people are far better at perc<sup>e</sup>iving objects in natural scenes and noting their relations , at understanding language and retrieving contextually appropriate information from memory, at making plans and carrying out contextually appropriate actions, and at a wide range of other natural c<sup>o</sup>gni <sup>t</sup> i <sup>v</sup>e tasks. People are also far better at learning to do these things more accurately and nuently through pro-

cessing <sup>e</sup>xperience.

What is the basis for these differences? One answer, perhaps the classic one we might <sup>e</sup>xpe<sup>c</sup>t from a<sup>r</sup>tificial intel ligence, is " <sup>s</sup>oftwar<sup>e</sup> ." If we only had the right computer program, the argument goes, we might be able to <sup>c</sup>apturethe nuidity and adaptability of human informatio<sup>n</sup>

processing.

Certainly this answer is partially correc<sup>t</sup> . There have been great breakthroughs in our understanding of <sup>c</sup>og<sup>n</sup>ition as a result of the dev<sup>e</sup>lop<sup>m</sup>e<sup>n</sup>t of expressive high- level computer languages and powerful algorithms. No doubt there will be more such breakth<sup>r</sup>oughs in the future. However, we do not think that software is the whole story.

In our view, people are smarter than today's computers because the brain employs abasic computational architecture that is more suited to deal with a central aspect of the natural information processing tasks that people are so good at. In this chapter, we will show through examples that these tasks generally requi <sup>r</sup>e the si<sup>m</sup>ultaneous consideration of many pieces of inf<sup>o</sup>rmation or const <sup>r</sup>aint<sup>s</sup> . Each constraint may be

decisive role in determining the outcome of processing. After examining these points, we will introduce a computational framework for modeling cognitive processes that seems well suited to exploiting these constaints and that seems closer than other frameworks to the style of computation as it might be done by the brain. We wi ll review several early examples of models developed in this framework, and we will show that the mechanisms these models employ can give rise to powerful emergent properties that begin to suggest attractive alternati ves to traditional accounts of various aspects of cognition. We wi ll also show that models of this class provide a basis for understanding how learning

can occur spontaneously, as a by-product of processing activity.

Multiple Simultaneous Constraints

Reaching and grasping. Hundreds of times each day we r�ach for things. We nearly never think about these acts of reaching. And yet, each time, a large number of different considerations appear to jointly determine exactly how we wi ll reach for the object. The position of the object, our. posture at the time, what else we may also be holding, the size, shape, and antici pated weight of the object, any obstacles that may be in the way-all of these factors jointly determine the exact method

we wil l use for reaching and grasping.

Consider the situation shown in Figure 1. Figure 1 A shows Jay McClelland's hand, in typing position at his terminal . Figure 1B indicates the position his hand assumed in reaching for a smal l knob on the desk beside the terminal. We will let him describe what happened in

the first person:

On the desk next to my terminal are several objects-a chi pped coffee mug, the end of a computer cable, a knob from a clock radio. I decide to pick the knob up. At first I hesitate, because it doesn't seem possible. Then I just reach for it, and find myself grasping the knob in what would normal ly be considered a very awkward position -but it solves all of the constraints. I'm not sure what all the details of the movement were, so I let myself try it a few times more. I observe that my right hand is carried up off the keyboard, bent at the elbow, until my forearm is at about a 30° angle to the desk top and parallel to the side of the terminal . The palm is facing downward through most of this. Then, my arm extends and lowers down more or less paral lel to the edge of the desk and parallel to the side of the terminal and, as it drops, it turns about 90° so that the

A

B

FIGURE I. A: An everyday situation in which it is necessary to take into account a large number of constraints to grasp a desired object. In this case the target object is the small knob to the left of the cup. B: The posture the arm arrives at in meeting these

palm is facing the cup and the thumb and index finger are below. The turning motion occurs just in time, as my hand drops, to avoid hitting the coffee cup. My index finger and thumb close in on the knob and grasp it, with my hand com-

pletely upside down.

Though the detai ls of what happened here might be quibbled with, the broad outl ines are apparent. The shape of the knob and its position on the table; the starting position of the hand on the keyboard; the positions of the terminal, the cup, and the knob; and the constraints imposed by the structure of the arm and the musculature used to control it -all these things conspired to lead to a solution which exactly suits the problem. If any of these constraints had not been included, the movement would have failed. The hand would have hit the cup or

the terminal -or it would have missed the knob.

The mutual itifluence of syntax and semantics. MUltiple constraints operate just as strongly in language processing as they do in reaching and grasping. Rumelhart (977) has documented many of these multiple constraints. Rather than catalog them here, we wi ll use a few examples from language to illustrate the fact that the constraints tend to be reci procal: The example shows that they do not run only from

syntax to semantics-they also run the other way.

It is clear, of course, that syntax constrains the assignment of meaning. Without the syntactic rules of English to guide us, we cannot correctly understand who has done what to whom in the fol lowing sen-

tence:

The boy the man chased kissed the girl.

But consider these examples (Rumelhart, 1977; Schank, 1973):

I saw the grand canyon flying to New York.

I saw the sheep grazing in the field.

Our knowledge of syntactic rules alone does not tell us what grammatical role is played by the prepositional phrases in these two cases. In the first, "flying to New York" is taken as descri bi ng the context in which the speaker saw the Grand Canyon-while he was flying to New York. In the second, "grazing in the field" could syntactically descri be an analogous situation, in which the speaker is grazing in the field, but this possi bility does not typical ly become available on first reading. Instead we assign "grazing in the field" as a modifier of the sheep (roughly, .. who were grazing in the field"). The syntactic structure of each of Copyrighted Material

these sentences, then, is determined in part by the semantic relations that the constituents of the sentence might plausibly bear to one another. Thus, the influences appear to run both ways, from the syn-

tax to the semantics and from the semantics to the syntax.

In these examples, we see how syntactic considerations influence semantic ones and how semantic ones influence syntactic ones. We

cannot say that one kind of constraint is primary.

Mutual constraints operate, not only between syntactic and semantic processing, but also within each of these domains as wel l. Here we consider an example from syntactic processing, namely, the assignment

of words to syntactic categories. Consider the sentences:

I like the joke. I like the drive. I like to joke. I like to drive.

In this case it looks as though the words the and to serve to determine whether the fol lowing word wi ll be read as a noun or a verb. This, of course, is a very strong constraint in English and can serve to force <sup>a</sup>

<sup>v</sup><sup>e</sup>rb interpretation of a word that is not ordinarily used this way:

I like to mud.

On the other hand, if the information specifying whether the function word preceding the final word is to or the is ambiguous, then the typical reading of the word that follows it wil l determine which way the function word is heard. This was shown in an experiment by Isenberg, Walker, Ryder, and Schweikert ( 1 980). They presented sounds halfway between to (actually/0) and the (actual ly / er f) and found that words like joke, which we tend to think of first as nouns, made subjects hear <sup>t</sup>he marginal sti<sup>m</sup>uli as the. while words like drive, which we tend to think of first as verbs, made subjects hear the marginal stimuli as to. General ly, then, it would appear that each word can help constrain the

syntactic role, and even the identity, of every other word.

Simultaneous mutual constraints in word recognition. Just as the syntactic role of one word can influence the role assigned to another in analyzing sentences, so the identity of one letter can influence the identity assigned to another in reading. A famous example of this, from Selfridge, is shown in Figure 2 . Along with this is a second example in which none of the letters, considered separately, can be identified unambiguously, but in which the possibil ities that the visual

TAE CAT

��8

S�OT

�ISH

DEQT

FIGURE 2. Some ambiguous displays. The first one is from Selfridge, 1955. The second line shows that th ree ambiguous characters can each constrain the identity of the others. The third, fourth, and fifth lines show that these characters are indeed ambiguous in that they assume other identities in other contexts. (The ink-blot technique of

making letters ambiguous is due to Lindsay and Norman, 1972).

information leaves open for each so constrain the possi ble identities of

the others that we are capable of identi fying all of them.

At first glance, the situation here must seem paradoxical: The identity of each letter is constrained by the identities of each of the others. But since in general we cannot know the identities of any of the letters

until we have established the identities of the others, how can we get

the process started?

The resolution of the paradox, of course, is simple. One of the different possible letters in each position fits together with the others. It appears then that our perceptual system is capable of exploring all these possibi lities without committing itself to one unti l all of the constraints

are taken into account.

Understanding through the interplay of mUltiple sources of knowledge. It is clear that we know a good deal about a large number of different standard situations. Several theorists have suggested that we store this knowledge in terms of structures called variously: scripts (Schank, 1976), frames (Minsky, <sup>197</sup> 5), or schemata (Norman & Bobrow, 1976; Rumelhart, 1975). Such knowledge st ructures are assumed to be the basis of comprehension. A great deal of progress

has been made within the context of this view.

However, it is important to bear in mind that most everyday situations cannot be rigidly assigned to just a single script. They general ly involve an interplay between a number of different sources of information. Consider, for example, a chi ld's birthday party at a restaurant. We know things about birthday parties, and we know things about restaurants, but we would not want to assume that we have explicit knowledge (at least, not in advance of our first restaurant bi rthday party) about the conjunction of the two. Yet we can imagine what such a party might be like. The fact that the party was being held in a restaurant would modify certain aspects of our expectations for birthday parties (we would not expect a game of Pin-the-Tail-on-the-Donkey, for example), while the fact that the event was a bi rthday party would inform our expectations for what would be ordered and who would pay

the bill.

Representations like scri pts, frames, and schemata are useful structures for encoding knowledge, although we believe they only approximate the underlying structure of knowledge representation that emerges from the class of models we consider in this book, as explained in Chapter 14. Our main point here is that any theory that tries to account for human knowledge using scri pt-like knowledge structures will have to allow them to interact with each other to capture the generati ve capacity of human understanding in novel situations. Achieving such interactions has been one of the greatest difficulties associated with implementing models that real ly think generatively using script- or

frame-like representations .

PARALLEL DISTRIBUTED PROCESSING

In the examples we have considered, a number of different pieces of information must be kept in mind at once. Each plays a part, constraining others and being constrained by them. What kinds of mechanisms seem well suited to these task demands? Intuitively, these tasks seem to require mechanisms in which each aspect of the information in the situation can act on other aspects, simultaneously influencing other aspects and being influenced by them. To articulate these intuitions, we and others have turned to a class of models we call Parallel Distributed Processing (POP) models. These models assume that information processing takes place through the interactions of a large number of simple processing elements called units, each sending excitatory and inhibitory signals to other units. In some cases, the units stand for possible hypotheses about such things as the letters in a particular display or the syntactic roles of the words in a particular sentence. In these cases, the activations stand roughly for the strengths associated with the different possible hypotheses, and the interconnections among the units stand for the constraints the system knows to exist between the hypotheses. In other cases, the units stand for possible goals and actions, such as the goal of typing a particular letter, or the action of moving the left index finger, and the connections relate goals to subgoals, subgoals to actions, and actions to muscle movements. In still other cases, units stand not for particular hypotheses or goals, but for aspects of these things. Thus a hypothesis about the identity of a word, for example, is itself distributed in the activations of

a large number of units.

PDP Models: Cognitive Science or Neuroscience?

One reason for the appeal of POP models is their obvious "physiological" flavor: They seem so much more closely tied to the physiology of the brain than are other kinds of information-processing models. The brain consists of a large number of highly interconnected elements (Figure 3) which apparently send very simple excitatory and inhibitory messages to each other and update their excitations on the basis of these simple messages. The properties of the units in many of the POP models we will be exploring were inspired by basic properties of the neural hardware. In a later section of this book, we will examine in

FIGURE 3. The arborizations of about I percent of the neurons near a vertica' ,lice th rough the cerebral cortex. The full height of the figure corresponds to the thickness of the cortex, which is in this instance about 2 mm. (From Mechanics of the Mind, p. 84, by C. Blakemore, 1977, Cambridge, England: Cambridge University Press. Copyright 1977

by Cambridge University Press. Reprinted by permission.)

Though the appeal of POP models is definitely enhanced by their physiological plausi bility and neural inspi ration , these are not the primary bases for their appeal to us . We are, after all, cognitive scientists, and POP models appeal to us for psychological and computational reasons. They hold out the hope of offering computational ly sufficient and psychologically accurate mechanistic accounts of the phenomena of human cognition which have eluded successful explication in conventional computational formalisms; and they have radically altered the way we think about the time-course of processing , the nature of

representation, and the mechanisms of learning.

The Microstructure of Cognition

The process of human cognition , examined on a time scale of seconds and minutes, has a distinctly sequential character to it. Ideas come, seem promising, and then are rejected; leads in the solution to a problem are taken up, then abandoned and replaced with new ideas. Though the process may not be discrete, it has a decidedly sequential character, with transitions from state-to-state occurring, say, two or three times a second. Clearly, any useful description of the overall organization of this sequential flow of thought wi ll necessarily describe

a sequence of states.

But what is the internal structure of each of the states in the sequence, and how do they come about? Serious attempts to model even the simplest macrosteps of cognition -say, recognition of single words-require vast numbers of microsteps if they are implemented sequentially. As Feldman and Ballard (I982) have pointed out, the biological hardware is just too sluggish for sequential models of the microstructure to provide a plausible account, at least of the microstructure of human thought. And the time limitation only gets worse, not better, when sequential mechanisms try to take large numbers of constraints into account. Each additional constraint requires more time in a sequential machine, and, if the constraints are imprecise, the constraints can lead to a computational explosion. Yet people get faster, not slower, when they are able to exploit additional

constraints.

Parallel distri buted processing models offer alternatives to serial models of the microstructure of cognition. They do not deny that there is a macrostructure, just as the study of subatomic particles does not deny the existence of interactions between atoms. What POP models do is describe the internal structure of the larger units, just as subatomic physics describes the internal structure of the atoms that

form the constituents of larger units of chemical structure.

We shal l show as we proceed through this book that the analysis of the microstructure of cognition has important implications for most of the central issues in cognitive science. In general , from the POP point of view, the objects referred to in macrostructural models of cognitive processing are seen as approximate descri ptions of emergent properties of the microstructure. Sometimes these approximate descri ptions may be sufficiently accurate to capture a process or mechanism well enough; but many ti mes, we wi ll argue, they fail to provide sufficiently elegant or tractable accounts that capture the very flexibility and openendedness of cognition that their inventors had originally intended to capt ure. We hope that our analysis of POP models wi ll show how an Copyrighted Material

for the sec<sup>o</sup>nd to inhibit the third and following letters, and so on. As a result of the interplay of activation and inhibition among these units, <sup>t</sup>he unit for the first letter was at first the most strongly active, and the

units for the other letters were partially <sup>a</sup>ctivated.

Each letter unit exerts influences on the hand and finger involved in typing the letter. The v unit, for example, tends to cause the index finger to move down and to cause the whole hand to move down with it. The e unit, on the other hand, tends to cause the middle finger on the left hand to move up and to cause the whole hand to move up also. The r unit also causes the left index finger to move up and the left

hand to move up with it.

The extent of the influences of each letter on the hand and finger it directs depends on the extent of the <sup>a</sup>ctivation of the letter. Therefore , at first, in typing the word very, the v exerts the greatest control.

Response System

Keypress Schemata

Word Schema THUMB

RESPONSE SYSTEM L1(.I, + 05) LM(+ 1 ·03) LJ(+ 1, ·03) RII+ 1. + 1 3)

> . target finger POSition current finger POSition

FIGURE 4. The interaction of activations in typing the word very. The very u<sup>n</sup>it is activated from outside the model. It in turn activates the units for each of the component letters. Each letter unit specifies the target finger positions, sp<sup>e</sup>cified in a keyboard coordinate system. L <sup>a</sup><sup>n</sup>d R stand for theleft and right ha<sup>n</sup>ds. and I and M for the index and middle fingers. The letter units receive information about the current finger position from the response system. Each letterunit inhibits the activation of all letter units that follow it in the word: inhibitory connections are indicated by the lines with solid dots at their terminations. (From "Simulating a Skilled Typist: A Study of Skilled Motor Performance" by D. E. Rumelhart and D. A. Norman. 1982, Cognitive Science. 6,

balance and bringing the tip closer to the target. After a number of iterations, the stick-person settled on postures that satisfied the goal of reaching the target and the goal of maintaining the center of gravity

over the" feeL"

Though the simulation was able to perform the task, eventually satisfying both goals at once, it had a number of inadequacies stemming from the fact that each joint processor attempted to achieve a solution in ignorance of what the other joints were attempting to do. This problem was overcome by using additional processors responsible for setting combinations of joint angles. Thus, a processor for flexion and extension of the leg would adjust the knee, hip, and ankle joints synergistical ly, while a processor for flexion and extension of the arm would adjust the shoulder and elbow together. With the addition of processors of this form, the number of iterations required to reach a sol ution was greatly reduced, and the form of the approach to the solution looked very natural. The sequence of configurations attained in one

processing run is shown in Figure 5.

Explicit attempts to program a robot to cope with the problem of maintaining balance as it reaches for a desired target have revealed the difficulty of deriving explicitly the right combinations of actions for each possible starting state and goal state. This simple model illustrates that we may be wrong to seek such an explicit solution. We see here that a solution to the problem can emerge from the action of a number of simple processors each attempting to honor the constraints

independently.

FIGURE S. A sequence of configurations <sup>a</sup>ssumed by the stick "person" performing the reaching task described in the text, from Hinton (I 984). The small circle represents the center of gravity of th: whole stick-figure, and the cross represents the goal to be

What kind of a mechanism might we propose to account for these facts? Marr and Poggio (976) began by explicitly represen ting the two views in two arrays, as human observers might in two di fferent reti nal images. They noted that corresponding black dots at different percei ved distances from the observer will be offset from each other by gi fferent amounts i n the two views. The job of the model is to determine which points correspond. This task is, of course, made difficult by the fact that there will be a very large number of spurious correspondences of individual dots. The goal of the mechanism, then, is to find those correspondences that represent real correspondences in depth and suppress those that represent spurious correspondences.

To carry out this task, Marr and Poggio assigned a processing unit to each possible .conjunction of a point in one image and a poi nt in the other. Since the eyes are offset horizontally, the possible conjunctions occur at various offsets or disparities along the horizontal dimension. Thus, for each point in one eye, there was a set of processing units with one unit assigned to the conjunction of that poi nt and the point at

each horizontal offset from it in the other eye.

Each processing unit recei ved activation whenever both of the points the unit stood for contained dots. So far, then, units for both real and spurious correspondences would be equal ly activated. To allow the mechani sm to find the right correspondences, they pointed out two general princi ples about the visual world: (a) Each point in each view generally corresponds to one and only one point in the other view, and Cb) neighbori ng points in space tend to be at nearly the same depth and therefore at about the sam( disparity in the two images. While there are discontinuities at the edges of things, over most of a twodimensional view of the world there will be continuity. These principles are called the uniqueness and continuity constraints , respectively. Marr and Poggio incorporated these princi ples into the interconnections between the processing units. The uniqueness constraint was captured by inhibitory connections among the units that stand for alternative correspondences of the same dot . The continuity principle was captured by excitatory connections among the units that stand for simi-

lar offsets of adjacent dots.

These additional connections allow the Marr and Poggio model to "sol ve" stereograms like the one shown in the figure. At first, when a pair of patterns is presented, the units for all possible correspondences of a dot in one eye with a dot in the other will be equally excited. However, the excitatory connections cause the units for the correct conjunctions to receive more excitation than units for spurious conjunctions, and the inhi bitory connections allow the units for the correct conjunctions to turn off the units for the spurious connections. Thus,

the model tends to settle down into a stable state in which only the

correct correspondence of each dot remains active.

There are a number of reasons why Marr and Poggio (I979) modified this model (see Marr, 1982, for a discussion) , but the basic mechanisms of mutual excitation between units that are mutually consistent and mut ual inhibition between units that are mutually incompatible provide a natural mechanism for settli ng on the right conjunctions of points and rejecting spurious ones. The model also il lustrates how general principles or rules such as the uniqueness and continuity principles may be embodied in the connections between processing uni ts, and how behavior in accordance with these princi ples can emerge from the

interactions determined by the pattern of these interconnections.

Perceptual completion of familiar patterns. Perception, of course, is influenced by familiarity. It is a well-known fact that we often misperceive unfamiliar objects as more famil iar ones and that we can get by with less time or with lower-quality information in perceiving fami liar items than we need for percei ving unfamil iar items. Not only does familiarity help us determine what the higher-level structures are when the lower-level information is ambiguous; it also allows us to fi ll in missing lower-level information within famil iar higher-order patterns. The well-known phonemic restoration effect is a case in point. [n this phenomenon, perceivers hear sounds that have been cut out of words as if they had actually been present . For example, Warren (I970) presented legi#lature to subjects, with a click in the location marked by the #. Not only did subjects correctly identify the word legislature; they also heard the missing /s/ just as though it had been presented. They had great difficulty localizing the click, which they tended to hear as a disembodied sound. Simi lar phenomena have been observed in

visual perception of words since the work of Pi lIsbury (897).

Two of us have proposed a model describing the role of familiarity in perception based on excitatory and inhibitory interactions among units standing for various hypotheses about the input at different levels of abstraction (McClelland & Rumelhart, 1981; Rumelhart & McClelland, 1982). The model has been applied in detail to the role of familiarity in the perception of letters in visually presented words, and has proved to provide a very close account of the results of a large number of

experiments.

The model assumes that there are units that act as detectors for the visual features which distinguish letters , with one set of units assigned to detect the features in each of the di fferent letter-positions in the word. For four-letter words, then , there are four such sets of detectors. There are also four sets of detectors for the letters themselves and a set

FIGURE 7. The unit for the letter T in the fi rst position of a four-letter array and some of its neighbors . Note that the feature and letter units stand only for the fi rst position; in a complete picture of the units needed from processing four-letter displays , there would be four fu ll sets of feature detectors and four full sets of letter detectors. (From " An Interactive Activation Model of Contex t Effects in Letter Perception: Part I. An Account of Basic Findings" by J. L. McClelland and D. E. Rumelhart, 1981, Psychological Review, 88, p. 380. Copyright 1981 by the American Psychological Association. Reprin ted by

permission .)

them and to inhibit detectors for words which do not have these letters. A number of words are partial ly consistent with the active letters, and receive some net excitation from the letter level , but only the word WORK matches one of the active letters in all four positions. As a result, WORK becomes more acti ve than any other word and inhibits the other words, thereby successfully dominating the pattern of activation among the word units. As it grows in strength, it sends feedback to the letter level , reinforci ng the activations of the W, 0, R, and K in the corresponding positions. In the fourth position, this feedback gi ves K the upper hand over R, and eventual ly the stronger activation of the Copyrighted Material

0.8

5 0.4 .� .2: o « 0.0

0.8

c: 0.4 .Q iii .� o « 0.0 Word Level

\_---- work

----------------------

---------- -- fork

weak

Letter Level ",-

\_ -- - --- - K

./" ,- // / / ./

'-------------------- ---- A

0

8 16 24 Time

32 40

FIGURE 8. A possible display which might be presented to the interactive activation model of word recognition, and the resulting activations of selected letter and word units. <sup>T</sup>he letter units are for <sup>t</sup>heletters indicated in the fo<sup>u</sup> <sup>r</sup>th position of a four-letter display.

K detector allows it to dominate the pattern of activation, suppressing

the R detector completely.

This example illustrates how PDP models can allow knowledge about what letters go together to form words to work together with natural constraints on the task (i .e., that there should only be one letter in one place at one time) , to produce perceptual completion in a simple and

direct way.

perception of letters in unfamili ar letter strings which are word - like but

not themselves actually familiar .

One way of accounting for such performances is to imagine that the percei ver possesses, in addition to detectors for famil iar words, sets of detectors for regular subword units such as fami l iar letter cl usters , or that they use abstract rules, specifying which classes of letters can go with which others in different contexts. It turns out, however, that the model we have already described needs no such additional structure to produce perceptual facilitation for word-l ike letter strings� to this extent it acts as if it "knows" the orthographi c struct ure of Engl ish . We illustrate this feature of the model with the example shown in Figure 9, where the nonword YEAD is shown in degraded form so that the second letter is incompletely visi ble. Gi ven the information about this letter, considered alone, either E or F would be possible in the second position . Yet our model wi ll tend to complete this letter as an E.

The reason for this behavior is that, when YEAD is shown , a number of words are partially acti vated. There is no word consistent with y. E or F. A. and D, but there are words which match YEA (YEAR, for example) and others which match EAD (BEAD. DEAD. HEAD. and

READ, for example) . These and- other near

' misses are partial ly

acti vated as a result of the pattern of activation at the letter level . While they compete with each other, none of these words gets strongly enough acti vated to completely suppress all the others. Instead, these units act as a group to reinforce particularly the letters E and A. There are no close partial matches which incl ude the letter F in the second position, so this letter recei ves no feedback support. As a result , E comes to dominate, and eventual ly suppress, the F in the second

position .

The fact that the word perception model exhibits perceptual faci litation to pronounceable non words as wel l as words illustrates once again how behavior in accordance with general princi ples or rules can emerge from the interactions of simple processing elements. Of course, the behavior of the word perception model does not implement exactly any of the systems of orthographic rules that have been proposed by linguists (Chomsky & Hal le , 1968; Venesky, 1970) or psychologists (Spoehr & Smith , 1975) . In this regard , it only approximates such rule-based descriptions of perceptual processing. However, rule systems such as Chomsky and Halle's or Venesky's appear to be only approximately honored in human performance as wel l (Smith & Baker, 1976) . Indeed , some of the discrepancies between human performance data and rule systems occur in exactly the ways that we would predict from the word perception model (Rumelhart & McClelland, 1982) . This ill ustrates the possibil ity that POP models may provide more accurate accounts of the detai ls of human performance than models Copyrighted Material

can access informat ion in memory based on nearly any attribute of the

representation we are trying to ret rieve.

Of course, some cues are much better than others. An attribute which is shared by a very large number of thi ngs we know about is not a very effective ret rieval cue, since it does not accurately pick out a particular memory representation. But, several such cues, in conjunction, can do the job. Th us, if we ask a friend who goes out with several women , "Who was that woman I saw you with?", he may not know which one we mean -but if we specify something else about her-say the color of her hai r, what she was wearing (in so far as he remembers this at all) , whe re we saw him with her-he wi ll likely be able to h i t

upon the right one .

It is, of course , possible to implement some kind of content addressabi lity of memory on a standard computer in a variety of different ways . One way is to search sequentially, examining each memory in the system to find the memory or the set of memories which has the particular content specified in the cue. An alternative, somewhat more efficient, scheme involves some form of index ingkeepi ng a list, for every con tent a memory might have, of which

memories have that content .

Such an indexing scheme can be made to work with error-free probes , but it wi ll break down if there is an error in the specification of the retrieval cue. There are possible ways of recovering from such errors , but they lead to the kind of combinatorial explosions which

plague this kind of computer implementation.

But suppose that we imagi ne that each memory is represented by a unit which has mutually excitatory interactions with units standing for each of its properties. Then , whenever any property of the memory became active, the memory would tend to be acti vated , and whenever the memory was acti vated , all of its contents would tend to become activated . Such a scheme would automatically produce content addressabi lity for us. Though it would not be immune to errors, it would not be devastated by an error in the probe if the remaining

properties specified the correct memory .

As descri bed thus far, whenever a property that is a part of a number of di fferent memories is activated, it will tend to acti vate al l of the memories it is in. To keep these other acti vities from swamping the "correct" memory unit, we simply need to add initial inhi bitory connections among the memory units. An additional desi rable feat ure would be mutually inhibi tory interactions among mutually incompatible property uni ts. For example, a person cannot both be single and married at the same time, so the units for di fferent marital states wou ld be

McClel land (1981) developed a simu lation model that ill ustrates how a system with these properties would act as a content addressable memory. The model i s obviously oversimplified, but it ill ust rates many of the characteristics of the more complex <sup>m</sup>odels that wi ll be con-

sidered in later chapters .

Consider the information represented in Figure 10, which lists a number of people we might meet if we went to live in an unsavory neighborhood, and so<sup>m</sup>e of thei r hypothetical characteristics. A subset

The Jets and The Sharks

Name Gang Age Edu Mar Occupation

Art Jets 40's J.H. Si ng. Pusher AI Jets 30's J.H. Mar. Burglar

Sam Jets 20's COL. Si

ng. Bookie

Clyde Jets 40's J.H. Sing. Bookie Mike Jets 30's J.H. Sing. Bookie Jim Jets 20's J.H. D<sup>i</sup> <sup>y</sup> . Burgtar Greg Jets 20's H.S. Mar. Push<sup>e</sup> <sup>r</sup> John Jets 20's J.H. Mar. Burglar Doug Jets 30's H.S. Sing. Bookie Lance Jets 20's J.H. Mar. Burglar George Jets 20's J.H. Di <sup>v</sup> . Burglar

Pete Jets 20's H.S. Si

ng. Bookie

Fred Jets 20's H.S. Sing. Pusher Gene Jets 20's COL. Sing. Pusher Ralph Jets 30's J.H. Sing. Pusher Phil Sharks 30's COL. Ma<sup>r</sup> . Pusher Ike Sharks 30's J.B. Sing. Bookie Nick Sharks 30's H . S. Si ng. Pusher Don Sharks 30's COL. Mar. Burglar Ned Sharks 30's COL. Mar. Bookie Karl Sharks 40's H. S. Ma<sup>r</sup> . Bookie Ken Sharks 20's H.S. Sing. Burglar Earl Sharks 40's H.S. Mar. Burglar Rick Sharks 30's H.S. Div. Burgl ar 01 Sharks 30's COL. Mar. Pusher Neal Sharks 30's H.S. Sing. Bookie Daye Sharks 30's H.S. Di y. Pusher

FIGURE 10. Characteristics of a number of individuals belonging to two gangs, the Jets and the Sharks. (From "Retrieving General and Specific Knowledge From Stored Knowledge of Specifics" by 1. L. McClelland, 1981, Proceedings of the Third Annual Conference of the Cognitive Science SOCiety, Berkeley, CA. Copyright 1981 by 1. L. McClel land

of the units needed to represent this information is shown in Figure 11 . In this network, there is an "instance unit" for each of the characters described in Figure 10, and that unit is linked by mutually excitatory connections to al l of the units for the fellow's properties. Note that we have included property units for the names of the characters, as well as

units for their other properties.

Now, suppose we wish to retrieve the properties of a particular individual , say Lance. And suppose that we know Lance's name. Then we can probe the network by activating Lance's name unit, and we can see what pattern of activation arises as a result. Assuming that we know of no one else named Lance, we can expect the Lance name unit to be hooked up only to the instance unit for Lance. This wi ll in turn activate the property units for Lance, thereby creating the pattern of

FIGURE 11. S o m e o f the units and interconnections needed to represent the indi viduals shown in Figure 10. The units connected with double-headed arrows are mutually excitatory. All the units within the same cloud are mutually inhibitory. (From "Retrieving General and Specific .Knowledge From Stored Knowledge of Specifics" by J. L. McClelland, 198 1, Proceedings of the Third Annual Coriference of the Cognitive Science Society, 8erkeley, CA. Copyright 1981 by J. L. McClelland. Reprinted by permission.)

activation corresponding to Lance. In effect, we have retrieved a representation of Lance. More wi ll happen than just what we have

described so far, but for the moment let us stop here.

Of course, sometimes we may wish to retrieve a name, given other information. In this case, we might start with some of Lance's properties, effectively asking the system, say "Who do you know who is a Shark and in his 20s?" by activating the Shark and 20s units. In this case it turns out that there is a single individual , Ken, who fits the description . So, when we actiyate these two properties, we wi ll acti vate the instance unit for Ken, and this in turn wi ll acti vate his name unit,

and fill in his other properties as well.

Graceful degradation. A few o f <sup>t</sup> h <sup>e</sup> desirable properties of this kind of model are visible from considering what happens as we vary the set of features we use to probe the memory in an attempt to retrieve a particular indi vidual 's name. Any set of features which is sufficient to uniquely characterize a particular item wi ll acti vate the instance node for that item more strongly than any other instance node. A probe which contains misleading features wi ll most strongly acti vate the node that it matches best. This wi ll clearly be a poorer cue than one which contains no misleading information -but it wi ll still be sufficient to activate the "right answer" more strongly than any other, as long as the introduction of misleading information does not make the probe closer to some other item. In general, though the degree of activation of a particular instance node and of the corresponding name nodes varies in this model as a function of the exact content of the probe, errors in the probe wi ll not be fatal unless they make the probe poi nt to the wrong memory. This kind of model's handl ing of incomplete or partial probes also requires no special error-recovery scheme to work -it is a natural by-product of the nature of the retrieval mechanism that it is capable of

graceful degradation.

These aspects of the behavior of the Jets and Sharks model deserve more detai led consideration than the present space allows. One reason we do not go into them is that we view this model as a stepping stone in the development of other models, such as the models using more distributed representations, that occur in other parts of this book. We do, however, have more to say about this simple model , for like some of the other models we have already examined, this model exhibits some useful properties which emerge from the interactions of the pro-

cessing uni ts.

Default assignment. It probably wi ll have occurred to the reader that in many of the situations we have been examining, there wi ll be other Copyrighted Material

activations occurring which may influence the pattern of acti vation which is retrieved. So, in the case where we retrieved the properties of Lance, those properties, once they become active, can begin to activate the units for other indi viduals with those same properties. The memory unit for Lance will be in competition with these units and wi ll tend to keep their activation down, but to the extent that they do become active, they wi ll tend to acti vate their own properties and therefore fill them in. In this way, the model can fi ll in properties of indi viduals based on what it knows about other, simi lar instances.

To ill ustrate how this might work we have simulated the case in which we do not know that Lance is a Burglar as opposed to a Bookie or a Pusher. It turns out that there are a group of indi viduals in the set who are very similar to Lance in many respects. When Lance's properties become acti vated, these other units become partially activated, and they start activating their properties. Since they all share the same "occupation," they work together to fill in that property for Lance. Of course, there is no reason why this should necessarily be the right answer, but generally speaking, the more simi lar two things are in respects that we know about, the more li kely they are to be similar in respects that we do not, and the model implements this heuristic.

Spontaneous generalization. The model we have been descri bing has another valuable property as well -it tends to retrieve what is common to those memories which match a retrieval cue which is too general to capture any one memory. Thus, for example, we could probe the system by activating the unit corresponding to membership in the Jets. This unit wi ll partially acti vate all the instances of the Jets, thereby causing each to send activations to its properties. In this way the model can retrieve the typical values that the members of the Jets have on each dimension-even though there is no one Jet that has these typical values. In the example, 9 of 15 Jets are single, 9 of 15 are in their 20s, and 9 of 15 have only a Junior High School education; when we probe by acti vating the Jet unit, all three of these properties dominate. The Jets are evenly divided between the three occupations, so each of these units becomes partially activated. Each has a different name, so that each name unit is very weakly activated, nearly cancel ling

each other out.

In the example just given of spontaneous general ization , it would not be unreasonable to suppose that someone might have explicitly stored a generalization about the members of a gang. The account just given would be an alternati ve to ·' explicit storage" of the generalization. It has two advantages, though, over such an account. First, it does not require any special generalization formation mechanism. Second, it can provide us with general izations on unanticipated lines, on demand.

Thus, if we want to know, for example, what people in their 20s with a junior high school education are l ike, we can probe the model by acti vating these two units. Since all such people are Jets and Burglars, these two units are strongly acti vated by the model in this case; two of them are divorced and two are married, so both of these units are par-

tially activated. I

The sort of model we are considering, then , is considerably more than a content addressable memory. In addition, it performs default assignment, and it can spontaneously retrieve a general concept of the individuals that match any specifiable probe. These properties must be explicitly implemented as complicated computational extensions of other models of knowledge retrieval , but in POP models they are

natural by-products of the retrieval process itself.

REPRESENTATION AND LEARNING IN PDP MODELS

In the Jets and Sharks model , we can speak of the model 's active representation at a particular time, and associate this with the pattern of activation over the units in the system. We can also ask: What is the stored knowledge that gi ves rise to that pattern of acti vation? In considering this question , we see immediately an important difference between POP models and other models of cognitive processes. In most models, knowledge is stored as a static copy of a pattern. Retrieval amounts to finding the pattern in long-term memory and copying it into a buffer or working memory. There is no real difference between the stored representation in long-term memory and the active representation in working memory. In PDP models, though , this is not the case. In these models, the patterns themselves are not stored. Rather, what is stored is the connection strengths between units that allow these patterns to be re-created. In the Jets and Sharks model, there is an instance unit assigned to each individual, but that unit does not contain a copy of the representation of that individual . Instead, it is simply the case that the connections between it and the other units in the system are such that activation of the unit wi ll cause the pattern for the

individual to be reinstated on the property units.

<sup>I</sup>In th<sup>i</sup>s and all other case<sup>s</sup> , there is a tendency for Ihe pattern of activation to be influ· enced by partially activated, near neigh bors, which do not quite match the probe. Thus, in this case, there is a Jet AI, who is a Mar <sup>r</sup>ied Burglar. The unit for AI gets slightly

activated, giving Married a slight edge over Divorced in the simulation .

This difference between POP models and conventional models has enormous implications, both for processing and for learning. We have already seen some of the implications for processing. The representation of the knowledge is set up in such a way that the knowledge necessari ly influences the course of processing. Using knowledge in processing is no longer a matter of finding the relevant information in memory and bringing it to bear� it is part and parcel of the processing itself.

For learning, the implications are equal ly profound. For if the knowledge is the strengths of the connections, learning must be a matter of finding the right connection strengths so that the right patterns of activation wi ll be produced under the right circumstances. This is an extremely important property of this class of models, for it opens up the possibil ity that an information processing mechanism could learn, as a result of tuning its connections, to capture the interdependencies between activations that it is exposed to in the

course of processing.

In recent years, there has been quite a lot of interest in learning in cognitive science. Computational approaches to learning fall predominantly into what might be called the " explicit rule formulation " tradition, as represented by the work of Winston (1975) , the suggestions of Chomsky, and the ACT\* model of 1. R. Anderson ( 1983) . All of this work shares the assumption that the goal of learning is to formulate explicit rules (propositions, productions, etc.) which capture powerful generalizations in a succinct way. Fairly powerful mechanisms, usually with considerable innate knowledge about a domain, and/or some starting set of primitive propositional representations, then formulate hypothetical general rules, e.g. , by comparing particular cases and for-

mulating explicit general izations.

The approach that we take in developing POP models is completely different. First, we do not assume that the goal of learning is the formulation of explicit rules. Rather, we assume it is the acquisition of connection strengths which allow a network of simple units to act as though it knew the rules. Second, we do not attri bute powerful computational capabi lities to the learning mechanism. Rather, we assume very simple connection strength modulation mechanisms which adjust the strength of connections between units based on information locally

available at the connection .

These issues wi ll be addressed at length in later sections of this book. For now, our purpose is to give a simple, ill ustrati ve example of the connection strength modulation process, and how it can produce net-

works which exhibit some interesting behavior.

Local vs. distributed representation. Before we turn to an explicit consideration of this issue, we raise a basic question about Copyrighted Material

representation. Once we have achieved the insight that the knowledge is stored in the strengths of the interconnections between units, a question arises. Is there any reason to assign one unit to each pattern that we wish to learn? Another possi bi l ity-one that we explore extensi vely in this book -is the possi bi l ity that the knowledge about any indi vidual pattern is not stored in the connections of a special unit reserved for that pattern, but is distributed over the connections among a large number of processing units. On this view, the Jets and Sharks model represents a special case in which separate units are reserved for each

instance.

Models in which connecion information is explicitly thought of as distributed have been proposed by a number of investigators. The units in these collections may themselves correspond to conceptual primiti ves, or they may have no particular meaning as indi viduals. In either case , the focus shifts to patterns of activation over these units and to mechanisms whose explicit purpose is to learn the right connection st rengths to allow the right patterns of activation to become

activated under the right circumstances.

In the rest of this section, we wi ll give a simple example of a POP model in which the knowledge is distributed. We wi ll first explain how the model would work, given pre-existing connections, and we wi l l then describe how it could come to acquire the right connection strengths through a very simple learning mechanism. A number of models which have taken this distributed approach have been discussed in this book's predecessor, Hinton and 1. A. Anderson's (1 98 1) Parallel Models of Associative Memory. We wi ll consider a simple version of a common

type of distributed model, a pattern associator.

Pattern associators are models in which a pattern of activation over one set of units can cause a pattern of acti vation over another set of units without any intervening units to stand for either pattern as a whole. Pattern associators WOUld, for example, be capable of associating a pattern of activation on one set of units corresponding to the appearance of an object with a pattern on another set corresponding to the aroma of the object, so that, when an object is presented visually, causing its visual pattern to become active, the model produces the

pattern corresponding to its aroma.

How a pattern associator works. For purposes of i llust ration, we present a very simple pattern associator in Figure 12. In this model , there are four units in each of two pools. The first pool , the A units, wi ll be the pool in which patterns corresponding to the sight of various objects might be represented. The second pool, the B units, will be the pool in which the pattern corresponding to the aroma wi ll be represented. We can pr�pyh�alMaflMiHe patterns of activation on

A Units

From Vision

�

�====�===\*====\*==�-V

-1 From V Ollaction

c::::==============:::::==::::::::8 ':;(

B Units

FIGURE <sup>12</sup> - A sim<sup>p</sup>le pattern associator\_ The example assumes that patterns of activation in the A units can be produced by the visual system and patterns in the B u<sup>n</sup>its can be produced by the olfactory system. The synaptic connections allow the outputs of the A units to influence the activations of the B units. The synaptic weights linking the A <sup>u</sup>nits to the B units were selected so as to al low <sup>t</sup> he pal lern of acti <sup>v</sup>ation shown on the <sup>A</sup> units to reproduce the pallern of activation shown on the B units without the need for

any olfactory input.

the A units are produced upon viewing a rose or a grilled steak, and alternative patterns on the B units are produced upon sniffing the same objects . Figure 13 shows two pai rs of patterns, as wel l as sets of interconnections necessary to allow the A member of each pai r to reproduce

the B member.

The detai ls of the behavior of the individual units vary among different versions of pattern associators . For present purposes, we'll assume that the units can take on positive or negative activation values, with 0 representing a kind of neutral intermediate value. The strengths of the interconnections between the units can be positive or negative

real numbers.

The effect of an A unit on a B unit is determined by multiplying the activation of the A unit times the strength of its synaptic connection with the B unit. For example, if the connection from a particular A unit to a particular B unit has a positive sign , when the A unit is Copyrighted Material

+1 -1 -1 +1 -1 +1 -1 +1

-.25 +.25 +.25 -.25 -

1 +.25 -.25 +.25 -. 25 -

1

-.25 +.25 +.25 -.25 -1 -.25 +.25 -.25 +.25 +1

+.25 -.25 -.25 +.25 +1 -.

25 +.25 -.25 +.25 +1

+.25 -.25 -.25 +25 +1 +.25 -.25 +.25 -.25 -1

FIGURE 13. Two simpleassociators represented as matrices. The weights in the first two matrices allow the A pattern shown above the matrix to produce the B pattern shown to the right of it. Note that the weights in the first matrix are the same as those shown in

the diagram in Figure 12.

excited (activation greater than 0) , it will excite the B unit. For this example, we'll simply assume that the activation of each unit is set to the sum of the excitatory and inhibitory effects operating on it. This is

one of the simplest possible cases.

Suppose, now, that we have created on the A units the pattern corresponding to the first visual pattern shown in Figure 13, the rose. How should we arrange the strengths of the interconnections between the A units and the B units to reproduce the pattern corresponding to the aroma of a rose? We simply need to arrange for each A unit to tend to excite each B unit which has a positive activation in the aroma pattern and to inhibit each B unit which has a negative activation in the aroma pattern. It turns out that this goal is achieved by setting the strength of the connection between a given A unit and a gi ven B unit to a value proportional to the product of the activation of the two units. In Figure 12 , the weights on the connections were chosen to allow the A pattern ill ustrated there to produce the illustrated B pattern according to this principle. The actual strengths of the connections were set to ± .25, rather than ± 1, so that the A pattern will produce the right magnitude, as well as the right sign , for the activations of the units in the B pattern . The same connections are reproduced in matri x form in Fig-

ure l3A.

Pattern associators like the one in Figure 12 have a number of nice properties. One is that they do not require a perfect copy of the input to produce the correct output, though its strength wi ll be weaker in this case. For example, suppose that the associator shown in Figure 12 were presented with an A pattern of (1,-1,0,1). This is the A pattern shown in the figure, with the acti vation of one of its elements set to 0. The B pattern produced in response will have the activations of all of the B units in the right direction; however, they wi ll be somewhat weaker than they would be, ha£qp;rightfffMfa�r4alttern been shown . Similar

effects are produced if an element of the pattern is distorted -or if the model is damaged, either by removing whole units, or random sets of connections, etc. Thus, thei r pattern retrieval performance of the model degrades gracefully both under degraded input and under

damage.

How a pattern associator learns. So far, we have seen how we as model builders can construct the right set of weights to allow one pattern to cause another. The interesting thing, though, is that we do not need to build these interconnection strengths in by hand. Instead, the pattern associator can teach itself the right set of interconnections through experience processing the patterns in conjunction with each

other.

A number of different rules for adjusting connection strengths have been proposed. One of the first -and definitely the best known -is due to D. O. Hebb (1949). Hebb's actual proposal was not sufficiently quantitative to build into an explicit model . However, a number of different variants can trace their ancestry back to Hebb. Perhaps the sim-

plest version is:

When unit A and unit B are simultaneously excited, increase

the strength of the connection between them.

A natural extension of this rule to cover the positive and negative

activation values allowed in our example is:

Adjust the strength o f t h e connection between units A and B in proportion to the product of their simultaneous activation.

In this formulation , if the product is positive, the change makes the connection more excitatory, and if the product is negative, the change makes the connection more inhibitory. For simplicity of reference, we will call this the Hebb rule, although it is not exactly Hebb's original

formulation.

With this simple learning rule, we could train a "blank copy " of the pattern associator shown in Figure 12 to produce the B pattern for rose when the A pattern is shown, simply by presenting the A and B patterns together and modulating the connection st rengths according to the Hebb rule. The size of the change made on every trial would, of course, be a parameter. We general ly assume that the changes made on each instance are rather smal l, and that connection strengths build up gradual ly. The values shown in Figure l3 A, then , would be acqui red as a result of a number of experiences with the A and B pattern pai r.

It is very important to note that the information needed to use the Hebb rule to determine the value each connection should have is locally available at the connection. All a given connection needs to consider is the acti vation of the units on both sides of it. Thus, it would be possible toactual ly implement such a connection modulation scheme local ly, in each connectio<sup>n</sup> , without requiring any programmer to reach into

each connection and set it to just the right val ue.

It turns out that the Hebb rule as stated here has some serious limitations, and, to our knowledge, no theorists continue to use it in this simple form. More sophisticated connection modulation �chemes h<sup>a</sup>ve been proposed by other workers; most important among these are the delta rule, discussed extensi vely in Chapters 8 and 11 ; the competi tive learning rule, discussed in Chapter 5 ; and the rules for learning in stochastic paral lel models, descri bed in Chapters 6 and 7. All of these learning rules have the property that they adjust the strengths of con<sup>n</sup>ections between units on the basis of information that can be assumed to be local ly avail able to the unit. Learning, then, in all of these cases, amounts to a very simpleprocess that can be implemented local ly at each connection without the need for any overall supervision. Thus , models which incorporate these learning rules train themselves to have the right interconnections in the course of processing the members of

an ensemble of patterns.

Learning multiple patterns in the same set of interconnections. Up to now, we have considered how we might teach our pattern associator to associate the visual pattern for one object with a pattern for the aroma of the same object . Obviously, different patterns of interconnections between the A and 8 unitsare appropriate for causing the visual pattern for a different object to gi ve rise to the pattern for its aroma. The same principles apply, however, and if we presented our pattem associator with the A and 8 patterns for steak , it would learn the ri&ht set of interconnections for that case instead (these are shown in Figure 138) . In fact, it turns out that we can actually teach the same pattern associator a number of different associations. The matrix representing the set of interconnections that would be learned if we taught the same pattern associator both the rose association and the steak association is show<sup>n</sup>in Figure 14. The reader can verify this by adding the two matrices for the indi vidual patternstogether. The reader can also verify that this set of connections wi ll allow the rose A pattern to produce the rose 8 pattern , and the steak A pattern to produce the steak 8 pattern: when either input pattern is presented , the correct corresponding output

The examples used here have the property that the two different visual patterns are ompl�<sup>t</sup> <sup>l</sup>yuncorrelated with each other. This being Copyrighted Material

+ + + + + + + +

+

+

+

+ +

+

+

+ +

++ --

-- ++

-- ++

++ --

FIGURE 14. The weights in the third matrix allow either A pattern shown in Figure \3 to recreate the corresponding B pattern . Each weight in this case is equal to the sum of

the weight for the A pattern and the weight for the B pattern , as illustrated.

the case , the rose pattern produces no effect when the interconnections for the steak have been establ ished, and the steak pattern produces no effect when the interconnections for the rose association are in effect. For this reason , it is possible to add together the pattern of interconnections for the rose association and the pattern for the steak association, and still be able to associate the sight of the steak with the smell of a steak and the sight of a rose with the smell of a rose. The two sets

of interconnections do not interact at al l.

One of the limitations of the Hebbian learning rule is that it can learn the connection strengths appropriate to an entire ensemble of patterns only when all the patterns are completely uncorrelated. This rest riction does not, however, apply to pattern associators which use

more sophisticated learning schemes.

Attractive properties of pattern associator models. Pattern associator models have the property that uncorrelated patterns do not interact with each other, but more similar ones do. Thus, to the extent that a new pattern of acti vation on the A units is similar to one of the old ones, it wi ll tend to have simi lar effects. Furthermore, if we assume that learning the in terconnections occurs in small increments, similar patterns wi ll essentially reinforce the strengths of the links they share in common with other patterns. Thus, if we present the same pai r of patterns over and over, but each time we add a little random noise to each element of each member of the pai r, the system wi ll automatical ly learn to associate the central tendency of the two patterns and wi ll learn to ignore the noise. What wi ll be stored will be an average of the similar patterns with the slight variations removed . On the other hand, when we present the system with completely uncorrelated patterns, they will not interact with each other in this way. Thus , the same pool of units can extract the central tendency of each of a number of pairs of unrelated patterns. This aspect of distributed models is exploited extensi vely

Extracting the structure of an ensemble of patterns. The fact that similar patterns tend to produce similar effects al lows di stributed models to exhibit a kind of spontaneous general ization , extending behavior appropriate for one pattern to other simi lar patterns. This property is shared by other POP models, such as the word perception model and the Jets and Sharks model described above; the main difference here is in the existence of simple, local , learning mechanisms that can allow the acquisition of the connection strengths needed to produce these general izations through experience with members of the ensemble of patterns . Distributed models have another interesting property as well: If there are regularities in the correspondences between pai rs of patterns , the model will natural ly extract these regularities. This property al lows distri buted models to acquire patterns of interconnections that lead them to behave in ways we ordinari ly take as

evidence for the use of linguistic rules.

A detailed example of such a model is described in Chapter 18. Here, we describe the model very briefly. The model is a mechanism that learns how to construct the past tenses of words from thei r root forms through repeated presentations of examples of root forms pai red with the corresponding past-tense form. The model consists of two pools of units. In one pool , patterns of activation representing the phonological structure of the root form of the verb can be represented, and, in the other , patterns representing the phonological structure of the past tense can be represented. The goal of the model is simply to learn the right connection strengths between the root units and the past-tense units, so that whenever the root form of a verb is presented the model wi ll construct the corresponding past· tense form. The model is trained by presenting the root form of the verb as a pattern of activation over the root units, and then using a simple, local , learning rule to adjust the connection strengths so that this root form wi ll tend to produce the correct pattern of activation over the past-tense units. The model is tested by simply presenting the root form as a pattern of activation over the root units and examining the pattern of activation

produced over the past-tense units.

The model is trained initially with a small number of verbs children learn early in the acquisition process . At this point in learning, it can only produce appropriate outputs for inputs that it has explicitly been shown. But as it learns more and more verbs, it exhibits two interest· ing behaviors. Fi rst, it produces the standard ed past tense when tested with pseudo-verbs or verbs it has never seen. Second, it "overregularizes " the past tense of irregular words it previously completed correctly. Often, the model wi ll blend the irregular past tense of the word with the regular ed ending, and produce errors like CAMED as the past of

COME. These phenomena mi rror those observed in the early phases of

acquisition of control over past tenses in young chi ldren.

The generativity of the chi ld's responses -the creation of regular past tenses of new verbs and the overregularization of the irregular verbs-has been taken as strong evidence that the child has induced the rule which states that the regular correspondence for the past tense in Engl ish is to add a final ed (Berko, 1958) . On the evidence of its performance, then, the model can be said to have acqui red the rule. However, no special rule-induction mechanism is used, and no special language-acquisition device is required. The model learns to behave in accordance with the rule, not by explicitly noting that most words take ed in the past tense in English and storing this rule away explicitly, but simply by building up a set of connections in a pattern associator through a long series of simple learning experiences . The same mechanisms of paral lel distributed processing and connection modification which are used in a number of domains serve, in this case, to produce implicit knowledge tantamount to a linguistic rule. The model also provides a fai rly detailed account of a number of the specific aspects of the error patterns children make in learning the rule. In this sense, it provides a richer and more detai led description of the acquisition process than any that falls out natural ly from the assumption that the child is building up a repertoi re of expl icit but inaccessible rules. There is a lot more to be said about distributed models of learning, about thei r strengths and thei r weaknesses , than we have space for in this prel iminary consideration. For now we hope mainly to have suggested that they provide dramatically different accounts of learning and acquisition than are offered by traditional models of these processes . We saw in earl ier sections of this chapter that performance in accordance with rules can emerge from the interactions of simple, interconnected units. Now we can see how the aquisition of performance that conforms to linguistic rules can emerge from a simple, local , connec-

tion strength modulation process.

We have seen what the properties of PDP models are in informal terms , and we have seen how these properties operate to make the models do many of the kinds of things that they do. The business of the next chapter is to lay out these properties more formal ly, and to introduce some formal tools for their descri ption and analysi s. Before we turn to this, however, we wish to describe some of the major

sources of inspi ration for the PDP approach .

ORIGINS OF PARALLEL DISTRIBUTED PROCESSING

The ideas behi nd the POP approach have a history that st retches back indefinitely. In this section, we mention briefly some of the people who have thought in these terms, particularly those whose work has had an impact on our own thinking. This section should not been seen as an authoritative review of the history, but only as a description of

our own sources of inspiration.

Some of the earl iest roots of the POP approach can be found in the work of the unique neurologi sts, Jackson 0869/ 1958) and Luria (1 966) . Jackson was a forceful and persuasive critic of the si mplistic localizationist doctrines of late nineteenth century neurology, and he argued convincingly for distributed, multilevel conceptions of processing systems . Luria, the Russian psychologist and neurologi st, put forward the notion of the dynamic fUnctional system. On this view, every behavioral or cogniti ve process resulted from the coordination of a large number of different components, each roughly local ized in different regions of the brain, but all working together in dynamic interaction. Neither Hughlings-Jackson nor Luria is noted for the clarity of his views , but we have seen in thei r ideas a rough characterization of the

kind of parallel distributed processing system we envision.

Two other contributors to the deep background of POP were Hebb (1949) and Lashley (1 950) . We al ready have noted Hebb' s contribution of the Hebb rule of synaptic modification; he also introduced the concept of cell assemblies -a concrete example of a limited form of di stributed processing-and discussed the idea of reverberation of activation within neural networks. Hebb's ideas were cast more in the form of speculations about neural functioning than in the form of concrete processing models, but his thinking captures some of the flavor of parallel distributed processing mechanisms. Lashley 's contribution was to insist upon the idea of distributed representation. Lashley may have been too radical and too vague, and his doct rine of equipotential ity of broad regions of cortex clearly overstated the case. Yet many of his insights into the difficulties of storing the "engram" locally in the brain are telling, and he seemed to capture quite precisely the essence of distributed representation in insisting that "there are no special cells

reserved for special memories" (Lashley, 1950, p. 500) .

In the 1950s, there were two major figures whose ideas have contributed to the development of our approach. One was Rosenblatt 0959, 1962) and the other was Selfridge (1955) . In his Principles of Neurodynamics (I 962) , Rosenblatt articulated clearly the promise of a neurally inspi red approach to computation, and he developed the perceptron convergence procedure, aeojJyW�i�I¥��mge over the Hebb rule for

changing synaptic connections. RosenblaU's work was very controversial at the time, and the specific models he proposed were not up to all the hopes he had for them. But his vision of the human information processing system as a dynamic, interactive, self-organizing system lies at the core of the POP approach. Selfridge's contri bution was his insistence on the importance of interactive processing, and the development of Pandemonium, an explicitly computational example of a dynamic, interactive mechanism appl ied to computational problems in

perception.

In the late 60s and early 70s, serial processing and the von Neumann computer dominated both psychology and artificial intel ligence, but there were a number of researchers who proposed neural mechanisms which capture much of the flavor of POP models. Among these figures, the most influential in our work have been J. A. Anderson, Grossberg, and Longuet-Higgins. Grossberg 's mathematical analysis of the properties of neural networks led him to many insights we have only come to appreciate through extensi ve experience with computer simulation, and he deserves credit for seeing the relevance of neural ly inspired mechanisms in many areas of perception and memory well before the field was ready for these kinds of ideas (Grossberg, 1978) . Grossberg (1 976) was also one of the first to analyze some of the properties of the competitive learning mechanism explored in Chapter 5. Anderson's work differs from Grossberg's in insisting upon distributed representation , and in showing the relevance of neural ly inspired models for theories of concept learning (Anderson, 1973, <sup>1977</sup>) ; the work in Chapters 17 and 25 on distributed memory and amnesia owes a great deal to Anderson's inspiration. Anderson's work also played a crucial role in the formulation of the cascade model (McClel land, 1979) , a step away from serial processing down the road to POP. Longuet-Higgins and his group at Edinburgh were also pursuing distri buted memory models during the same period, and Oavid Wi llshaw, a member of the Edinburgh group, provided some very elegant mathematical analyses of the properties of various distri buted representation schemes (Wi <sup>l</sup> lshaw, 1981). His insights provide one of the sources of the idea of coarse coding described at length in Chapter 3. Many of the contri butions of Anderson, Wi llshaw, and others distributed modelers may be found in Hinton and Anderson (981 ) . Others who have made important contributions to learning in POP models incl ude Amari ( 1 977a) , Bienenstock, Cooper, and Munro (I 982) , Fukushima (I <sup>97</sup> 5) , Kohonen ( <sup>1</sup> <sup>9</sup><sup>77</sup> , 1984) , and von der

Malsburg (I 973) .

Toward the middle of the 1970s, the idea of paral lel processing began to have something of a renaissance in computational circles. We have already mentioned the Marr and Poggio (976) model of stereoscopic Copyrighted Material

depth perception. Another model from this period , the HEARSA Y model of speech understanding, played a prominent role in the development of our thinking. Unfortunately , HEARSAY's computational architecture was too demandi ng for the avai lable computational resources , and so the model was not a computational success . But its basically parallel , interactive character inspi red the interactive model of reading (Rumelhart, 1977) , and the interactive activation model of word recogni <sup>t</sup>ion (McClel land & Rumelhart, 1981; Rumelhart &

McClelland, 1982) .

The ideas represented in the interactive activation model had other <sup>p</sup>recursors as wel l . Morton's logogen model (Morton , 1 969) was one of the first models to capture concretely the pri nciple of interaction of different sources of information, and Marslen-Wi lson (e.g., Marslen-Wi lson & Welsh, 1978) provided important empi rical demonstrations of interaction between different levels of language processing. Levin's (1 976) Proteus model demonst rated the virtues of activationcompetition mechanisms, and Glushko (1 979) hel ped us see how conspi racies of partial activations could account for certain aspects of

apparently rule-guided behavior.

Our work also owes a great deal to a number of col leagues who have been working on related ideas in recent years . Many of these col leagues appear as authors or coauthors of chapters in this book . But there are others as well. Several of these people have been very influential in the development of the ideas in this book. Feldman and Ballard (1 982) laid out many of the computational pri nciples of the POP approach (under the name of connectionism), and stressed the biological implausibility of most of the prevai l ing computational models in artificial intelligence. Hofstadter (1979, 1985) deserves credit for s<sup>t</sup> ressing the exi stence of a subcognitive-what we call microstructural -level , and pointing out how important it can be to del ve into the microstructure to gain insight . A sand dune, he has said, is not a grain of sand. Others have contributed crucial technical insights. Sutton and Barto (981) provided an insightful analysis of the connection modification scheme we call the delta rule and i l l <sup>u</sup>st rated the power of the rule to account for some of the subtler properties of classical conditioning. And Hopfield's (982) contri bution of the idea that network models can be seen as seeking minima in energy landscapes played a prominent role in the development of the Boltzmann machine (Chapter 7) , and in the crystallization of the ideas presented in

Chapters 7 and 14 on harmony theory and schemata.

The power of paral lel distributed processing is becoming more and more apparent, and many others have recently joined in the exploration of the capabi lities of these mechanisms. We hope this book represents Copyrighted Material

the nature of the enterprise we are al l involved in, and that it does jus-

tice to the potential of the PDP approach.

ACKNOWLEDGMENTS

This research was supported by Contract N00014-79-C-03 23, NR 667-437 with the Personnel and Training Research Programs of the Office of Naval Research, by grants from the System Development Foundation, and By a NIMH Career Development Award (MH00385 )

to the first author.

CHAPTER 2

A General Framework for

Parallel Distributed Processing

D. E. RUMELHART, G. E. HINTON, and 1. L. McCLELLAND

In Chapter 1 and throughout this book, we describe a large number of models, each different in detail-each a variation on the parallel distributed processing (PDP) idea. These various models, and indeed many in the literature, clearly have many features in common, but they are just as clearly distinct models. How can we characterize the general model of which these specific models are instances? In this chapter we propose a framework sufficiently general so that all of the various models discussed in the book and many models in the literature are special cases. We will proceed by first sketching the general framework and then by showing properties of certain specific realizations of the

general model. I

The General Framework

It is useful to begin with an analysis of the various components of our models and then describe the various specific assumptions we can

I We are, of course, not the first to attempt a general characterization of this general class of models. Kohonen 0977, 1984), Amari (l977a), and Feldman and Ballard (1982)

are papers with similarly general aims.

make about these components. There are eight major aspects of a

parallel distributed processing model:

• A set of processing units • A state of activation

• An output junction for each unit • A pattern of connectivity among units

• A propagation rule for propagating patterns of activities through

the network of connectivities

• An activation rule for combining the inputs impinging on a unit with the current state of that unit to produce a new level of

activation for the unit.

• A learning rule whereby patterns of connectivity are modified by

experience

• An environment within which the system must operate

Figure 1 illustrates the basic aspects of these systems. There is a set of processing units generally indicated by circles in our diagrams; at each point in time, each unit Uj has an activation value, denoted in the diagram as aj (t); this activation value is passed through a function fj to produce an output value OJ (t). This output value can be seen as passing through a set of unidirectional connections (indicated by lines or arrows in our diagrams) to other units in the system. There is associated with each connection a real number, usually called the weight or strength of the connection designated wi) which determines the amount of effect that the first unit has on the second. All of the inputs must then be combined by some operator (usually addition) -and the combined inputs to a unit, along with its current activation value, determine, via a function F, its new activation value. The figure shows illustrative examples of the function f and F. Finally, these systems are viewed as being plastic in the sense that the pattern of interconnections is not fixed for all time; rather, the weights can undergo modification as a function of experience. In this way the system can evolve. What a unit represents can change with experience, and the system can come to perform in substantially different ways. In the fol lowing sections we develop an explicit notation for each of these components and describe some of the alternate assumptions that have been made concerning

each such component.

A set of processing units. Any parallel activation model begins with a set of processing units. Specifying the set of processing units and what they represent is typically the first stage of specifying a PDP model. In some models these units may represent particular conceptual objects such as features, letters, words, or concepts; in others they are Copyrighted Material

Threshold OU'pu' Function

o

·lZJ ·m 0

nor, = �"'JOJ(I) Sigmoid ActiValion Function

FIGURE 1. The basic components of a parallel distributed processing system.

simply abstract elements over which meaningful patterns can be defined. When we speak of a distributed representation, we mean one in which the units represent small, feature-like entities. In this case it is the pattern as a whole that is the meaningful level of analysis. This should be contrasted to a one-unit-one-concept representational system in which single units represent entire concepts or other large meaning-

ful entities.

We let N be the number of units. We can order the units arbitrarily and designate the ith unit Ui' All of the processing of a PDP model is carried out by these units. There is no executive or other overseer. There are only relatively simple units, each doing it own relatively simple job. A unit's job is simply to receive input from its neighbors and, as a function of the inputs it receives, to compute an output value which it sends to its neighbors. The system is inherently parallel in that

Within any system we are modeling, it is useful to characterize three types of units: input, output, and hidden. Input units receive inputs from sources external to the system under study. These inputs may be either sensory input or inputs from other parts of the processing system in which the model is embedded. The output units send signals out of the system. They may either directly affect motoric systems or simply influence other systems external to the ones we are modeling. The hidden units are those whose only inputs and outputs are within the system we are modeling. They are not "visible" to outside systems.

The state of activation. In addition, to the set of units, we need a representation of the state of the system at time t. This is primarily specified by a vector of N real numbers, a (t), representing the pattern of activation over the set of processing units. Each element of the vector stands for the activation of one of the units at time t. The activation of unit Ui at time t is designated 0i (t ). It is the pattern of activation over the set of units that captures what the system is representing at any time. It is useful to see processing in the system as the evolu-

tion, through time, of a pattern of activity over the set of units.

Different models make different assumptions about the activation values a unit is allowed to take on. Activation values may be continuous or discrete. If they are continuous, they may be unbounded or bounded. If they are discrete, they may take binary values or any of a small set of values. Thus in some models, units are continuous and may take on any real number as an activation value. In other cases, they may take on any real value between some minimum and maximum such as, for example, the interval [0,11. When activation values are restricted to discrete values they most often are binary. Sometimes they are restricted to the values 0 and I where I is usually taken to mean that the unit is active and 0 is taken to mean that it is inactive. In other models, activation values are restricted to the values {- I,+ I} (often denoted simply (-,+}). Other times nonbinary discrete values are involved. Thus, for example, they may be restricted to the set {- I,O,+I}, or to a small finite set of values such as {I,2,3,4,5,6,7,8,9}. As we shall see, each of these assumptions leads to a model with slightly different characteristics. It is part of the program of research represented in this book to determine the implications of these various

assumptions.

Output of the units. Units interact. They do so by transmitting signals to their neighbors. The strength of their signals, and therefore the degree to which they affect their neighbors, is determined by their degree of activation. Associated with each unit, Ui, there is an output function, Ii (0, (1», which maps the current state of activation 0i (I) to Copyrighted Material

CD � Q. CD ::::r- c5' � �

Network Representation

Input

u6 Us u4 u3 u2 U,

Output

u, Us

Matrix Representation

U, Uz u3 u4 U

s Us u7 uB

+3

-4

0 0 +1

+4

0 0

+4

0 0 +6

-1

0 -2

0 +4

§:

FIGURE

2.

The connectivity of a network represented by a network

drawing

and in a matrix. The figure shows an

eight-unit

network with

units umbered

from 1 to 8. Units 1 to 4

are input units. They receive

inputs

from

he outside world and feedback

from the output units-

units 5 through

8.

The connections among the units are indicated

by the ope

n and

filled

disks. The size of

the

disk indicates the strength of

connection. Thus,

the

large black disk on the line connecting unit

1 to unit 8

indicates a strong

inhibitory connection

from 1 o 8. Simirythe

lar

ge open disk on the

output line

from uit 8

to unit 2 idicas

that unit

8 strongly

excites unit 2. The

same connections

are shown in the

matrix representation on

the left.

The +6 in he

column

for Us

and the r

w for u2 indices that unit 8

strongly ex

cits unit 2. It

should be

noted that

whenever there is a disk

on a lin

e connecting t

he output of

one unit

to the input of another

in the netwr

k diagram there is a

corresponding nonzero

entry

in the matrix. If

the

disk is filled, the

entry in

he matrix is

negative. If the

disk is

open, the

entry

is positive.

The

arger l the

disk the greater the magnitude of

he entry

in the matrix.

It

might also be noted

hat the conectins in the

network have been

laid out to correspond to the

entries of

the matrix. The black disk in

he upper

left corner of the

network corresponds

to the -6 in

the upper

left corner of the

matrix.

Each

disk in the network is in the corresponding position of its location in the matrix. The network would not

have

to be drawn

in this way,

of course, and the matrix would s

ill

capture all

of

the connectivity

information in

the

network. In general,

because

network drawings are

difficult

to work with we wi

ll

often simply

use the matri

x representation

(0

specify (he

pattern of c

nnectiviy

inhibitory connection and an excitatory connection. When the models assume simple addition of inhibition and excitation they do not constitute different types of connections in our present sense. They only constitute distinct types when they combine through some more complex

rules.

The pattern of connectivity is very important. It is this pattern which determines what each unit represents. As we shall see below, many of the issues concerning whether top-down or bottom-up processing systems are correct descriptions or whether a system is hierarchical and if so how many levels it has, etc., are all issues of the nature of the connectivity matrix. One important issue that may determine both how much information can be stored and how much serial processing the network must perform is the fan-in and fan-out of a unit. The fan-in is the number of elements that either excite or inhibit a given unit. The fanout of a unit is the number of units affected directly by a unit. Note, in some cases we need more general patterns of connectivity. Specifying such a pattern in the general case is complex and will be addressed

in a later section of this chapter.

The rule of propagation. We also need a rule which takes the output vector, 0 (t), representing the output values of the units and combines it with the connectivity matrices to produce a net input for each type of input into the unit. We let netij be the net input of type i to unit u). Whenever only one type of connectivity is involved we suppress the first subscript and use net) to mean the net input into unit u). In vector notation we can write net; (I) to represent the net input vector for inputs of type i. The propagation rule is generally straightforward. For example, if we have two types of connections, inhibitory and excitatory, the net excitatory input is usually the weighted sum of the excitatory inputs to the unit. This is given by the vector product nete = Weo (I). Similarly, the net inhibitory effect can be written as net j<sup>=</sup>W j<sup>O</sup>(t). When more complex patterns of connectivity are involved, more complex rules of propagation are required. We treat this in the final section

of the chapter.

Activation rule. We also need a rule whereby the net inputs of each type impinging on a particular unit are combined with one another and with the current state of the unit to produce a new state of activation. We need a function, F, which takes a (t) and the vectors net) for each different type of connection and produces a new state of activation. In the simplest cases, when F is the identity function and when all connections are of the same type, we can write a (t+ 1) = Wo (t } = net (t}. Sometimes F is a threshold function so that the net input must exceed

Virtually all learning rules for models of this type can be considered a variant of the Hebbian learning rule suggested by Hebb in his classic book Organization of Behavior (949) . Hebb's basic idea is this: If <sup>a</sup> unit, U;, receives a input from another unit, Uj; then, if both are highly active, the weight, wij, from Uj to U; should be strengthened. This idea has been extended and modified so that it can be more generally stated

as

<sup>a</sup>W;j = g (a; (t) ,1; (I» h(Oj (t), w;),

where t; (t) is a kind of leaching input to U;. Simply stated, this equation says that the change in the connection from Uj to U; is given by the product of a function, gO, of the activation of U; and its teaching input I; and another function, hO, of the output value of Uj and the connection strength wij. In the simplest versions of Hebbian learning there is no teacher and the functions g and h are simply proportional to their

first arguments. Thus we have

where TJ is the constant of proportionality representing the learning rate. Another common variation is a rule in which h (OJ (t), wij) = OJ (I) and g(a; (I) ,1; (I» = TJ (I; (t )-a; (I». This is often called the Widrow-Hoff rule (Sutton & Barto, 1981). However, we call it the delta rule because the amount of learning is proportional to the difference (or delta) between the actual activation achieved and the target activation provided by a teacher. (The delta rule is discussed at length in

Chapters 8 and 11.) In this case we have

<sup>a</sup>Wij = TJ (tj (I)-Oj (t ))OJ (t).

This is a generalization of the perceptron learning rule for which the famous perception convergence theorem has been proved. Still another

variation has

<sup>a</sup>Wij = TJOi (t) (OJ (t )- Wi) ).

This is a rule employed by Grossberg (1976) and a simple variant of which has been employed in Chapter 5. There are many variations on this generalized rule, and we will describe some of them in more detail

when we discuss various specific models below.

Representation of the environment. It is crucial in the development of any model to have a clear model of the environment in which this model is to exist. In PDP models, we represent the environment as a time-varying stochastic function over the space of input patterns. That Copyrighted Material

is, we imagine that at any point in time, there is some probability that any of the possible set of input patterns is impinging on the input units. This probability function may in general depend on the history of inputs to the system as well as outputs of the system. In practice, most PDP models involve a much simpler characterization of the environment. Typically, the environment is characterized by a stable probability distribution over the set of possible input patterns independent of past inputs and past responses of the system. In this case, we can imagine listing the set of possible inputs to the system and numbering them from 1 to M. The environment is then characterized by a set of probabilities, Pi for i = 1, ... , M. Since each input pattern can be considered a vector, it is sometimes useful to characterize those patterns with nonzero probabi lities as constituting orthogonal or linearly independent sets of vectors. 2 Certain PDP models are restricted in the kinds of patterns they are able to learn: some being able to learn to respond correctly only if the input vectors form an orthogonal set; others if they form a linearly independent set of vectors; and still others are able to

learn to respond to essentially arbitrary patterns of inputs.

CLASSES OF PDP MODELS

There are many paradigms and classes of PDP models that have been developed. In this section we describe some general classes of assumptions and paradigms. In the following section we describe some specific PDP models and show their relationships to the general framework out-

lined here.

Paradigms of Learning

Although most learning rules have roughly the form indicated above, we can categorize the learning situation into two distinct sorts. These

are:

• Associative learning, in which we learn to produce a partIcular pattern of activation on one set of units whenever another particular pattern occurs on another set of units. In general , such a learning scheme must allow an arbitrary pattern on one set of

units to produce another arbitrary pattern on another set of

units.

• Regularity discovery, in which units learn to respond to "interesting" patterns in their input. In general, such a scheme should be able to form the basis for the development of feature detectors and therefore the basis for knowledge representation in a

PDP system.

In certain cases these two modes of learning blend into one another, but it is valuable to see the different goals of the two kinds of learning. Associative learning is employed whenever we are concerned with storing patterns so that they can be re-evoked in the future. These rules are primarily concerned with storing the relationships among subpatterns. Regularity detectors are concerned with the meaning of a single units response. These kinds of rules are used when feature discovery is

the essential task at hand.

The associative learning case generally can be broken down into two subcases-pattern association and auto-association. A pattern association paradigm is one in which the goal is to build up an association between patterns defined over one subset of the units and other patterns defined over a second subset of units. The goal is to find a set of connections so that whenever a particular pattern reappears on the first set of units, the associated pattern will appear on the second set. In this case, there is usually a teaching input to the second set of units during training indicating the desired pattern association. An auto-association par<sup>a</sup>digm is one in which an input pattern is associated with itself. The goal here is pattern completion. Whenever a portion of the input pattern is presented , the remainder of the pattern is to be filled in or completed. This is similar to simple pattern association, except that the input pattern plays both the role of the teaching input and of the pattern to be associated. It can be seen that simple pattern association is a special case of auto-association. Figure 3 illustrates the two kinds of learning paradigms. Figure 3A shows the basic structure of the pattern association situation. There are two distinct groups of units-a set of input units and a set of output units. Each input unit connects with each output unit and each output unit receives an input from each input unit. <sup>D</sup><sup>u</sup>r<sup>i</sup>ng training, patterns are presented to both the input and output units. The weights connecting the input to the output units are modified during this period. During a test, patterns are presented to the input units and the response on the output units is measured. Figure 3B shows the connectivity matrix for the pattern associator. The only modifiable connections are from the input units to the output units. All other connection<sup>s</sup>ate fixed tal, zero .. Fl igure 3C shows the hasic (;opyngTi au Matena

A B Matrix at Cannectivities OJ

> Inpul UnllS

OUlpul Unll.

SeIDl Inpul Unils Sel 01 Oulpul Unils

c 0

Inpul & Oulput Units

Inputs

Inpul Unlls Oulpul Unll.

Modlflable Welghls

> Connacllvlly Malrl. lor Pall.rn Assoclalor

> > All weights .r. Modifi abl.

Connecllvlly IIIllrl. lor AulO Auoetslor

FIGURE 3. A: The basic structure of the pattern association situation. There are two distinct groups of units-a set of input units and a set of output units. Each input unit connects with each output unit and each output unit receives an input from each input unit. During training, patterns are presented to both the input and output units. The weights connecting the input to the output units are modified during this period. During a test, patterns are presented to the input units and the response on the output units is measured. (After Anderson, 1977.) B: The connectivity matrix for the pattern associ ator. The only modifiable connections are from the input units to the output units. All other connections are fixed at zero. C: The basic structure of the auto-association situation. All units are both input and output units. The figure shows a group of 6 units feeding back on itself through modifiable connections. Note that each unit feeds back on itself as well as on each of its neighbors. (After Anderson, Silverstein , Ritz , & Jones, 1977.) D: The connectivity matrix for the auto-associator. All units connect to all other

units with modifiable weights.

structure of the auto-association situation. All units are both input and output units. The figure shows a group of 6 units feeding back on itself through modifiable connections. Note that each unit feeds back on itself as well as on each of its neighbors . Figure 3D shows the connectivity matrix for the auto-associator. All units connect to all other units with modifiable weights. In the case of auto-association, there is

potentially a modifiable connection from every unit to every other unit. In the case of pattern association, however, the units are broken into two subpatterns, one representing the input pattern and another representing the teaching input. The only modifiable connections are those from the input units to the output units receiving the teaching input. In other cases of associative learning the teaching input may be more or less indirect. The problem of dealing with indirect feedback is difficult, but central to the development of more sophisticated models of learning. Barto and Sutton (I 981) have begun a nice analysis of

such learning situations.

In the case of regularity detectors, a teaching input is not expl icitly provided; instead, the teaching function is determined by the unit itself. The form of the internal teaching function and the nature of its input patterns determine what features the unit will learn to respond to. This is sometimes called unsupervised learning. Each different kind of unsupervi sed learning procedure has its own evaluation function . The particular evaluation procedures are mentioned when we treat these models. The three unsupervi sed learning models discussed in this book

are addressed in Chapters 5, 6, and 7.

Hierarchical Organizations of PDP Networks

It has become commonplace in cognitive science to describe such processes as top-down , bottom-up, and interactive to consist of many stages of processing, etc. It is useful to see how these concepts can be represented in terms of the patterns of connectivity in the PDP framework. It is also useful to get some feel ing for the processing conse-

quences of these various assumptions.

Bottom-Up Processing

The fundamental characteristic of a bottom-up system is that units at level i may not affect the activity of units at levels lower than i. To see how this maps onto the current formulation, it is useful to partition the coal itions of units into a set of discrete categories corresponding to the levels their inputs come from. There are assumed to be no coal itions with inputs from more than one level . Assume that there are L; units at level i in the system. We then order the units such that those

in level L 1 are numbered u 1, ••. , UL

' those in level L 2 are numbered

bottom-up system is equivalent to the constraint that the connectivity matrix, W, has zero entries for wi} in which Uj is the member of a level no higher than Uj. This amounts to the requirement that the upper right-hand region of W contains zero entries. Table 1 shows this constraint graphical ly. The table shows an example of a three-level system with four units at each level. 3 This leads to a 12 x 12 connectivity matrix and an a vector of length 12. The matrix can be divided up into 9 regions. The upper-left region represents interactions among Level 1 units. The entries in the left-middle region of the matrix represents the effects of Level 1 units on Level 2 units. The lower-left region represents the effects of Level 1 units on Level 3 units. Often bottom-up models do not allow units at level i effect units at level i+ 2. Thus, in the diagram we have left that region empty representing no effect of Level 1 on Level 3. It is typical in a bottom-up system to assume as well that the lowest level units (Level 1) are input units and that the highest level units (Level 3) are output units. That is, the lowest level of the system is the only one to receive direct inputs from outside of this module and only the highest level units affect other

units outside of this module.

TABLE I

<sup>L</sup>evell Level 2 Level <sup>3</sup> Input Units Hidd<sup>e</sup>nUnit<sup>s</sup>Output Units ul u2 u3 u4 uS u6 u7 u8 u9 ulO ull u12

ul within

Levell u2 Level I U<sup>n</sup>its u3 effects u4

uS Levell within

Level 2 u6 affecting Level 2 Units u7 Level 2 effects

u8

u9 Level 2 withi<sup>n</sup>

Level 3 ulO affecting Level 3 Units ull Level 3 effects

ul2

Top-Down Processing

The generalization to a hierarchical top-down system should be clear enough. Let us order the units into levels just as before. A top-down model then requires that the lower-left regions of the weight matrix be empty-that is, no lower level unit affects a higher level unit. Table 2 illustrates a simple example of a top-down processing system. Note, in this case, we have to assume a top-down input or "message" that is propagated down the system from higher to lower levels as well as any

data input that might be coming directly into Levell units.

Interactive Models

Interactive models are simply models in which there can be both top-down and bottom-up connections. Again the generalization is straightforward. In the general interactive model, any of the cells of the weight matrix could be nonzero. The more restricted models in which information flows both ways, but in which information only flows between adjacent levels, assume only that the regions of the matrix more than one region away from the main diagonal are zero. Table 3 illustrates a simple three-level interactive model with both topdown and bottom-up input. Most of the models that actually have been

suggested count as interactive models in this sense.

TABLE 2

Levell Level 2 Level 3 Input Units Hidden Units Output Units ul u2 u3 u4 u5 u6 u7 u8 u9 ulO u11 ul2

ul within Level 2

Level I u2 Level I affecting Units u3 effects Levell

u4

u5 within Level 3

Level 2 u6 Level 2 affecting <sup>U</sup>n<sup>i</sup>ts u7 effects Level 2

u8

u9 within

Level 3ulO Level 3 Units u11 effects

ul2 G6-{JYfighied MtJf� jar

ul

Level I u2 Units u3

> u4 uS

Level 2 u6 Units u7

> uS u9

Level 3 ulO Units ull

ul2

TABLE 3

Level I Level 2 Level 3 Input Units Hidden Units Output Units ul u2 u3 u4 uS u6 u7 uS u9 ulO ull u12

within Level 2 Level I affecting effects Level I

Levell within Level 3 affecting Level 2 affecting Level 2 effects Level 2

> Level 2 within affecting Level 3 Level 3 effects

It is sometimes supposed that a "single level" system with no hierarchical structure in which any unit can communicate with any other unit is somehow less powerful than these multi level hierarchical systems. The present analysis shows that , on the contrary, the existence of levels amounts to a restriction, in general , of free communication among all units. Such nonhierarchical systems actually form a superset of the kinds of layered systems discussed above. There is, however, something Ito the view that having multi ple levels can increase the power of certain systems. In particular, a "one-step" system consisting of only input and output units and no communication between them in which there is no opportunity for feedback or for hidden units is less powerful than systems with hidden units and with feedback. Since, in general , hierarchical systems involve many hidden units, some intralevel communication, and some feedback among levels, they are more powerful than systems not involving such hidden units. However, a system with an equal number of hidden units, but one not characterizable as hierarchical by the communication patterns is, in general , of more potential computational power. We address the issue of hidden units and "single-step" versus "multiple-step" systems in our discussion of

speci fic models below.

Synchronous Versus Asynchronous Update

Even given all of the components of the PDP models we have described so far, there is still another important issue to be resolved in the development of specific models; that is the timing of the application of the activation rule. In some models, there is a kind of central timing pulse and after each such clock tick a new value is determined simultaneously for all units. This is a synchronous update procedure. It is usually viewed as a discrete, difference approximation to an underlying continuous, differential equation in which al l units are continuously updated. In some models, however, units are updated asynchronously and at random. The usual assumption is that at each point in time each unit has a fixed probability of evaluating and applying its activation rule and updating its activation value. This later method has certain theoretical advantages and was developed by Hopfield (1982) and has been employed in Chapters 6, 7, and 14. The major advantage is that since the units are independently being updated, if we look at a short enough time interval, only one unit is updating at a time. Among other things, this system can help the stability of the network by keeping it out of oscil lations that are more readily entered into with

synchronous update procedures .

SPECIFIC VERSIONS OF THE GENERAL PARALLEL

ACTIV ATION MODEL

In the following sections we will show how specification of the particular functions involved produces various kinds of these models. There have been many authors who have contributed to the field and whose work might as well have been discussed. We discuss only a representa-

tive sample of this work.

Simple Linear Models

Perhaps the simplest model of this class is the simple linear model. In the simple linear model, activation values are real numbers without restriction. They can be either positive or negative and are not bounded. The output function, f (a/), in the linear model is just equal to the activation level ai' Typically, linear models consist of two sets of

below, there is no need for hidden units since all computation possible with a multiple-step linear system can be done with a single-step linear system.) In general , any unit in the input layer may connect to any unit in the output layer. All connections in a linear model are of the same type. Thus, only a single connectivity matrix is required. The matrix consists of a set of positive, negative, and zero values, for excitatory values, inhibitory values, and zero connections, respectively. The new value of activation of each unit is simply given by the weighted sums of the inputs. For the simple linear model with connectivity matrix W we

have

a(t+1) = Wa(t).

In general , it can be shown that a linear model such as this has a number of limitations. In particular, it can be shown that nothing can be computed from two or more steps that cannot be computed by a

single step. This follows because the above equation implies

a (t) = W' a (0).

We can see this by proceeding step by step. Clearly,

<sup>a</sup>(2) = Wa (1) = W (Wa (0» = W2a (0) .

It should be clear that similar arguments lead to a (t) = W' a (0). From this, it follows that for every linear model with connectivity matrix W that can attain a particular state in t steps, there is another linear model with connectivity matrix W ( that can reach the same state in one step. This means, among other things, that there can never be any computational advantage in a linear model of multi ple-step systems, nor can there ever be any advantage for allowing feedback.

The pattern association paradigm is the typical learning situation for a linear model. There is a set of input units and a set of output units. In general , each input unit may be connected to any output unit. Since this is a linear network, there is no feedback in the system nor are there hidden units between the inputs and outputs. There are two sources of input in the system. There are the input patterns that establish a pattern of activation on the input units, and there are the teaching units that establish a pattern of activation on the output units. Any of several learning rules could be employed with a linear network such as this, but the most common are the simple Hebbian rule and the delta rule. The linear model with the simple Hebbian rule is called the simple linear associator (cf. Anderson, 1970; Kohonen, 1977, 1984). In this case, the increment in weight wI} is given by A wi} = TJaj t;. In matrix notation, this means that A W = TJT aT. The system is then tested by presenting an input pattern without a teaching input and Copyrighted Material

seeing how close the pattern generated on the output layer matches the original teaching input. It can be shown that if the input patterns are orthogonal,4 there will be no interference and the system will perfectly produce the relevant associated patterns exactly on the output layer. If they are not orthogonal , however, there wi ll be interference among the input patterns. It is possible to make a modification in the learning rule and allow a much larger set of possible associations. In particular, it is possible to build up correct associations among patterns whenever the set of input patterns are linearly independent. To achieve this, an error correcting rule must be employed. The delta rule is most commonly

employed. In this case, the rule becomes !:1wij = ",(tj-aj)aj

' What is

learned is essentially the difference between the desired response and that actually attained at unit Uj due to the input. Although it may take many presentations of the input pattern set, if the patterns are linearly independent the system wi ll eventually be able to produce the desired outputs . Kohonen 0977, 1984) has provided an importantanalysis of

this and related learning rules.

The examples described above were for the case of the pattern associator. Essentially the same results hold for the auto-associator version of the linear model . In this case, the input patterns and the teaching patterns are the same, and the input layer and the output layer are also the same. The tests of the system involve presenting a portion of the input pattern and having the system attempt to reconstruct the missing

parts.

Linear Threshold Units

The weaknesses of purely linear systems can be overcome through the additio<sup>n</sup>of nonlinearities . Perhaps the simplest of the nonlinear system consists of a network of linear threshold units. The linear threshold unit is a binary unit whose activation takes on the values {O, I} . The activation value of unit Uj is I if the weighted sum of its inputs is greater than some threshold 9 jand is 0 otherwise. The connectivity matri x for a network of such units, as in the linear system, is a matrix consisting of positive and negative numbers. The output function, /, is the identity function so that the output of a unit is equal to

its activation value.

It is useful to see some of the kinds of functions that can be computed with linear threshold units that cannot be computed with simple linear models. The classic such function is the exclusive or (XOR) illustrated in Figure 4. The idea is to have a system which responds (I} if it receives a ( 0, l) or a (1 ,OJ and responds (O} otherwise. The figure shows a network capable of this pattern. In this case we require two

Output Unit

Internal Units

> Input Units

XOR Netwo rk

+1 +1

Thresholds = .01

Input Output

1 1 J 0 00

10] <sup>1</sup> 01

layers of units. Each unit has a zero threshold and responds just in case its input is greater than zero. The weights are ± 1. Since the set of stimulus patterns is not linearly independent, this is a discrimination that can never be made by a simple linear model and cannot be done in

a single step by any network of linear threshold units.

Although multi layered systems of linear threshold units are very powerful and, in fact, are capable of computing any boolean function, there is no generally known learning algorithm for this general case (see Chapter 8) . There is, however, a wel l-understood learning algorithm for the special case of the perceplron. A perceptron is essentially a single-layer network of linear threshold units without feedback. The learning situation here is exactly the same as that for the linear model . An input pattern is presented along with a teaching input. The perceptron learning rule is precisely of the same form as the delta rule for error correcting in the linear model, namely, �wij = 'Y/(/;-a; )aj . Since the teaching input and the activation values are only 0 or 1, the rule

reduces to the statements that:

1. Weights are only changed on a given input line when that line

is turned on (Le., aj = O.

2. If the system is correct on unit i (i .e., I; = a; ), make no change

on any of the input weights.

3. If the unit j responds 0 when it should be I, increase weights

on all active lines by amount 'Y/.

4. If the unit j responds 1 when it should be 0, decrease weights

on all active lines by amount 'Y/.

There is a theorem, the perceptron convergence theorem, that guarantees that if the set of patterns are learnable by a perceptron, this learning procedure wi ll find a set of weights which allow it to respond correctly to all input patterns. Unfortunately, even though multilayer linear threshold networks are potentially much more powerful than the linear associator, the percept ron for which a learning result exi sts can learn no patterns not learnable by the linear associator. It was the limitations on what perceptrons could possibly learn that led to Minsky and Papert's (969) pessimistic evaluation of the perceptron. Unfortunately that evaluation has incorrectly tainted more interesting and powerful networks of linear threshold and other nonlinear units. We have now developed a version of the delta rule -the generalized delta rule-which is capable of learning arbitrary mappings. It does not work for linear threshold units but does wor foL <sup>t</sup><sup>h</sup><sup>e</sup> class of semilinear activation , (.;<sup>o</sup><sup>p</sup><sup>y</sup>ng teu <sup>M</sup><sup>a</sup><sup>t</sup><sup>e</sup>nal

functions (i.e. , differentiable activation functions) . See Chapter 8 for a full discussion. As we shall see in the course of this book, the limitations of the one-step perceptron in no way apply to the more complex

networks.

Brain State in a Box

The brain state in a box model was developed by J. A. Anderson (I 977) . This model too is a close relative of the simple linear associator. There is, however, a maximum and minimum activation value associated with each unit. Typically, units take on activation values in the interval [- I, lL The brain state in a box (BSB) models are organized so that any unit can, in general , be connected to any other unit. The auto-associator illustrated in Figure 3 is the typical learning paradigm for BSB. Note that with this pattern of interconnections the system feeds back on itself and thus the activation can recycle through the system in a positive feedback loop. The positive feedback is especially evident in J. A. Anderson and Mozer's (1980 version. Their acti vation

rule is gi ven by

aj (t+ I) = aj (1 )+ L wij a; (I )

if aj is less than I and greater than -I. Otherwise, if the quantity is greater than I, aj = I and if it is less than -I , aj = -1. That is, the activation state at time t+ 1 is given by the sum of the state at time t and the activation propagated through the connectivity matrix provided that total is in the interval [- I,ll. Otherwise it simply takes on the maximum or minimum value. This formulation wi ll lead the system to a state in which all of the units are at either a maximum or minimum value. It is possible to understand why this is cal led a brain state in a box model by considering a geometric representation of the system. Figure 5 illustrates the "activation space " of a simple BSB system consisting of three units. Each point in the box corresponds to a particular value of activation on each of the three units. In this case we have a three-dimensional space in which the first coordinate corresponds to the activation value of the first unit, the second coordinate corresponds to the activation val ue of the second unit, and the third coordinate corresponds to the activation value of the third unit. Thus, each point in the space corresponds to a possi ble state of the system. The feature that each unit is limited to the region (- I,l <sup>l</sup>means that all points must lie somewhere within the box whose vertices are given by the points (- 1,-1,- 0, (- 1,-1,+ I), (-1,+ 1,-0 , (- 1,+1,+ 0, (+ 1 ,- 1,-0, (+ 1,-1,+0 , (+ 1,+1,-0 , and (+ 1,+1,+0 . Moreover, since the Copyrighted Material

(-,+,+)

( -, -,+) I-----'"�-----..{

Activation ! of Unit 2

(-, -,-) ----------.1

Activation of Unit 1

(+ .+.-)

FIGURE 5. The state space for a three-unit version of a BSB model . Each dimension of the box represents the activation value of one unit. Each unit is bounded in activation between [- I,ll. The curving arrow in the box represents the sequence of states the system moved through. It began at the black spot near the middle of the box and, as processing proceeded, moved to the (- ,+ ,+ ) corner of the box. BSB systems always end up in one or another of the corners. The particular corner depends on the start state of the network, the input to the system, and the pattern of connections among the units.

system involves positive feedback, it is eventually forced to occupy one of these vertices . Thus , the state of the system is constrained to lie within the box and eventually , as processing continues, is pushed to one of the vertices. Of course, the same geometric analogy carries over to higher dimensional systems. If there are N units, the state of the system can be characterized as a point within this N-dimensional hyper-

cube and eventually the system ends up in one of the 2

N corners of the

hypercube.

Learning in the BSB system involves auto-association. In different applications two different learning rules have been applied. J. A. Anderson and Mozer (1 98 1) appl ied the simplest rule. They simply allowed the system to settle down and then employed the simple Hebbian learning rule. That is, � wI} = 'Y/a/aj' The error correction rule has also been applied to the BSB model . In this case we use the input as the teaching input as well as the source of activation to the system. The learning rule thus becomes � wI} = 'Y/ (t,-a, )aj where t, is the input to .unit i and where a, and aj are the activation values of the system

after it has stabi lized in one of the corners of the hypercube.

Thermodynamic Models

Other more recent developments are the thermodynamic models. Two examples of such models are presented in the book. One, harmony theory, was developed by Paul Smolensky and is described in detai l in Chapter 6. The other, the Boltzmann machine, was developed by Hinton and Sejnowski and is described in Chapter 7. Here we describe the basic .idea behind these models and show how they relate to the general class of models under discussion. To begin, the thermodynamic models employ binary units which take on the values {O, 1) . The units are divided into two categories: the visible units corresponding to our input and output units and the hidden units. In general , any unit may connect to any other unit. However, there is a constraint that the connections must be symmetric. That is, the wi} = wji ' In these models, there is no distinction between the output of the unit and its activation value. The activation values are, however, a stochastic func-

tion of the inputs. That is,

where 'Y/, is the input from outside of system into unit i, 0, is the threshold for the unit, and T is a parameter, called temperature, which determines the slope of the probability function. Figure 6 shows how the probabilities vary with various values of T. It should be noted that as T approaches zero, the individual units become more and more like linear threshold units. In general , if the unit exceeds threshold by a great enough margin it wi ll always attain value 1. If it is far enough below threshold, it always takes on value O. Whenever the unit is above threshold, the probabi lity that it wi ll turn on is greater than 112.

apply so that Il Wjj = .,.,ajaj

. Note, since activations take on values of 0

and 1 this says that the weight is incremented by an amount .,., whenever unit ; and j are on, otherwise no change occurs. During the second phase of learning, the system is al lowed to respond for an equal period of time in a so-cal led free-running state in which no inputs are presented. Since the system is stochastic, it wi ll continue to respond even though no actual stimuli are presented. During this phase, a sim-

ple anti-Hebbian rule is employed, Il wi} = -.,.,ajQj

' The intuition is

roughly that the performance during the environmentally driven phase is determined by both the pattern of interconnections and by the environment. The performance during the free-running phase is determined only by the internal set of connections. To correctly reflect the environment, we should look at its performance due to the environment plus internal structure and then subtract out its performance due to internal structure alone. This is actually quite a powerful learning scheme. It can be shown that if a portion of the input units are turned on after the system has learned, it wi ll complete the remaining portion of the visible units with the probability that those units had been present in the stimulus patterns given the subpattern that had been

turned on. These issues are again addressed in Chapter 7.

Grossberg

Stephen Grossberg has been one of the major contributors to models of this class over the years. His work is complex and contains many important details which we cannot review here. We will instead describe some of the central aspects of his work and show how it relates to the general framework. Perhaps the clearest summary of Grossberg's work appears in Grossberg (1980) . Grossberg's units are allowed to take on any real activation value between a minimum and a maximum value. The output function is, in many of Grossberg's applications , a threshold function so that a given unit wi ll affect another unit only if its activation level is above its threshold. Moreover, Grossberg argues that the output function must be a sigmoid or S-shaped function of the activation value of the unit. Grossberg's activation rule is rather more complex than the others we have discussed thus far in that excitatory and inhibitory inputs don't simply sum, but appear separately in the activation rule. Grossberg has presented a number of possible

activation rules, but they typically have the form

aj (1+ 1) = aj (r ) O-A ) + (B-aj (r ) netej (t) - (aj (t }+C)neti) (t )

where A is the decay rate, B represents the maximal degree of excitation of the unit, and C is much smaller in magnitude than B and represents the maximal amount the unit can be inhibited below the resting value of O. Grossberg generally assumes that the inhibitory inputs come from a kind of recurrent inhibitory field in which the unit is embedded and the excitatory inputs come from the unit itself and

from another level of the system.

Grossberg has studied learning in these networks over a number of years and has studied several different learning schemes. The learning rule he has studied most, however, is similar to the one analyzed in

Chapter 5 and is given by

A wi) = "f)a; (OJ- wi) .

Grossberg has applied this and similar learning rules in a number of cases, but a review of these appl ications is beyond the scope of the

present discussion.

Interactive Activation Model

The interactive activation model of McClelland and Rumelhart (98 1) and Rumelhart and McClelland (982) had units which represented visual features, letters and words. Units could take on any value in the range [min,max] . The output function was a threshold function such that the output was 0 if the activation was below threshold and was equal to the difference of the activation value and the threshold if the activation was above threshold. The interactive activation model involves a connectivity pattern in which units are organized in layers , such that an element in a layer connects with excitatory connections with all elements in the layers above and below that are consistent with that unit, and connects negatively to all units in the layers above and below that are inconsistent with that unit. In addition, each unit inhibits all units in its own layer that are inconsistent with the unit in question. Thus, the interactive activation model is a kind of positive feedback systerr.. with maximum and minimum values for each unit, like the BSB model. The information coming into each unit is weighted (by the interconnection strengths) and summed algebraically to yield a "net input" to the unit. Let net) = r. Wi) a; be the net input to unit j. This net input is then combined with the previous activation Copyrighted Material

value to produce the new activation value according to the fol lowing

activation rule:

where e is the decay rate of the activation given no input. In other words, the new activation value is given by the old activation value properly decayed, plus (or minus) a factor that pushes toward the minimum or maximum value depending on the magnitude of the net input into the unit. This activation rule is similar to that employed by Grossberg, except in this formulation the excitation and inhibition are

algebraically combined.

The interactive activation model was designed as a model for a processing system and our goals were to show how we could account for specific aspects of word perception. Thus, there was no specific model of learning proposed to explain where the particular network we assumed came from. As we shall see, much of the work on learning reported in this book has been aimed at giving plausible accounts of how such a network might have been learned. (See especial ly Chapters

5 and 6.)

Feldman and Ballard

Feldman and Bal lard (982) have proposed a framework they cal <sup>l</sup> connectionist modeling. The units have continuous activation values, which they cal l potential which can take on any value in the range [- 10,10] . Their output function is a kind of threshold functIOn which is allowed to take on a smal l number of discrete integer values (O� OJ � 9) . They have proposed a number of other unit types each with a somewhat different activation rule. Their simplest unit type is what they call the P-unit . In this case the activation rule is given by

aj (t+ 1) = aj (t ) + (3 netj (r ).

Once the activation reaches its maximum or minimum value i t i s simply pinned to that value. Decay is implemented by self inhibition. Feldman and Bal lard also have a cOrUunctive unit similar to our sigma-pi units described below. Feldman 098I) has also considered learning. In general , the approach to learning offers more machinery than is avai lable within our current framework. In practice, however, the learning rules actually examined are of the same class we have already

SIGMA-PI UNITS

Before completing our section on a general framework , it should be mentioned that we have sometimes found it useful to postulate units that are more complex than those described up to this point in this chapter. In our descriptions thus far, we have assumed a simple additive unit in which the net input to the unit is given by L wij a/ . This is certainly the most common form in most of our models. Sometimes, however, we want multiplicative connections in which the output val ues of two (or possibly more) units are mUltiplied before entering into the sum. Such a multiplicative connection allows one unit to gate another. Thus, if one unit of a multipl icative pair is zero, the other member of the pair can have no effect, no matter how strong its output. On the other hand, if one unit of a pair has value 1, the output of the other is passed unchanged to the receiving unit. Figure 7 illustrates several such connections. In this case, the input to unit A is the weighted sum of the products of units B and C and units D and E. The pairs, BC and DE are called co,yuncts. In this case we have conjuncts of size 2. In general , of course , the conjuncts could be of any size. We have no appl ications, however, which have required conjuncts larger than size 2. In general , then, we assume that the net input to a unit is given by the weighted sum of the products of a set of individual inputs. That is, the

net input to a unit is given by L wijIIai

\

ai2 •.. aik where i indexes the

conjuncts impinging on unit j and Ui \

' Ui2 ••••• Uik are the k units in

the conjunct. We call units such as these sigma-pi units.

In addition to their use as gates, sigma-pi units can be used to convert the output level of a unit into a signal that acts like a weight connecting two units. Thus, assume we have the pattern of connections illustrated in the figure. Assume further that the weights on those connections are all 1. In this case, we can use the output levels of units B and D to, in effect, set the weights from C to A and E to A respectively. Since, in general , it is the weights among the units that determine the behavior of the network, sigma-pi units allow for a dynamically programmable network in which the activation value of some units

determine what another network can do.

In addition to its general usefulness in these cases, one might ask whether we might not sometime need still more complex patterns of interconnections. Interestingly, as described in Chapter 10, we wi ll never be forced to develop any more complex interconnection type, since sigma-pi units are sufficient to mimic any function monotonic of

Sigma Pi Units

Con ) - QB . QC Con 2- aD ' aE

FIGURE 7. Two conjunctive inputs to unit A from the conjunct Band C and D and E. The input to unit A is the sum of the product of the outputs of units BC and DE.

CONCLUSION

We have provided a very general mathematical and conceptual framework within which we develop our models. This framework provides a language for expressing PDP models, and, though there is a lot of freedom within it, it is at least as constrained as most computational formalisms , such as production systems or high-level languages such as

Lisp.

We must take note of the fact, however, that the framework does not specify a/l of the constraints we have imposed on ourselves in our model building efforts. For example, virtually any computing device, serial or parallel, can be described in the framework we have described

There is a further set of considerations which has guided our particular formulations. These further considerations arise from two sources: our beliefs about the nature of the hardware avai lable for carrying out mental processes in the brain and our beliefs about the essential character of these mental processes themselves. We discuss below the additional constraints on our model bui lding which arise from these two

beliefs .

.

First, the operations in our models can be characterized as .. neurally inspired." We wish to replace the " computer metaphor" as a model of mind with the "brain metaphor" as model of mind. This leads us to a number of considerations which further inform and constrain our model building efforts. Perhaps the most crucial of these is time. Neurons are remarkably slow relative to components in modern computers. Neurons operate in the time scale of mi l l iseconds whereas computer components operate in the time scale of nanoseconds-a factor of 106 faster. This means that human processes that take on the order of a second or less can involve only a hundred or so time steps. Since most of the processes we have studied- perception, memory retrieval , speech processing , sentence comprehension, and the l ike - take about a second or so, it makes sense to impose what Feldman ( 985) cal <sup>l</sup>s the " 1 00 step program " constraint. That is, we seek explanations for these mental phenomena which do not require more than about a hundred elementary sequential operations. Given that the processes we seek to characterize are often quite complex and may involve consideration of large numbers of simultaneous constraints, our algorithms must involve considerable paral lelism. Thus, although a serial computer could be created out of the kinds of components represented by our units, such an implementation would surely violate the 100-step program constraint

for any but the simplest processes .

A second consideration differentiates our models from those inspired by the computer metaphor: that is, the constraint that all the knowledge is in the connections. From conventional programmable computers we are used to thinking of knowledge as being stored in the state of certain units in the system. In our systems we assume that only very short term storage can occur in the states of units; long term storage takes place in the connections among units. Indeed, it is the connections -or perhaps the rules for forming them through experience-which primari ly differentiate one model from another. This is a profound difference between our app<sup>r</sup>oach and other more conventional approaches, for it means that almost al l knowledge is implicit in the structure of the device that carries out the task rather than explicit in the states of units themselves. Knowledge is not directly accessible to interpretation by some separate processor, but it is bui lt into the processor itself and d rectly determines the course of Copyrighted Material

processing. It is acquired through tuning of connections as these are used in processing, rather than formulated and stored as declarative

facts.

In addition to these two neurally inspired working assumptions, there are a number of other constraints that derive rather directly from our understanding of the nature of neural information processing. These

assumptions are discussed more fully in Chapter 4.

The second class of constraints arises from our beliefs about the nature of human information processing considered at a more abstract, computational level of analysis. We see the kinds of phenomena we have been studying as products of a kind of constraint satisfaction procedure in which a very large number of constraints act simultaneously to produce the behavior. Thus , we see most behavior not as the product of a single, separate component of the cognitive system, but as the product of large set of interacting components, each mutually constraining the others and contributing in its own way to the global ly observable behavior of the system. It is very difficult to use serial algorithms to implement such a conception, but very natural to use highly paral lel ones . These problems can often be characterized as best match or optimization problems. As Minsky and Papert (I969) have pointed out, it is very difficult to solve best match problems serially. However, this is precisely the kind of problem that is readily implemented using highly parallel algorithms of the kind we consider in this book. See Kanerva (I 984) for a discussion of the best match problem and its

solution with parallel processing systems.

To summarize, the PDP framework consists not only of a formal language, but a perspective on our models. Other qual itative and quantitative considerations arising from our understanding of brain processing and of human behavior combine with the formal system to form what might be viewed as an aesthetic for our model building enterprises. The remainder of our book is largely a study of this aesthetic in

practice.

ACKNOWLEDGMENTS

This research was supported by Contract N0001 4-79-C-03 23 , NR 667-437 with the Personnel and Training Research Programs of the Office of Naval Research, by grants from the System Development Foundation, and by a NIMH Career Development Award (MHOO385)

to the second author.

CHAPTER 3

Distributed Representations

G. E. HINTON, 1. L. McCLELLAND, and D. E. RUMELHART

Given a network of simple computing elements and some entities to be represented, the most straightforward scheme is to use one computing element for each entity. This is cal led a local representation. It is easy to understand and easy to implement because the structure of the physical network mirrors the structure of the knowledge it contains. The naturalness and simplicity of this relationship between the knowledge and the hardware that implements it have led many people to simply assume that local representations are the best way to use parallel hardware. There are, of course, a wide variety of more compl icated implementations in which there is no one-to-one correspondence between concepts and hardware units, but these implementations are only worth considering if they lead to increased efficiency or to interesting emergent properties that cannot be conveniently achieved

using local representations.

This chapter describes one type of representation that is less familiar and harder to think about than local representations. Each entity is represented by a pattern of activity distributed over many computing elements, and each computing element is involved in representing many different entities. The strength of this more complicated kind of representation does not lie in its notational convenience or its ease of implementation in a conventional computer, but rather in the efficiency with which it makes use of the processing abi lities of networks of sim-

Every representational scheme has its good and bad points. Distributed representations are no exception. Some desirable properties arise very naturally from the use of patterns of activity as representations. Other properties, like the ability to temporarily store a large set of arbitrary associations, are much harder to achieve. As we shall see, the best psychological evidence for distributed representations is the degree to which their strengths and weaknesses match those of the human

mind.

The first section of this chapter stresses some of the virtues of distributed representations. The second section considers the efficiency of distributed representations, and shows clearly why distributed representations can be better than local ones for certain classes of problems. A final section discusses some difficult issues which are often avoided by advocates of distributed representations, such as the representation of constituent structure and the sequential focusing of

processing effort on different aspects of a structured object.

Disclaimers. Before examining the detailed arguments in favor of distributed representations, it is important to be clear about their status within an overall theory of human information processing. It would be wrong to view distributed representations as an alternative to representational schemes like semantic networks or production systems that have been found useful in cognitive psychology and artificial intelligence. It is more fruitful to view them as one way of implementing these more abstract schemes in parallel networks, but with one proviso: Distributed representations give rise to some powerful and unexpected emergent properties. These properties can therefore be taken as primitives when working in a more abstract formalism. For example, distributed representations are good for content-addressable memory, automatic generalization, and the selection of the rule that best fits the current situation. So if one assumes that more abstract models are implemented in the brain using distributed representations, it is not unreasonable to treat abilities like content-addressable memory, automatic generalization, or the selection of an appropriate rule as primitive operations, even though there is no easy way to implement these operations in conventional computers. Some of the emergent properties of distributed representations are not easily captured in higher-level formalisms. For example, distributed representations are consistent with the simultaneous application of a large number of partially fitting rules to the current situation, each rule being applied to the degree that it is relevant. We shall examine these properties of distributed representations in the chapter on schemata (Chapter 14). There we will see clearly that schemata and other higher-level constructs provide only approximate characterizations of mechanisms which rely on dist ributed Copyrighted Material

representations. Thus, the contribution that an analysis of distributed representations can make to these nigher-level formalisms is to legitimize certain powerful, primitive operations which would otherwise appear to be an appeal to magic; to enrich our repertoire of primitive operations beyond those which can conveniently be captured in many higher-level formalisms; and to suggest that these higher-level formalisms may only capture the coarse features of the computational capabi l i-

ties of the underlying processing mechanisms.

Another common source of co<sup>n</sup>fusion is the idea that distributed representations are somehow in conflict with the extensive evidence for localization of function in the brain (Luria, 1973). A system that uses distributed representations still requires many different modul('s for representing completely different kinds of thing at the same time. The distributed representations occur within these localized modules. For example, different modules would be devoted to things as different as mental images and sentence structures, but two different mental images would correspond to alternative patterns of activity in the same module. The representations advocated here are local at a global scale but global

at a local scale.

VIRTUES OF DISTRIBUTED REPRESENTATIONS

This section considers three important features of distributed representations: (a) their essentially constructive character; (b) their ability to generalize automatically to novel situations; and (c) their tunability to changing environments. Several of these virtues are shared by certain local models, such as the interactive activation model of word perception, or McClelland's (1980 model of generalization and

retrieval described in Chapter 1.

Memory as Inference

People have a very flexible way of accessing their memories: They can recall items from partial descriptions of their contents (Norman & Bobrow, 1979). Moreover, they can do this even if some parts of the partial description are wrong. Many people, for example, can rapidly retrieve the item that satisfies the following partial description: It is an actor, it is intelligent, it is a politician. This kind of content-addressable memory is very useful and it .is very hard. to implement on a conven-

tional computer becaus'i

°eKf\$!!£efSM�o�J each item at a particular

address, and to retrieve an item they must know its address. If all the combinations of descriptors that will be used for access are free of errors and are known in advance, it is possible to use a method called hash coding that quickly yields the address of an item when given part of its content. In general , however, content-addressable memory requires a massive search for the item that best fits the partial description. The central computational problem in memory is how to make this search efficient. When the cues can contain errors, this is very difficult because the failure to fit one of the cues cannot be used as a fi lter

for quickly eliminating inappropriate answers.

Distributed representations provide an efficient way of using parallel hardware to implement best-fit searches. The basic idea is fairly simple, though it is quite unlike a conventional computer memory. Different items correspond to different patterns of activity over the very same group of hardware units. A partial description is presented in the form of a partial activity pattern, activating some of the hardware units. 1 Interactions between the units then al low the set of active units to influence others of the units, thereby completing the patteft1, and generating the item that best fits the description. A new item is "stored" by modifying the interactions between the hardware units so as to create a new stable pattern of activity. The main difference from a conventional computer memory is that patterns which are not active do not exist anywhere. They can be re-created because the connection strengths between units have been changed appropriately, but each connection strength is involved in storing many patterns, so it is impossible to point to a particular place where the memory for a particular item is

stored.

Many people are surprised when they understand that the connections between a set of simple processing units are capable of supporting a large number of different patterns. Illustrations of this aspect of distributed models are provided in a number of papers in the literature (e.g., Anderson, 1977; Hinton, 1981a) ; this property is illustrated in the model of memory and amnesia described in Chapters 17 and 25. One way of thinking about distributed memories is in terms of a very large set of plausible inference rules. Each active unit represents a "microfeature" of an item, and the connection strengths stand for plausible "microinferences" between microfeatures. Any particular pattern

<sup>I</sup>This is easy if the partial description is simply a set of features, but it is much more difficult if the partial description mentions relationships to other objects. If, for example, the system is asked to retrieve John's father, it must represent John, but if John and his father are represented by mutually exclusive patterns of activity in the very same group of units , it is hard to see how this can be done without preventing the representation of

John's father. A distributed solution to this problem is described in the text.

of activity of the units will satisfy some of the microinferences and violate others. A stable pattern of activity is one that violates the plausible microinferences less than any of the neighboring patterns. A new stable pattern can be created by changing the inference rules so that the new pattern violates them less tha<sup>n</sup>its neighbors. This view of memory makes it clear that there is no sharp distinction between genuine memory and plausible reconstruction. A genuine memory is a pattern that is stable because the inference rules were modified when it occurred before. A "confabulation" is a pattern that is stable because of the way the inference rules have been modified to store several different previous patterns. So far as the subject is concerned, this may

be indistinguishable from the real thing.

The blurring of the distinction between veridical recall and confabulation or plausible reconstruction seems to be characteristic of human memory (Bartlett, 1932� Neisser, 1981). The reconstructive nature of human memory is surprising only because it conflicts with the standard metaphors we use. We tend to think that a memory system should work by storing literal copies of items and then retrieving the stored copy, as in a filing cabinet or a typical computer database. Such sys-

tems are not naturally reconstructive.

If we view memory as a process that constructs a pattern of activity which represents the most plausible item that is consistent with the given cues, we need some guarantee that it will converge on the representation of the item that best fits the description, though it might be tolerable to sometimes get a good but not optimal fit. It is easy to imagine this happening, but it is harder to make it actually work. One recent approach to this problem is to use statistical mechanics to analyze the behavior of groups of interacting stochastic units. The analysis guarantees that the better an item fits the description, the more likely it is to be produced as the solution. This approach is described in Chapter 7, and a related approach is described in Chapter 6. An alternative approach, using units with continuous activations (Hopfield,

1984) is described in Chapter 14.

Similarity and Generalization

When a new item is stored, the modifications in the connection strengths must not wipe out existing items. This can be achieved by modifying a very large number of weights very slightly. If the modifications are all in the directio<sup>n</sup>that helps the pattern that is being stored, there will be a conspiracy effect: The total help for the intended pattern will be��t@f1�aNJ;;§Jl1all separate modifications.

For unrelated patterns, however, there will be very little transfer of effect because some of the modifications will help and some will hinder. Instead of all the small modifications conspiring together, they will mainly cancel out. This kind of statistical reasoning underpins most distributed memory models, but there are many variations of the basic idea (See Hinton & Anderson, 1981, for several examples).

It is possible to prevent interference altogether by using orthogonal patterns of activity for the various items to be stored (a rudimentary example of such a case is given in Chapter O. However, this eliminates one of the most interesting properties of distributed representations: They automatically give rise to generalizations. If the task is simply to remember accurately a set of unrelated items, the generalization effects are harmful and are called interference. But generalization is normally a helpful phenomenon. It allows us to deal effectively with situations that are similar but not identical to previously experienced

situations.

People are good at generalizing newly acquired knowledge. If you learn a new fact about an object, your expectations about other similar objects tend to change. If, for example, you learn that chimpanzees like onions you will probably raise your estimate of the probability that gorillas like onions. In a network that uses distributed representations, this kind of generalization is automatic. The new knowledge about chimpanzees is incorporated by modifying some of the connection strengths so as to alter the causal effects of the distributed pattern of activity that represents chimpanzees. 2 The modifications automatically change the causal effects of all similar activity patterns. So if the representation of gorillas is a similar activity pattern over the same set

of units, its causal effects will be changed in a similar way.

The very simplest distributed scheme would represent the concept of onion and the concept of chimpanzee by alternative activity patterns over the very same set of units. It would then be hard to represent chimps and onions at the same time. This problem can be solved by using separate modules for each possible role of an item within a larger structure. Chimps, for example, are the" agent" of the liking and so a pattern representing chimps occupies the" agent" module and the pattern representing onions occupies the "patient" module (see Figure I).

<sup>2</sup>The internal structure of this pattern may also change. There is always a choice between changing the weights on the outgoing connections and changing the pattern itself so that different outgoing connections become relevant. Changes in the pattern itself alter its similarity to other patterns and thereby alter how generalization will occur in the future. It is generally much harder to figure out how to change the pattern that represents an item than it is to figure out how to change the outgoing connections so that a particu-

Each module can have alternative patterns for all the various items, so this scheme does not involve local representations of items. What is

localized is the role.

If you subsequently learn that gibbons and orangutans do not like onions your estimate of the probability that gorillas like onions will fall, though it may still remain higher than it was initially. Obviously, the combination of facts suggests that liking onions is a peculiar quirk of chimpanzees. A system that uses distributed representations will automatically arrive at this conclusion, provided that the alternative patterns that represent the various apes are related to one another in a particular way that is somewhat more specific than just being similar to one another: There needs to be a part of each complete pattern that is identical for all the various apes. In other words, the group of units used for the distributed representations must be divided into two

RELATIONSHIP

AGENT PATIENT

FIGURE I. In this Simplified scheme there are two different modules, one of which represents the agent and the other the patient. To incorporate the fact that chimpanzees like onions, the pattern for chimpanzees in one module must be associated with the pattern for onions in the other module. Relationships other than "liking" can be implemented by having a third group of units whose pattern of activity represents the relationship. This pattern must then "gate" the interactions between the agent and patient groups. Hinton (1981a) describes one way of doing this gating by using a fourth group of

subgroups, and all the various apes must be represented by the same pattern in the first subgroup, but by different patterns in the second subgroup. The pattern of activity over the first subgroup represents the type of the item, and the pattern over the second subgroup represents additional microfeatures that discriminate each instance of the type from the other instances. Note that any subset of the microfeatures can be considered to define a type. One subset might be common to all apes, and a different (but overlapping) subset might be common to all pets. This allows an item to be an instance of many different types

simultaneously.

When the system learns a new fact about chimpanzees, it usually has no way of knowing whether the fact is true of all apes or is just a property of chimpanzees. The obvious strategy is therefore to modify the strengths of the connections emanating from all the active units, so that the new knowledge will be partly a property of apes in general and partly a property of whatever features distinguish chimps from other apes. If it is subsequently learned that other apes do not like onions, correcting modifications will be made so that the information about onions is no longer associated with the subpattern that is common to all apes. The knowledge about onions will then be restricted to the subpattern that distinguishes chimps from other apes. If it had turned out that gibbons and orangutans also liked onions, the modifications in the weights emanating from the subpattern representing apes would have reinforced one another, and the knowledge would have become associated with the subpattern shared by all apes rather than with the patterns

that distinguish one ape from another.

A very simple version of this theory of generalization has been implemented in a computer simulation (Hinton, 1981a). Several applications that make use of this property can be found in Part IV of this

book.

There is an obvious generalization of the idea that the representation of an item is composed of two parts, one that represents the type and another that represents the way in which this particular instance differs from others of the same type. Almost all types are themselves instances of more general types, and this can be implemented by dividing the pattern that represents the type into two subpatterns, one for the more general type of which this type is an instance, and the other for the features that discriminate this particular type from others instances of the same general type. Thus the relation between a type and an instance can be implemented by the relationship between a set of units and a larger set that includes ·it. Notice that the more general the type, the smaller the set of units used to encode it. As the number of terms in an intensional description gets smaller, the corresponding

In traditional semantic networks that use local representations, generalization is not a direct consequence of the representation. Given that chimpanzees like onions, the obvious way of incorporating the new knowledge is by changing the strengths of connections belonging to the chimpanzee unit. But this does not automatically change connections that belong to the gorilla unit. So extra processes must be invoked to implement generalization in a localist scheme. One commonly used method is to allow activation to spread from a local unit to other units that represent similar concepts (Collins & Loftus, 1975; Quillian, 1968). Then when one concept unit is activated, it will partially activate its neighbors and so any knowledge stored in the connections emanating from these neighbors will be partially effective. There are many variations of this basic idea (Fahlman, 1979; Levin, 1976;

McClelland, 1981).

It is hard to make a clean distinction between systems that use local representations plus spreading activation and systems that use distributed representations. In both cases the result of activating a concept is that many different hardware units are active. The distinction almost completely disappears in some models such as McClelland's (981) generalization model, where the properties of a concept are represented by a pattern of activation over feature units and where this pattern of activation is determined by the interactions of a potentially very large number of units for instances of the concept. The main difference is that in one case there is a particular individual hardware unit that acts as a "handle" which makes it easy to attach purely conventional properties like the name of the concept and easier for the theorist who constructed the network to know what the individual parts of the network

stand for.

If we construct our networks by hand-specifying the connections between the units in the network, a local representation scheme has some apparent advantages. First, it is easier to think one understands the behavior of a network if one has put in all the "knowledge" -all the connections-oneself. But if it is the entire, distributed pattern of interacting influences among the units in the network that is doing the work, this understanding can often be illusory. Second, it seems intuitively obvious that it is harder to attach an arbitrary name to a distributed pattern than it is to attach it to a single unit. What is intuitively harder, however, may not be more efficient. We will see that one can actually implement aribitrary associations with fewer units using distributed representations. Before we turn to such considerations, however, we examine a different advantage of distributed representations: They make it possible to create new concepts without allocating new

Creating New Concepts

Any plausible scheme for representing knowledge must be capable of learning novel concepts that could not be anticipated at the time the network was initially wired up. A scheme that uses local representations must first make a discrete decision about when to form a new concept, and then it must find a spare hardware unit that has suitable connections for implementing the concept involved. Finding such a unit may be difficult if we assume that, after a period of early development, new knowledge is incorporated by changing the strengths of the existing connections rather than by growing new ones. If each unit only has connections to a small fraction of the others, there will probably not be any units that are connected to just the right other ones to implement a new concept. For example, in a collection of a million units each connected at random to ten thousand others, the chance of there being any unit that is connected to a particular set of 6 others is only one in a

million.

In an attempt to rescue local representations from this problem, several clever schemes have been proposed that use two classes of units. The units that correspond to concepts are not directly connected to one another. Instead, the connections are implemented by indirect pathways through several layers of intermediate units (Fahlman, 1980; Feldman, 1982). This scheme works because the number of potential pathways through the intermediate layers far exceeds the total number of physical connections. If there are k layers of units, each of which has a fan-out of n connections to randomly selected units in the follow-

ing layer, there are n

k potential pathways. There is almost certain to be

a pathway connecting any two concept-units, and so the intermediate units along this pathway can be dedicated to connecting those two concept-units. However, these schemes end up having to dedicate several intermediate units to each effective connection, and once the dedication has occurred, all but one of the actual connections emanating from each intermediate unit are wasted. The use of several intermediate units to create a single effective connection may be appropriate in switching networks containing elements that have units with relatively small fan-out, but it seems to be an inefficient way of using the

hardware of the brain.

The problems of finding a unit to stand for a new concept and wiring it up appropriately do not arise if we use distributed representations. All we need to do is modify the interactions between units so as to create a new stable pattern of activity. If this is done by modifying a large number of connections very slightly, the creation of a new pattern need not disrupt the existing representations. The difficult problem is Copyrighted Material

to choose an appropriate pattern for the new concept. The effects of the new representation on representations in other parts of the system wi ll be determined by the units that are active, and so it is important to use a collection of active units that have roughly the correct effects. Fine-tuning of the effects of the new pattern can be achieved by slightly altering the effects of the active units it contains, but it would be unwise to choose a random pattern for a new concept because major changes would then be needed in the weights, and this would disrupt other knowledge. Ideally, the distributed representation that is chosen for a new concept should be the one that requires the least modification of weights to make the new pattern stable and to make it have the

required effects on other representations.

Naturally, it is not necessary to create a new stable pattern all in one step. It is possible for the pattern to emerge as a result of modifications on many separate occasions. This alleviates an awkward problem that arises with local representations: The system must make a discrete allor-none decision about when to create a new concept. If we view concepts as stable patterns, they are much less discrete in character. It is possible, for example, to differentiate one stable pattern into two closely related but different variants by modifying some of the weights slightly. Unless we are allowed to clone the hardware units (and all their connections) , this kind of gradual, conceptual differentiation is

much harder to achieve with local representations.

One of the central problems in the development of the theory of distributed representation is the problem of specifying the exact procedures by which distributed representations are to be learned. All such procedures involve connection strength modulation, following "learning rules" of the type outlined in Chapter 2. Not all the problems have been solved, but significant progress is being made on these prob-

lems. (See the chapters in Part II.)

DISTRIBUTED REPRESENTATIONS THAT

WORK EFFICIENTLY

In this section, we consider some of the technical details about the implementation of distributed representations. First, we point out that certain distributed representation schemes can fail to provide a sufficient basis for differentiating different concepts, and we point out what is required to avoid this limitation. Then, we describe a way of using distributed representations to get the most information possible out of a simple network of connected units. The central result is a surprising one' If you want to eru:.ode features\_ a�curately using as few units as . (;opyngntea Matenal

possible, it pays to use units that are very coarsely tuned, so that each feature activates many different units and each unit is activated by many different features. A specific feature is then encoded by a pattern of activity in many units rather than by a single active unit, so coarse

coding is a form of distributed representation.

To keep the analysis simple, we shall assume that the units have only two values, on and off.3 We shall also ignore the dynamics of the system because the question of interest, for the time being, is how many units it takes to encode features with a given accuracy. We start by considering the kind of feature that can be completely specified by giving a type (e.g. , line-segment, corner, dot) and the values of some continuous parameters that distinguish it from other features of the same type (e. g., position, orientation, size.) For each type of feature there is a space of possible instances. Each continuous parameter defines a dimension of the feature space, and each particular feature corresponds to a point in the space. For features like dots in a plane, the space of possible features is two-dimensional . For features like stopped, oriented edge-segments in three-dimensional space, the feature space is si x-dimensional . We shall start by considering twodimensional feature spaces and then generalize to higher dimensionali-

ties.

Suppose that we wish to represent the position of a single dot in a plane, and we wish to achieve high accuracy without using too many units. We define the accuracy of an encoding scheme to be the number of different encodings that are generated as the dot is moved a standard distance through the space. One encoding scheme would be to divide the units into an X group and a Y group, and dedicate each unit to encoding a particular X or Y interval as shown in Figure 2. A given dot would then be encoded by activity in two units, one from each group, and the accuracy would be proportional to the number of units used. Unfortunately , there are two problems with this. First, if two dots have to be encoded at the same time, the method breaks down. The two dots will activate two units in each group, and there will be no way of telling, from the active units, whether the dots were at (x 1, y 1) and

(x 2, y 2) or at (x 1, Y 2) and (x 2

, y 1). This is called the binding prob-

lem. It arises because the representation does not specify what goes

with what .

<sup>3</sup>Similar arguments apply with multi valued activity levels, but it is important not to allow activity levels to have arbitrary precision because this makes it possible to represent an infinite amount of information in a single activity level. Units that transmit a discrete impulse with a probability that varies as a function of their activation seem to approximate the kind of precision that is possible in neural circuitry (see Chapters 20 and 21). Copyrighted Material

Y group

Y group

o o

o

•

o

1 •

X group

X group

FIGURE 2. A: A simple way of using two groups of binary units to encode the position of a point in a two-dimensional space. The active units in the X and Y groups represent the x- and y-coordinates. B: When two points must be encoded at the same time, it is

impossible to tell which x-coordinate goes with which y-coordinate.

The second problem arises even if we allow only one point to be represented at a time. Suppose we want certain representations to be associated with an overt response, but not others: We want (x I, y 1) and (x 2, y 2) to be associated with a response, but not (x I, y 2) or (x 2, y 1). We cannot implement this association using standard weighted connections to response units from units standing for the values on the two dimensions separately. For the unit for x 1 and the

unit for x 2 would both have o activ.a.te

y 1 and the unit for y 2 would both have to activate the response. There would be no way of preventing the response from being activated when the unit for x 1 and the unit for y 2 were both activated. This is another aspect of the binding problem since, again, the representation fails to

specify what must go with what.

In a conventional computer it is easy to solve the binding problem. We simply create two records in the computer memory. Each record contains a pair of coordinates that go together as coordinates of one dot, and the binding information is encoded by the fact that the two coordinate values are sitting in the same record (which usually means they are sitting in neighboring memory locations). In parallel networks

it is much harder to solve the binding problem.

Conjunctive Encoding

One approach is to set aside, in advance, one unit for each possible combination of X and Y values. This amounts to covering the plane with a large number of small, nonoverlapping zones and dedicating a unit to each zone. A dot is then represented by activity in a single unit so this is a local representation. The use of one unit for each discriminable feature solves the binding problem by having units which stand for the conjunction of values on each of two dimensions. In general, to permit an arbitrary association between particular combinations of features and some output or other pattern of activation, some conjunc-

tive representation may be required.

However, this kind of local encoding is very expensive. It is much less efficient than the previous scheme because the accuracy of pinpointing a point in the plane is only proportional to the square root of the number of units. In general, for a k -dimensional feature space, the local encoding yields an accuracy proportional to the kth root of the number of units. Achieving high accuracy without running into the

binding problem is thus very expensive.

The use of one unit for each discriminable feature may be a reasonable encoding if a very large number of features are presented on each occasion, so that a large fraction of the units are active. However, it is a very inefficient encoding if only a very small fraction of the possible features are presented at once. The average amount of information conveyed by the state of a binary unit is 1 bit if the unit is active half the time, and it is much less if the unit is only rarely active.4 It would

therefore be more efficient to use an encoding in which a larger fraction of the units were active at any moment. This can be done if we abandon the idea that each discriminable feature is represented by

activity in a single unit.

Coarse Coding

Suppose we divide the space into larger, overlapping zones and assign a unit to each zone. For simplicity, we will assume that the zones are circular, that their centers have a uniform random distribution throughout the space, and that all the zones used by a given encoding scheme have the same radius. The question of interest is how accurately a feature is encoded as a function of the radius of the zones. If we have a given number of units at our disposal is it better to use large zones so that each feature point falls in many zones, or is it better to use small zones so that each feature is represented by activity in

fewer but more finely tuned units?

The accuracy is proportional to the number of different encodings that are generated as we move a feature point along a straight line from one side of the space to the other. Every time the line crosses the boundary of a zone, the encoding of the feature point changes because the activity of the unit corresponding to that zone changes. So the number of discriminable features along the line is just twice the number of zones that the line penetrates. 5 The line penetrates every zone whose center lies within one radius of the line (see Figure 3). This number is proportional to the radius of the zones, r, and it is also proportional to their number, n. Hence the accuracy, a, is related to

the number of zones and to their radius as follows:

aa: nr.

In general, for a k-dimensional space, the number of zones whose centers lie within one radius of a line through the space is proportional to the volume of a k-dimensional hypercylinder of radius r. This volume is equal to the length of the cylinder (which is fixed) times its (k - 1) -dimensional cross-sectional area which is proportional to <sup>r</sup>

k - I.

<sup>5</sup>Problems arise if you enter and leave a zone without crossing other zone borders in between because you revert to the same encoding as before, but this effect is negligible if the zones are dense enough for there to be many zones containing each point in the

space.

FIGURE 3. The number of zone boundaries that are cut by the line is proportional to

the number of zone centers within one-zone radius of the line.

Hence, the accuracy is given by

So, for example, doubling the radius of the zones increases by a factor of 32, the linear accuracy with which a six-dimensional feature like a stopped oriented three-dimensional edge is represented. The intuitive idea that larger zones lead to sloppier representations is entirely wrong because distributed representations hold information much more efficiently than local ones. Even though each active unit is less specific in its meaning, the combination of active units is far more specific. Notice also that with coarse coding the accuracy is proportional to the number of units, which is much better than being proportional to the

kth root of the number.

Units that respond to complex features in retinotopic maps in visual cortex often have fairly large receptive fields. This is often interpreted as the first step on the way to a translation invariant representation. However, it may be that the function of the large fields is not to achieve translation invariance but to pinpoint accurately where the

feature is!

Limitations on coarse coding. So far, only the advantages of coarse coding have been mentioned, and its problematic aspects have been ignored. There are a number of limitations that cause the coarse coding strategy to break de'tYIJyrYf}Hl�dt�iew5FPtive fields" become too

large. One obvious limitation occurs when the fields become comparable in size to the whole space. This limitation is generally of little interest because other, more severe, problems arise before the recep-

tive fields become this large.

Coarse coding is only effective when the features that must be represented are relatively sparse. If many feature points are crowded together, each receptive field will contain many features and the activity pattern in the coarse-coded units will not discriminate between many alternative combinations of feature points. (If the units are allowed to have integer activity levels that reflect the number of feature points faIling within their fields, a few nearby points can be tolerated, but not many.) Thus there is a resolution/accuracy trade-off. Coarse coding can give high accuracy for the parameters of features provided that features are widely spaced so that high resolution is not also required. As a rough rule of thumb, the diameter of the receptive fields should be of the same order as the spacing between simultaneously present

feature points.6

The fact that coarse coding only works if the features are sparse should be unsurprising given that its advantage over a local encoding is that it uses the information capacity of the units more efficiently by making each unit active more often. If the features are so dense that the units would be active for about half the time using a local encoding,

coarse coding can only make things worse.

A second major limitation on the use of coarse coding stems from the fact that the representation of a feature must be used to affect other representations. There is no point using coarse coding if the features have to be recoded as activity in finely tuned units before they can have the appropriate effects on other representations. If we assume that the effect of a distributed representation is the sum of the effects of the individual active units that constitute the representation, there is a strong limitation on the circumstances under which coarse coding can be used effectively. Nearby features will be encoded by similar sets of active units, and so they will inevitably tend to have similar effects. Broadly speaking, coarse coding is only useful if the required effect of a feature is the average of the required effects of its neighbors. At a fine enough scale this is nearly always true for spatial tasks. The scale at which it breaks down determines an upper limit on the size of the

receptive fields.

6 It is interesting that many of the geometric visual illusions illustrate interactions between features at a distance much greater than the uncertainty in the subjects' knowledge of the position of a feature. This is just what would be expected if coarse cod-

Another limitation is that whenever coarse-coded representations interact, there is a tendency for the coarseness to increase. To counteract this tendency, it is probably necessary to have lateral inhibition operating within each representation. This issue requires further

research.

Extension to noncontinuous spaces. The principle underlying coarse coding can be generalized to noncontinuous spaces by thinking of a set of items as the equivalent of a receptive field. A local representation uses one unit for each possible item. A distributed representation uses a unit for a set of items, and it implicitly encodes a particular item as

the intersection of the sets that correspond to the active units.

In the domain of spatial features there is generally a very strong regularity: Sets of features with similar parameter values need to have similar effects on other representations. Coarse coding is efficient because it allows this regularity to be expressed in the connection strengths. In other domains, the regularities are different, but the efficiency arguments are the same: It is better to devote a unit to a set of items than to a single item, provided that the set is chosen in such a way that membership in the set implies something about membership in other sets. This implication can then be captured as a connection strength. Ideally, a set should be chosen so that membership of this set has strong implications for memberships of other sets that are also

encoded by individual units.

We illustrate these points with a very simple example. Consider a microlanguage consisting of the three-letter words of English made up of w or I, followed by i or e, followed by g or r. The strings wig and leg are words, but weg, fig, and all strings ending in r are not. Suppose we wanted to use a distributed representation scheme as a basis for representing the words, and we wanted to be able to use the distributed pattern as a basis for deciding whether the string is a word or a nonword. For simplicity we wi ll have a single " decision " unit. The problem is to find connections from the units representing the word to the decision unit such that it fires whenever a word is present but does not

fire when no word is present. 7

7 Note that the problem remains the same if the decision unit is replaced by a set of units and the task of the network is to produce a different pattern for the word and nonword decisions. For when we examine each unit, it either takes the same or a different value in the two patterns; in the cases where the value is the same, there is no problem, but neither do such units differentiate the two patterns. When the values are different,

the unit behaves just like the single decision unit discussed in the text.

Figure 4 shows three representation schemes: a distributed scheme that does not work, a distributed scheme that does work, and a local scheme. In the first scheme, each letter/positi on combination is represented by a di fferent uni t. Since there are only five letter/ position possibilities, only five units have connections to the output unit. Each word and nonword produces a different and unique pattern over these five units, but the connections from the five units to the decision unit cannot be set in such a way as to make the decision uni t fire whenever one of the words is present and fail to fire whenever one of the non-

words is present.

The reason for the problem is simply that the connections between the letter/ position units and the decision units can only capture the degree to which each letter indicates whether the string is a word or not. The g tends to indicate that a word is present, whereas the r indicates that the item is not a word; but each of the other letters , taken

individually, has absolutely no predictive ability in this case.

Whether a letter string is a word or not cannot be determined conclusively from the individual letters it contains; it is necessary to consider also what combinations of letters it contains. Thus, we need a representation that captures what combinations of letters are present in a way that is sufficient for the purposes of the network. One could capture this by using local representations and assigning one node to each word, as in the third panel of Figure 4. However, it is important to see that one need not go all the way to local representations to solve the

FIGURE 4. Three networks applied to the problem of determining which of the strings that can be made from w or I, followed by i or e, followed by g or r form words. Numbers on the connections represent connection strengths; numbers on the units represent the units' thresholds. A unit will take on an activation equal to I if its input

exceeds it threshold; otherwise, its activation is O.

problem facing our network . Conjunctive distributed representations

will suffice.

The scheme illustrated in the second panel of the figure provides a conjunctive distributed representation. In this scheme, there are units for pairs of letters which, in this limited vocabulary, happen to capture the combinations that are essential for determining whether a string of letters is a word or not. These are, of course, the pairs wi and Ie. These conjunctive units, together with direct input to the decision unit from the g uni t, are sufficient to construct a network which cOlTectly classifies all strings consisting of a w or an I, followed by an i or an e,

followed by a g or r.

This example illustrates that conjunctive coding is often necessary if distributed representations are to be used to solve problems that might easily be posed to networks. This same point could be illustrated with many other examples-the exclusive or problem is the classic example (Minsky & Papert, 1969) . Other examples of problems requiring some sort of conjunctive encoding can be found in Hinton (1981 a) and in Chapters 7 and 8. An application of conjunctive coding to a psychologi-

cal model is found in Chapter 18.

Some problems (mostly very simple ones) can be solved without any conjunctive encoding at all, and others will require conjuncts of more than two units at a time. In general, it is hard to specify in advance just what "order" of conjunctions will be required. Instead, it is better to search for a learning scheme that can find representations that are adequate. The mechanisms proposed in Chapters 7 and 8 represent two

steps toward this goal .

Implementing an Arbitrary Mapping Between Two Domains

The attentive reader will have noticed that a local representation can always be made to work in the example we have just considered. However , we have already discussed several reasons why distributed representations are preferable. One reason is that they can make more

efficient use of parallel hardware than local representations.

This section shows how a distributed representation in one group of units can cause an appropriate distributed representation in another group of units. We consider the problem of implementing an arbitrary pairing between representations in the two groups, and we take as an example an extension of the previous one: the association between the visual form of a word and its meaning. The reason for considering an arbitrary mapping is that this is the case in which local representations seem most helpful. If distributed representations are better in this Copyrighted Material

case, then they are certainly better in cases where there are underlying regularities that can be captured by regularities in the patterns of activation on the units in one group and the units in another. A discussion of the benefit distributed representations can provide in such cases can

be found in Chapter 18.

If we restrict ourselves to monomorphemic words, the mapping from strings of graphemes onto meanings appears to be arbitrary in the sense that knowing what some strings of graphemes mean does not help one predict what a new string means. 8 This arbitrariness in the mapping from graphemes to meanings is what gives plausibility to models that have explicit word units. It is obvious that arbitrary mappings can be implemented if there are such units. A grapheme string activates exactly one word unit, and this activates whatever meaning we wish to associate with it (see Figure SA) . The semantics of simi lar grapheme strings can then be completely independent because they are mediated by separate word units. There is none of the automatic general ization

that is characteristic of distributed representations.

Intuitively, it is not at all obvious that arbitrary mappings can be implemented in a system where the intermediate layer of units encodes the word as a distributed pattern of activity instead of as activity in <sup>a</sup> single local unit. The distri buted alternative appears to have a serious drawback. The effect of a pattern of activity on other representations is the combined result of the individual effects of the active units in the pattern. So similar patterns tend to have similar effects. It appears that we are not free to make a given pattern have whatever effect we wish on the meaning representations without thereby altering the effects that other patterns have. This kind of interaction appears to make it difficult to implement arbitrary mappings from distributed representations of words onto meaning representations. We shall now show that these intuitions are wrong and that distributed representations of words can work perfectly well and may even be more efficient than single word

units.

Figure SB shows a three-layered system in which grapheme/ position units feed into word-set units which, in turn, feed into semantic or sememe units. Models of this type, and closely related variants, have been analyzed by Willshaw (1 98 0 , V. Dobson (personal communication, 1984) , and by David Zipser (personal communication, 1981) ; some further relevant analyses are discussed in Chapter 12. For simpli-

8 Even for monomorphemic words there may be particular fragments that have associated meaning. For example, words starting with sn usually mean something unpleasant to do with the lips or nose (sneer, snarl, snigger) , and words with long vowels are more likely to stand for large, slow things than words with short vowels (George Lakoff, per-

sonal communication) . Much of Lewis Carroll's poetry relies on such effects. Copyrighted Material

FIGURE S. A: A three-layer network. The b<sup>O</sup>llom layer contains units that represent particular graphemes in particular positions within the word. The middle layer contains units that recognize complete words, and the top layer contains units that represent semantic features of the meaning of the word. This network uses local representations of words in the middle layer. B: The top and bollom layers are the same as in (A), but the middle layer uses a more distributed representation. Each unit in this layer can be activated by the graphemic representation of any one of a whole set of words. The unit then provides input to every semantic feature that occurs in the meaning of any of the words that activate it. Only those word sets containing the word cat are shown in this example. Notice that the only semantic features which receive input from all these word

sets are the semantic features of cat.

city, we shall assume that each unit is either active or inactive and that there is no feedback or cross-connections. These assumptions can be relaxed without substantially affecting the argument. A word-set unit is activated whenever the pattern of the grapheme/ position units codes a word in a particular set. The set could be all the four-letter words starting with HE, for example, or all the words containing at least two T's. All that is required is that it is possible to decide whether a word is in Copyrighted Material

the set by applying a simple test to the acti vated grapheme/ position units. So, for example, the set of all words meaning "nice" is not al lowed as a word set. There is an implicit assumption that word meanings can be represented as sets of sememes. This is a contentious issue. There appears to be a gulf between the componential view in which a meaning is a set of features and the structural ist view in which the meaning of a word can only be defined in terms of its relationships to other meanings. Later in this chapter we consider one way of integrating these two views by allowing articulated representations to be

built out of a number of different sets of active features.

Returning to Figure 5B, the question is whether it is possible to implement an arbitrary set of associations between grapheme/ position vectors and sememe vectors when the word-set units are each activated by more than one word. It wi ll be sufficient to consider just one of the many possible specific models. Let us assume that an active word-set unit provides positive input to all the sememe units that occur in the meaning of any word in the word set. Let us also assume that each sememe unit has a variable threshold that is dynamically adjusted to be just slightly less than the number of active word-set units. Only sememe units that are receiving input from every active word-set unit

will then become active.

All the sememes of the correct word wi ll be activated because each of these sememes will occur in the meaning of one of the words in the active word sets. However, additional sememes may also be activated because, just by chance, they may receive input from every active word-set unit. For a sememe to receive less input than its threshold, there must be at least one active word set that does not contain any word which has the sememe as part of its meaning. For each active

word set the probabi lity, i, of this happening is

i= (l \_ p)(w- O

where p is the proportion of words that contain the sememe and wis the number of words in the word set of the word-set unit. The reason for the term <sup>w</sup>- 1 is that the sememe is already assumed not to be part of the meaning of the correct word, so there are only <sup>w</sup>- 1

remaining words that could have it in their meaning.

Assume that when a word is coded at the graphemic level it activates <sup>u</sup>units at the word-set level . Each sememe that is not part of the word's meaning has a probabi lity i of failing to receive input from each word-set unit. The probabi lity, f, that al l of these word-set units wi ll

provide input to it is therefore

<sup>f</sup>= (I - j)u

== [I - (I - p )

(w - I)]u

.

By inspection, this probability of a "false-positive" sememe reduces to zero when w is 1. Table 1 shows the value of f for various combinations of values of p, u, and w. Notice that if p is very small, f can remain negligible even if w is quite large. This means that distributed representations in which each word-set unit partici pates in the representation of many words do not lead to errors if the semantic features are relatively sparse in the sense that each word meaning contains only a small fraction of the total set of sememes. So the word-set units can be fairly nonspecific provided the sememe units are fairly specific (not shared by too many different word meanings) . Some of the entries in the table make it clear that for some values of p, there can be a negl igible chance of error even though the number of word-set units is considerably less than the number of words (the ratio of words to word-set

units is w/ u ).

The example described above makes many simpl ifying assumptions. For example, each word-set unit is assumed to be connected to every relevant sememe unit. If any of these connections were missing, we could not afford to give the sememe units a threshold equal to the number of active word-set units. To allow for missing connections we could lower the threshold. This would increase the false-positive error rate, but the effect may be quite small and can be compensated by adding word-set units to increase the specificity of the word-level representations (Willshaw, 1981) . Alternati vely, we could make each word-set unit veto the sememes that do not occur in any of its words. This scheme is robust against missing connections because the absence of one veto can be tolerated if there are other vetos (V. Dobson, per-

sonal communication, 1984) .

There are two more simplifying assumptions both of which lead to an underestimate of the effectiveness of distributed representations for the arbitrary mapping task. First, the calculations assume that there is no fine-tuning procedure for incrementing some weights and decrementing others to improve performance in the cases where the most frequent errors occur. Second, the calculations ignore cross-connections among the sememes. If each word meaning is a familiar stable pattern of sememes, there will be a strong "clean-up" effect which tends to suppress erroneous sememes as soon as the pattern of activation at the sememe level is sufficiently close to the famil iar pattern for a particular word meaning. Interactions among the sememes also provide an explanation for the abi lity of a single grapheme string (e.g. , bank) to elicit two quite different meanings. The bottom-up effect of the activated Copyrighted Material

w p 5 .2 10 .2 20 .2 40 2 80 .2

0.07 1 0.49 0.93 1.0 1.0

10 10 .2 0.24 10 20 .2 0.86 10 40 .2 1.0 10 80 .2 1.0 10 160 .2 1.0

40 40 .2 0.99 40 80 .2 1.0 <sup>40</sup>16<sup>0</sup>.2 1.0 40 320 .2 1.0 40 640 .2 1.0

100 100 .2 1.0 <sup>100</sup><sup>200</sup>.2 1.0 100 400 .2 1.0 100 800 .2 1.0

w

10 100 10 200 100 400 100 800

TABLE <sup>1</sup>

p .1 .1 . 1 .1 .1 .1 .1 .1 .1 .1 .1 .1 .1 .1 .I .1 .1 .1 .1

f

0.0048 0.086 0.48 0.92 1.0

0.0074 0.23 0.85 1.0 1.0 0.52 0.99 1.0 1.0 1.0

0.99 1.0 1.0 1.0

u

p

.0 1 .01 .0 1 .01 .0 1

9.5x 10-8 4.8x 10-6 0.00016 0.0036 0.049 .01 2.3x 10-11 .01 2.5x 10-8 .01 I. 3x Hr5 .01 0.0024 .01 0. <sup>10</sup>

.01 2.7x 10-20 .01 3 .

5x 10-11

.01 0.00012 .01 0. 1 9 .01 0.94 .01 9.0x 10-21 .01 4.8x 10-7 .0 1 0.16 .01 0.97

The probability,!, of a false-positive sememe as a function of the number of active wordset units per word, U, the number of words in each word-set, w, and the probability, p, of

a sememe being part of a word meaning.

word-set units helps both sets of sememes, but as soon as top-down factors give an advantage to one meaning, the sememes in the other meaning wi ll be suppressed by competitive interactions at the sememe

level (Kawamoto & Anderson, 1984) .

A simulation. As soon as there are cross-connections among the sememe units and tine-tuning of individual weights to avoid frequent errors, the relatively straightforward probabilistic analysis given above breaks down. To give the cross-connections time to clean up the output, it is necessary to use an iterative procedure instead of the simple "straight-through " processing in which each layer completely determines the states of all the units in the subsequent layer in a single, synchronous step. Systems containing cross-connections, feedback, and asynchronous processing elements are probably more realistic, but they are generally very hard to analyze. However, we are now beginning to discover that there are subclasses of these more complex systems that behave in tractable wayf00Vi4§"d�fttaJj'falhis subclass is described in

more detai l in Chapter 7. It uses processing elements that are inherently stochastic. Surprisingly , the use of stochastic elements makes these networks better at performing searches, better at learning,

and easier to analyze.

A simple network of this kind can be used to illustrate some of the claims about the ability to " clean Uph the output by using interactions among sememe units and the ability to avoid errors by fine - tuning the appropriate weights. The network contains 30 grapheme units, 20 word - set units, and 30 sememe units. There are no direct connections between grapheme and sememe units, but each word-set unit is connected to all the grapheme and sememe units. The grapheme units are divided into three sets of ten, and each three-letter word has one active unit in each group of ten (units can only have activity levels of 1 or 0) . The "meaning" of a word is chosen at random by selecting each sememe unit to be active with a probability of 0.2. The network shown in Figure 6 has learned to associated 20 different grapheme strings with their chosen meanings. Each word-set unit is involved in the representation of many words, and each word involves many word-set units. The detai ls of the learning procedure used to create this network and the search procedure which is used to settle on a set of active sememes when given the graphemic input are described in Chapter 7. Here we

simply summarize the main results of the simulation .

After a long period of learning, the network was able to produce the correct pattern of sememes 99.9% of the time when given a graphemic input. Removal of any one of the word-set units after the learning typical ly caused a slight rise in the error rate for several different words rather than the complete loss of one word. Similar effects have been observed in other distributed models (Wood, 1 978) . In our simulations, some of the erroneous responses were quite interesting. In 10,000 tests with a missing word-set unit there were 140 cases in which the model failed to recover the right sememe pattern. Some of these consisted of one or two missing or extra sememes, but 83 of the errors were exactly the pattern of sememes of some other word. This is a result of the cooperative interactions among the sememe units. If the input coming from the word-set units is noisy or underspecified as it may be when units are knocked out, the clean-up effect may settle on a

similar but incorrect meaning.

This effect is reminiscent of a phenomenon called deep dyslexia which occurs with certain kinds of brain damage in adults. When shown a word and asked to read it, the subject wil l sometimes say a different word with a very similar meaning. The incorrect word sometimes has a very different sound and spelling. For example, when shown the word PEACH, the subject might say APRICOT. (See Coltheart , Patterson, & Marshall , 1980, for more information about acquired dyslexia.) Copyrighted Material

FIGURE 6. A compact display that shows all the connection strengthsof the 20 units in the middle layer of a three-layer network. The network can map from a pattern of acti vity over the 30 units in the bottom layer (representing graphemes) to an associated pal\ern of acti vity over the 30 <sup>u</sup> <sup>n</sup>its of the lop layer (representing sememes) . Wi thi <sup>n</sup> each of the large rectangles that are used to depict middle-layer units, <sup>I</sup> he 30 black and white rectangles at the top depict the weights of the connections to the top layer, and the 30 rectangles at the bottom depict the <sup>w</sup>eightsfrom the bottom layer. White rectangles are positiveweights, black are negative, and the area of a recta<sup>n</sup> gle depicts the magn itude of the weight . The singl<sup>e</sup>weight that occurs somewhere in the middle of a unit is its

th

<sup>r</sup>eshold (black m<sup>e</sup>ans a positive threshold). The weights between the 30 units in the

top layer are not sh<sup>o</sup>w<sup>n</sup>in this display .

Semantic errors of this kind seem bizarre because it seems as if the subject must have accessed the lexical item PEACH in order to make the semantically related error, and if he can get to the lexical item why can't he say it ? (These subjects may know and be able to say the words that they misread.) Distributed representations allow us to dispense wi th the rigid distinction between accessing a word and not accessing i<sup>t</sup> . In a network that has learned the word PEACH, the graphemic representation of PEACH wi ll cause approximately the right input to the sememe units, and interactions at the sememe level can then cau<sup>s</sup>e exactly the pattern of sememe<sup>s</sup>for APRICOT. Another psychologically interesti��ftf&l1fIB�WBJl the network relearns after

it has been damaged. The network was damaged by adding noise to every connection that involved a word-set unit. This reduced the performance from 99. 3% correct to 64.3%. 9 The network was then retrained and it exhibited very rapid relearning, much faster than its original rate of learning when its performance was 64. 3% correct. This rapid recovery was predicted by a geometrical argument which shows that there is something special about a set of connection strengths that is generated by adding noise to a near-perfect set. The resulting set is very different from other sets of connection strengths that exhibit the

same performance. (See Chapter 7 for further discussion.)

An even more surprising effect occurs if a few of the words are omitted from the retraining. The error rate for these words is substantially reduced as the retraining proceeds, even though the other graphemesememe pairings have no intrinsic relation to them because all the pairings were selected randomly. The "spontaneous" recovery of words that the network is not shown again is a result of the use of distributed representations. All the weights are involved in encoding the subset of the words that are shown during retraining, and so the added noise tends to be removed from every weight. A scheme that used a separate unit for each word would not behave in this way, so one can view spontaneous recovery of unrehearsed items as a qualitative signature of dis-

tributed representations.

STRUCTURED REPRESENTATIONS AND PROCESSES

In this section we consider two extensions of distributed representations. These extensions illustrate that the idea of distributed representations is consistent with some of the major insights from the field of artificial intelligence concerning the importance of structure in representations and processes. Perhaps because some proponents of distributed· representations have not been particularly attuned to these issues, it is often unclear how structure is to be captured in a distributed representational scheme. The two parts of this section give some indication of the directions that can be taken in extending distributed

representations to deal with these important considerations.

<sup>9</sup>The error rate was 99.3% rather than 99.9% in this example because the network was forced to respond faster, so the cooperative effects had less time to settle on the optimal

Representing Constituent Structure

Any system that attempts to implement the kinds of conceptual structures that people use has to be capable of representing two rather different kinds of hierarchy. The first is the "IS-A" hierarchy that relates types to instances of those types. The second is the part/whole hierarchy that relates items to the constituent items that they are composed of. The most important characteristics of the IS-A hierarchy are that known properties of the types must be "inherited " by the instances, and properties that are found to apply to all instances of a type must normally be attributed to the type. Earlier in this chapter we saw how the IS-A hierarchy can be implemented by making the distributed representation of an instance include, as a subpart, the distributed representation for the type. This representational trick automatically yields the most important characteristics of the IS-A hierarchy, but the trick can only be used for one kind of hierarchy. If we use the part/whole relationship between patterns of activity to represent the type/instance relationship between items, it appears that we cannot also use it to represent the part/whole relationship between items. We cannot make the representation of the whole be the sum of the representa-

tions of its parts.

The question of how to represent the relationship between an item and the constituent items of which it is composed has been a major stumbling block for theories that postulate distributed representations. In the ri val , local ist scheme , <sup>a</sup>whole is a node that is linked by labeled arcs to the nodes for its parts. But the central tenet of the distributed scheme is that different items correspond to alternative patterns of activity in the same set of units, so it seems as if a whole and its parts

cannot both be represented at the same time.

Hinton 098 1a) descri bed one way out of this dilemma. It relies on the fact that wholes are not simply the sums of their parts. They are composed of parts that play particular roles within the whole structure. A shape, for example, is composed of smaller shapes that have a particular size, orientation, and position relative to the whole. Each constituent shape has its own spatial role, and the whole shape is composed

of a set of shape/ role pairs.

1O Similarl<sup>y</sup> , <sup>a</sup>proposition is composed of

objects that occupy particular semantic roles in the whole propositional

<sup>10</sup>Relationships between parts are important as well. One advantage of explicitly representing shape/ role pairs is that it allows different pairs to support each other. One can view the various different locations within an object as slots and the shapes of parts of an object as the fillers of these slots. Knowledge of a whole shape can then be imple-

structure. This suggests a way of implementing the relationship between wholes and parts: The identity of each part should first be combined with its role to produce a single pattern that represents the combination of the identity and the role, and then the distributed representation for the whole should consist of the sum of the distributed representations for these identity/ role combinations (plus some additional "emergent" features) . This proposal differs from the simple idea that the representation of the whole is the sum of the representations of its parts because the subpatterns used to represent identity/ role combinations are quite different from the patterns used to represent the identities

alone. They do not , for example, contain these patterns as parts.

Naturally, there must be an access path between the representation of an item as a whole in its own right and the representation of that same item playing a particular role within a larger structure. It must be possible, for example, to generate the identity/ role representation from two separate, expl icit, distributed patterns one of which represents the identity and the other of which represents the role. It must also be possible to go the other way and generate the explicit representations of the identity and role from the single combined representation of the

identity / role combination (see Figure 7).

The use of patterns that represent identity/ role combinations al lows the part/ whole hierarchy to be represented in the same way as the type/ instance hierarchy. We may view the whole as simply a particular instance of a number of more general types, each of which can be defined as the type that has a particular kind of part playing a particular

role (e.g. , men with wooden legs) .

Sequential Symbol Processing

If constituent structure is implemented in the way described above, there is a serious issue about how many structures can be active at any one time. The obvious way to al locate the hardware is to use a group of units for each possible role within a structure and to make the pattern of activity in this group represent the identity of the constituent that is currently playing that role. This implies that only one structure can be represented at a time, unless we are wi lling to postulate multiple copies of the entire arrangement. One way of doing this, using units with programmable rather than fixed connections, is described in Chapter 16. However, even this technique runs into difficulties if more than a few modules must be "programmed " at once. However, people do seem to suffer from strong constraints on the number of structures of the same general type that they can process at once. The Copyrighted Material

AG EN T

IDENTITY

PATIENT LOCATION

FIGURE 7. A sketch of the apparatus that might be necessary for combining separate representations of an identity and a role into a single pattern. Only one identity and only one role can be explicitly represented at a time because the identity and role groups can each have only one pattern of activity at a time. However, the various role groups allow many identity/role combinations to be encoded simultaneously. The small triangular symbols represent the ability of the pattern of activity in the group that expl ictly represents a role to determine which one of the many role groups is currently interacting with the identity group. This allows the identity occupying a particular role to be "read out " as well as allowing the reverse operation of combining an identity and a role.

sequentiality that they exhibit at this high level of descri ption is initially surprising given the massively parallel architecture of the brain, but it becomes much easier to understand if we abandon our localist predelictions in favor of the distributed alternative which uses the parallelism to give each active representation a very rich internal structure that allows the right kinds of generalization and content-addressability. There may be some truth to the notion that people are sequential symbol processors if each "symbolic representation " is identified with a Copyrighted Material

successive state of a large interactive network. See Chapter 14 for

further discussion of these issues.

One central tenet of the sequential symbol processing approach (Newell, 1980) is the ability to focus on any part of a structure and to expand that into a whole that is just as rich in content as the original whole of which it was a part. The recursive abi lity to expand parts of a structure for indefinitely many levels and the inverse ability to package up whole structures into a reduced form that allows them to be used as constituents of larger structures is the essence of symbol processing. It allows a system to build structures out of things that refer to other whole structures without requiring that these other structures be

represented in all their cumbersome detail.

In conventional computer implementations, this ability is achieved by using pointers. These are very convenient, but they depend on the use of addresses. In a parallel network, we need something that is functional ly equivalent to arbitrary pointers in order to implement symbol processing. This is exactly what is provided by subpatterns that stand for identity/ role combinations. They allow the full identity of the part to be accessed from a representation of the whole and a representation of the role that the system wishes to focus on, and they also allow explicit representations of an identity and a role to be combined into a less cumbersome representation , so that several identity/role combinations can be represented simultaneously in order to form the represen-

tation of a larger structure.

SUMMARY

Given a parallel network, items can be represented by activity in a single, local unit or by a pattern of activity in a large set of units with each unit encoding a microfeature of the item. Distributed representations are efficient whenever there are underlying regularities which can be captured by interactions among microfeatures. By encoding each piece of knowledge as a large set of interactions, it is possible to achieve useful properties like content-addressable memory and automatic generalization, and new items can be created without having to create new connections at the hardware level . In the domain of continuously varying spatial features it is relatively easy to provide a mathematical analysis of the advantages and drawbacks of using distri-

buted representions.

Distributed representations seem to be unsuitable for implementing purely arbitrary mappings because there is no underlying structure and so general ization only causes unwanted interference. However, even Copyrighted Material

for this task, distributed representations can be made fairly efficient and they exhibit some psychologically interesting effects when damaged. There are several difficult problems that must be solved before distributed representations can be used effectively. One is to decide on the pattern of activity that is to be used for representing an item. The similarities between the chosen pattern and other existing patterns wi ll determine the kinds of general ization and interference that occur. The search for good patterns to use is equivalent to the search for the underlying regularites of the domain. This learning problem is

addressed in the chapters of Part II.

Another hard problem is to clarify the relationship between distributed representations and techniques used in artificial intelligence like schemas, or hierarchical structural descriptions. Existing artificial intelligence programs have great difficulty in rapidly finding the schema that best fits the current situation. Parallel networks offer the potential of rapidly applying a lot of knowledge to this best-fit search, but this potential wi ll only be realized when there is a good way of implementing schemas in parallel networks. A discussion of how this might be

done can be found in Chapter 14.

ACKNOWLEDGMENTS

This chapter is based on a technical report by the first author, whose work is supported by a grant from the System Development Foundation . We thank Jim Anderson, Dave Ackley, Dana Bal lard, Francis Crick, Scott Fahlman, Jerry Feldman , Chri stopher Longuet-Higgins, Don Norman, Terry Sejnowski , and Tim Shallice for helpful

discussions.

FIGURE I. The one-layer perceptron analyzed by Minsky and Papert. (From Perceprrons by M. L. Minsky and S. Papert, 1969, Cambridge, MA: MIT Press. Copyright 1969 by

MIT Press. Reprinted by permission.)

Minsky and Papert's analysis of the li mitations of the one-layer perceptron, coupled with some of the early successes of the symbolic processing approach in artificial intell igence, was enough to suggest to a large number of workers in the field that there was no future in perceptron-l ike computational devices for artificial intell igence and cognitive psychology. The problem is that although Minsky and Papert were perfectly correct in their analysis, the results apply only to these simple one-layer perceptrons and not to the larger class of perceptronlike models. In particular (as Minsky and Papert actually conceded) , it can be shown that a multi layered percept ron system, including several layers of predicates between the retina and the decision stage, can compute functions such as parity, using reasonable numbers of units each computing a very local predicate. (See Chapters 5 and 8 for examples of multilayer networks that compute parity). Similarly, it is not difficult to develop networks capable of solving the connectedness or inside/outside problem. Hinton and Sejnowski have analyzed a version

of such a network (see Chapter 7) .

Essentially, then, although Minsky and Papert were exactly correct in thei r analysis of the one-layer perceptron, the theorems don't apply to systems which are even a little more complex. In particular, it doesn't apply to multilayer systems nor to systems that allow feedback loops . Minsky and Papert argued that there would not be much value to multilayer perceptrons. First, they argued that these systems are sufficiently unrestricted as to be vacuous. They pointed out, for example, that a universal computer could be built out of linear threshold units. Copyrighted Material

Therefore, restricting consideration of machines made out of linear

threshold units is no restriction at all on what can be computed.

We don't, of course, believe that the class of models sketched in Chapter 2 is a smal l or restrictive class. (Nor, for that matter, are the languages of symbol processing systems especially restrictive.> The real issue, we bel ieve, is that different algorithms are appropriate to different architectural designs. We are investigating an architecture in which cooperative computation and paral lelism is natural. Serial symbolic systems such as those favored by Minsky and Papert have a natural domain of algorithms that di ffers from those in PDP models. Not everything can be done in O:1e step without feedback or layering (both of which suggest a kind of "seriality"). We have been led to consider models that have both of these features. The real point is that we seek algorithms that are as parallel as possible. We bel ieve that such algorithms are going to be closer in form to the algorithms which could be employed by the hardware of the brain and that the kind of parallelism we employ allows the exploitation of multiple information sources

and cooperative computation in a natural way .

A further argument advanced by Minsky and Papert against perceptron-l ike models with hidden units is that there was no indication how such multilayer networks were to be trained. One of the appealing features of the one-layer perceptron is the existence of a powerful learning procedure, the perceptron convergence procedure of Rosenblatt. In Minsky and Papert's day, there was no such powerful learning procedure for the more complex multilayer systems. This is no longer true. Chapters 5, 6, 7, and 8 all provide schemes for learning in systems with hidden units. Indeed, Chapter 8 provides a direct generalization of the perceptron learning procedure which can be applied to arbitrary networks with multiple layers and feedback among layers. This procedure can , in principle, learn arbitrary functions including, of

course, parity and connectedness .

The problem of stimulus equivalence. A second problem with early PDP models-and one that is not necessari ly completely overcome by multi layer systems-is the problem of invariance or stimulus equivalence. An A is an A is an A, no matter where on the retina it appears or how large it is or how it is oriented; and people can, in general, recognize patterns rather well despite various transformations. It has always seemed elegant and natural to imagine that an A, no matter where it is presented, is normalized and then processed for recognition using stored knowledge of the appearance of the letter (Marr, 1982; Neisser,

1967).

Letter Units [

Canonical Feature Units

Retinocentric Feature

Units 8

Mapping Units

FIGURE 2. Hinton's (I981b) scheme for mapping patterns in one coordinate system into patterns in another coordinate system. At the top are two letter-detector units, with mutual excitatory connections to the six canonical feature units (the position and orientation of the line segment each of these detectors represents is indicated by the line segment in the "body" of each unit). At the bottom are six retinocentric feature units, and al the right are units corresponding to each of three different mappings from retinocentric to canonical features. (The arrows on the units indicate which direction in the retinocentric frame corresponds to upright in the canonical frame, and the arrow outside the uni<sup>t</sup> indicates the nature of the transformation imposed on the retinocentric pattern). Each canonical unit receives three pairs of inputs, with each pair arriving at a multiplicative

<sup>c</sup>onnectio<sup>n</sup>. These inputs are illustrated for one canonical unit only.

recetvmg unit. In this case, if a particular retinocentric feature is on and the 90° clockwise mapping unit is on, then the canonical feature corresponding to the active retinal feature will receive an excitatory input. If just one of the two inputs to the connection is on , no activation wi ll flow to the central unit. In this way, when a mapping unit is active, it effectively programs the multipl icati ve connections needed to implement the corresponding mapping by activating one of the two

inputs to each of the programmable connections.

Using this mechanism, it is possible to map from retinal to central coordinates if the mapping is known in advance. Object recognition can now proceed as follows: A mapping is chosen (perhaps on the basis of processing the preceding stimulus> , and this is used to map a retinal input onto the canonCCopyMl)iUbd \_tetia�ystem involving variable

Recursion. There are many other specific points that have been raised with respect to existing PDP models. Perhaps the most common one has to do with recursion. The ability to perform recursive function calls is a major feature of certain computational frameworks, such as augmented transition network (ATN) parsers (Woods, 1973; Woods & Kaplan, 1971), and is a property of such frameworks that gives them the capability of processing recursively defined struct ures such as sentences, in which embedding may produce dependencies between elements of a surface string that are indefinitely far removed from each other (Chomsky , 1957). It has often been suggested that PDP mechanisms lack the capacity to perform recursi ve computations and so are simply incapable of providing mechanisms for processing sentences

and other recursively defined structures.

As before, these suggestions are simply wrong. As we have already seen, one can make an arbitrary computational machine out of linear threshold units, including, for example, a machine that can carry out all the operations necessary for implementing a Turing machine; the one limitation is that real biological systems cannot be Turing machines because they have finite hardware. In Chapter 14, however, we point out that with external memory aids (such as paper and pencil and a

notational system) such limitations can be overcome as well.

We have not dwelt on PDP implementations of Turing machines and recursive processing engines because we do nut agree with those who would argue that such capabilities are of the essence uf human computation. As anyone who has ever attempted to process sentences like "The man the boy the girl hit kissed moved" can attest, our ability to process even moderate degrees of center-embedded structure is grossly impaired relative to that of an ATN parser. And yet, the human ability to use semantic and pragmatic contextual information to facilitate comprehension far exceeds that of any existing sentence processing

machine we know of.

What is needed, then, is not a mechanism for flawless and effortless processing of center-embedded constructions. Compilers of computer languages generally provide such facilities, and they are powerful tools, but they have not demonstrated themselves sufficient for processing natural language. What is needed instead is a parser built from the kind of mechanism which faci litates the simultaneous consideration of large numbers of mutual and interdependent constraints. The challenge is to show how those processes that others have chosen to explain in terms of recursive mechanisms can be better explained by the kinds of

processes natural for PDP networks.

This challenge is one that has not yet been fully met. However, some initial steps toward a PDP model of language processing are

described in Chapter �pyriJfh�d ��Hr)ia/

whose implementation is

A related claim that some people have made is that our models appear to share much in common with behaviorist accounts of behavior. While they do involve simple mechanisms of learning, there is acrucial difference between our models and the radical behaviorism of Skinner and his followers. In our models, we are expl icitly concerned with the problem of internal representation and mental processing, whereas the radical behaviorist explicitly denies the scientific utility and even the validity of the consideration of these constructs. The training of hidden units is , as is argued in Chapters 5 to 8, the construction of internal representations. The models described throughout the book all concern internal mechanisms for activating and acquiring the ability to activate appropriate internal representations. In this sense, our models must be seen as completely antithetical to the radical behaviorist program and strongly committed to the study of representa-

tion and process.

PDP Models Are the Wrong Level of Analysis

It is sometimes said that although PDP models are perfectly correct, they are at the wrong level of analysis and therefore not relevant to psychological data. 2 For example, Broadbent (1985) has argued that psychological evidence is irrelevant to our argument about distributed memory because the distribution assumption is only meaningful at what Marr (I982) has called the implementational (physiological) level and that the proper psychological level of description is the computational

level .

The issues of levels of analysis and of theorizing is difficult and requires a good deal of careful thought. It is, we bel ieve, largely an issue of scientific judgement as to what features of a lower level of analysis are relevant to a higher one. We are quite sure that it is not a matter for prescri ption. We begin our response to this objection with <sup>a</sup> review of Marr's analysis and his three levels of description. We then suggest that indeed our models are stated at the same level (in Marr's sense) as most traditional models from cognitive science. We then describe other senses of levels, including one in which higher level accounts can be said to be convenient approximations to lower level accounts. This sense comes closest to capturing our view of the

2 The following discussion is based on a paper (Rumelhart & McClelland, 1985) written in response to a critique by Donald Broadbent (1985) on our work on distributed

relation between our PDP models and other traditional information

processing models.

Marr's Notion of Levels

David Marr (1982) has provided an influential analysis of the issue of levels in cognitive science. Although we are not sure that we agree entirely with Marr's analysis, it is thoughtful and can serve as a starting point. Whereas Broadbent acknowledges only two levels of theory, the computational and the implementational, Marr actual ly proposes three, the computational, the algorithmic, and the implementational levels. Table 1 gives a description of Marr's three levels. We believe that PDP models are generally stated at the algorithmic level and are primarily aimed at specifying the representation of information and the processes or procedures involved in cognition. Furthermore, we agree with Marr's assertions that "each of these levels of descri ption will have their place" and that they are "logically and causal ly related." Thus, no particular level of descri ption is independent of the others. There is an impl icit computational theory in PDP models as wel l as an appeal to certain implementational (physiological) considerations. We believe this to be appropriate. It is clear that different algorithms are more naturally implemented on different types of hardware and, therefore, information about the implementation can inform our hypotheses at the

algorithmic level.

TABLE 1 THE THREE LEVELS AT WHICH ANY MACHINE CARRYING OUT INFORMA TION PROCESSING TASKS MUST DE UNDERSTOOD

Computational Theory What is the goal of the computation, why is it appropriate, and what is the logic of the strategy by which it can be carried

out?

Representation and Algorithm How can this computational theory be implemented? In particular, what is the representation for the input and output, and what is the algorithm for the transformation?

Hardware Implementation How can the representation and algorithm be realized physically?

Computational models, according to Marr, are focused on a formal analysis of the problem the system is solving-not the methods by which it is solved. Thus, in linguistics, Marr suggests that Chomsky's (1965) view of a competence model for syntax maps most closely onto a compulationa/level theory, whereas a psycholinguistic theory is more of <sup>a</sup>performance theory concerned with how grammatical structure might actually be computed. Such a theory is concerned with the algorithmic level of description. It is the algorithmic level at which we are concerned with such issues as efficiency, degradation of performance under noise or other adverse conditions, whether a particular problem is easy or difficult, which problems are solved quickly and which take a long time to solve, how information is represented, etc. These are all questions to which psychological inquiry is directed and to which psychological data is relevant. Indeed, it would appear that this is the level to which psychological data speaks most strongly. At the computational level, it does not matter whether the theory is stated as a program for a Turing machine, as a set of axioms, or as a set of rewrite rules. It does not matter how long the computation takes or how performance of the computation is affected by "performance" factors such as memory load, problem complexity, etc. It doesn't matter how the information is represented, as long as the representation is rich enough, in princi ple, to support computation of the required function. The question is simply what junction is being computed, not how is it being computed.

Marr recommends that a good strategy in the development of theory is to begin with a careful analysis of the goal of a particular computation and a formal analysis of the problem that the system is trying to solve. He believes that this top-down approach will suggest plausible algorithms more effectively than a more bottom-up approach. Thus, the computational level is given some priority. However, Marr certainly does not propose that a theory at the computational level of description

is an adequate psychological theory,

As psychologists, we are committed to an elucidation of the algorithmic level. We have no quarrel with Marr's top-down approach as a strategy leading to the discovery of cognitive algorithms, though we have proceeded in a different way. We emphasize the view that the various levels of description are interrelated. Clearly, the algorithms must, at least roughly, compute the function specified at the computational level . Equally clearly, the algorithms must be computable in amounts of time commensurate with human performance, using the kind and amount of hardware that humans may reasonably be assumed to possess . For example, any algorithm that would require more specific events to be stored separately than there are synapses in the brain should be given a lower plausibility rating than those that require

4. GENERAL ISSUES 127

such primitives wi ll change the shape of higher level theory

considerably.

PDP mechanisms may also place some constraints on what we might realistically ask for in the way of computational primiti ves because of the costs of implementing certain kinds of computations in parallel hardware in a single relaxation search. The paral lel matching of variablized productions is one case in point. Theories such as ACT\* (J. R. Anderson, 1983) assume that this can be done without worrying about the implementation and, therefore, provide no principled accounts of the kinds of crosstalk exhibited in human behavior when processing multiple patterns simultaneously . However, it appears to be a quite general property of PDP mechanisms that they wi ll exhibit crosstal k when processing multiple patterns in parallel (Hinton & Lang, 1985;

Mozer, 1984; see Chapters 12 and 16) .

High-level languages often preserve some of the character of the lower level mechanisms that implement them, and the resource and time requirements of algorithms drastically depends on the nature of the underlying hardware. Higher level languages that preserve the character of PDP mechanisms and exploit the algorithms that are effective descriptions of parallel networks are not here yet, but we expect such things to be coming along in the future. This wi ll be a welcome development, in our view, since certain aspects of cognitive theory have been too strongly influenced by the discrete, sequential algorithms

available for expression in most current high-level languages.

As we look closely, both at the hardware in which cognitive algorithms are implemented and at the fine structure of the behavior that these algorithms are designed to capture, we begin to see why it may be appropriate to formulate models which come closer to describing the microstructure of cognition. The fact that our microstructural models can account for many of the facts about the representation of general and specific information, for example, as discussed in Chapter 18, makes us ask why we should view constructs like logogens, prototypes, and schemata as anything other than convenient approximate descri p-

tions of the underlying structure of memory and thought.

Reductionism and Emergent Properties

A slightly different, though related, argument is that the PDP enterprise is an exercise in reductionism-an exercise in which all of psychology is reduced to neurophysiology and ultimately to physics. It is argued that coherent phenomena which emerge at any level (psychol-

ogy or physics or sociolW'Jyh���� �9g

grn language of descri ption

it has to do with the fact that we can't know everything and find out everything at once. The approach we have been arguing for suggests that to understand something thoroughly at some level requires knowledge at that level , plus knowledge of the lower levels. Obviously, this is impractical . In practice, even though there might be effects of lower levels on higher levels, one cannot always know them. Thus, attempting to formulate a description at this higher level as a first order of approximation is an important research strategy. We are forced into it if we are to learn anything at all . It is possible to learn a good deal about psychology without any reference whatsoever to any lower levels. This practical strategy is not, however, an excuse for ignoring what is known about the lower levels in the formulation of our higher level theories . Thus, the economist is wrong to ignore what we might know about individuals when formulating his theories. The chemist would be wrong to ignore what is known about the structure of the carbon atom in explaining the hardness of diamonds. We argued above that the view that the computational level is correct derives from experience with a very special kind of device in which the higher level was designed to give the right answers-exactly. In describing natural intelligence that can't, we suspect, be right-exactly. It can be a first order of approximation. As we learn more about a topic and as we look at it in more and more detai l we are going to be forced to consider more and more how it might emerge (in the above sense) from the interactions among its constituents. Interaction is the key word here. Emergent properties occur whenever we have nonli near interactions. In these cases the princi ples of interaction themselves must be formulated and the real theory at the higher level is, like chemistry, a theory of interac-

tions of elements from a theory one level lower.

Not Enough Is Known From Neuroscience to Seriously

Constrain Cognitive Theories

Many cognitive scientists believe that there will eventually be an understanding of the relationships between cognitive phenomena and brain functioning. Many of these same people feel, however, that the brain is such an exceptionally powerful computational device that it is capable of performing just about any computation. They suppose that facts now known from neuroscience place little or no restriction on what theories are possible at a cognitive level . In the meantime, they suppose, a top-down analysis of possible mechanisms of cognition can lead to an understanding of cognition that will stand independently of whatever might be discqsf)pyJ-;�m# �;fHnctioning. Moreover, they

constraint. Moreover, note that individual neurons probably don ' t compute very complicated functions. It seems unlikely that a single neuron computes a function much more complex than a single instruction in a digital computer. Imagine, again, writing an interesting program in even 1000 operations of this limited complexity of a serial computer. Evidently, the brain succeeds through massive parallelism. Thus, we conclude , the mechanisms of mind are most likely best understood as resulting from the cooperative acti vity of very many relatively simple

processing units operating in paral lel .

There is a very large number of neurons. Another self-evident, but important, aspect of brain - style processing is the very large number of processing units involved. Conventional estimates hold that there are on the order of 1010 to 1011 neurons in the brain. Moreover, each neuron is an active processing unit. This suggests parallelism on <sup>a</sup>very large scale indeed. An understanding of parallel computation involving a few hundred reasonably complex processors provides the wrong model . It may well be that it is the massive scale of the parallel ism of

the brain that gives it its amazing power.

Although the human brain is large, the number of neurons is not unl imited. It happens that our models sometimes push the limits of plausibility because of the large number of units they require. This is a real constraint , one that we and others have begun to take into account in evaluating our models (see Chapter 12 for a discussion of this issue).

Neurons receive inputs from a large number of other neurons. Another important feature of brain processing is the large fan-in and fan - out to and from each unit. Estimates vary, but single cortical neurons can have from 1,000 to 100,000 synapses on their dendrites and, likewise, can make from 1,000 to 100,000 synapses on the dendrites of other neurons. Generally, one or a small number of action potentials received are not enough to generate an action potential (see, for example, Chapter 20) . This suggests that human computation does not involve the kind of logic circuits out of which we make our digital computers, but that it involves a kind of statistical process in which the single units do not make decisions, but in which decisions are the product of the cooperative action of many somewhat independent processing units. Reliability derives from the stabi lity of the statistical behavior of large numbers of units . Again, this degree of connectivity should be contrasted with the number of immediate neighbors of processors in current parallel computers . Usually these numbers are measured in the tens (or less) rather than in the thousands . Moreover, this large degree of connectivity suggests that no neuron is very many synapses away

modalities. But higher level functions seem very much to be character-

ized by distributed, rather than central control .

This point has been made most clearly by the Russian neuropsychologist Luria (I 966; 1973) . Luria's investigations show that for every integrated behavioral function (e.g., visual perception, language comprehension or production, problem solving, reading) , many different parts of the cortex play a role so that damage to any part influences performance but is not absolutely crucial to it. Even the frontal lobes , most frequently associated with executive functions , are not absolutely necessary in Luria's view, in that some residual function is generally observed even after massive frontal damage (and mild frontal damage may result in no detectable symptomatology at al l) . The frontal lobes have a characteristic role to play, facilitating strategy shifts and inhibiting impulsive responding, but the overall control of processing can be as severely impaired by damage to parietal lobe structures that appear to be responsible for maintaining organized representations that

support coordinated and goal-directed activity.

Our view of the overall organization of processing is similar to Luria's. We have come to believe that the notion of subroutines with one system "cal ling" another is probably not a good way to view the operation of the brain. Rather, we believe that subsystems may modulate the behavior of other subsystems , that they may provide constraints to be factored into the relaxation computation . An elaboration of some

aspects of these ideas may be found in Chapter 14.

Relaxation is the dominant mode oj computation. Although there is no specific piece of neuroscience which compels the view that brainstyle computation involves relaxation, all of the features we have just discussed have led us to bel ieve that the primary mode of computation in the brain is best understood as a kind of relaxation system (cf. Chapters 6, 7, 14, 15, and 21) in which the computation proceeds by iteratively seeking to satisfy a large number of weak constraints. Thus, rather than playing the role of wires in an electric circuit, we see the connections as representing constraints on the co-occurrence of pairs of units. The system should be thought of more as settling into a solution than calculating a solution. Again, this is an important perspective change which comes out of an interaction of our understanding of how the brain must work and what kinds of processes seem to be required

to account for desired behavior.

As can be seen, this list does not depend on specific discoveries from neuroscience. Rather, it depends on rather global considerations. Although none of these general properties of the brain tell us in any detai l how the brain functions to support cognitive phenomena, together they lead to aoo��fIAA9�Rhl how the brain works that

the fact that we normally assume that units communicate via numbers. These are sometimes associated with mean firing rates. In fact, of course, neurons produce spi kes and this spiking itself may have some computational significance (see Chapters 7 and 21 for discussions of the possible computational significance of neural spiking) . Another example of possibly important facts of neuroscience which have not played a role in our models is the diffuse pattern of communication which occurs by means of the dispersal of chemicals into various regions of the brain through the blood stream or otherwise . We generally assume that communication is point-to-point from one unit to another. However. we understand that diffuse communication can occur through chemical means and such communication may play an important role i <sup>n</sup> setting parameters and modulating the networks so that they can perform rather di fferent tasks in different situations. We have employed the idea of diffuse distribution of chemicals in our account of amnesia (Chapter 25) , but, in general , we have not otherwise integrated such assumptions into our models. Roughly, we imagine that we are studying networks in which there is a fixed setting of such parameters, but the situation may well be much more complex than that. (See Chapter 24 for some discussion of the role of norepinephrine and other neuro-

modulators.)

Most of our models are homogeneous with respect to the functioning of our units. Some of them may be designated as inhibitory and others as excitatory, but beyond that, they are rarely differentiated. We understand that there are perhaps hundreds of kinds of neurons (see Chapter 20) . No doubt each of these kinds play a slightly different role in the information processing system. Our assumptions in this regard are obviously only approximate. Simi larly, we understand that there are many different kinds of neurotransmitters and that there are different systems in which different of these neurotransmitters are dominant. Again, we have ignored this difference (except for excitatory and inhibitory connections) and presume that as more is understood about the information processing impl ications of such facts we will be

able to determine how they fit into our class of models.

It is also true that we have assumed a number of mechanisms that are not known to exist in the brain (see Chapter 20) . In general , we have postulated mechanisms which seemed to be required to achieve certain important functional goals, such as, for example, the development of internal representations in multilayer networks (see Chapter 8) . It is possible that these hypothesized mechanisms do exist in the brain but have not yet been recognized. In that sense our work could be considered as asource of hypotheses for neuroscience. It is also possible that we are correct about the computations that are performed, but that they are perfo�&JKf6MfM1 kind of neural mechanism

certain patterns of behavior which, in evolutionary time, have proven to be useful (such as sucking, reaching, or whatever) we can bui ld them in, but we leave the organism free to modify or completely reverse any of these behavioral predispositions.7 At the same time, we have the best of the empiricist view- namely, we place no a priori limitations on how the organism may adapt to its environment. We do, however, throw out the weakest aspect of the empiricist dogmanamely, the idea of the tabula rasa (or totally random net) as a starting point. The organism could start at whatever initial state its evolutionary

history prepared it for.

Perhaps, at this stage, all of this seems painfully obvious. It seems obvious to us too, and nevertheless, it gives us a new perspective on the nativism /empiricism issue. The issue is not what is the set of predetermined modules as some would suggest (cf. Fodor, 1983) . On this view it seems quite reasonable, we submit, that to the degree that there are modules, they are co-determined by the start state of the system (the genetic predisposition) and by the environment. (We take a module to be roughly a set of units whi ch are powerfully interconnected among themselves and relatively weakly connected to units outside of the set; of course, this concept admits all gradations of modularity, just as our view of schemata allows al l degrees of schematization of knowledge.) There is, on this view, no such thing as "hardwiring." Neither is there any such thing as "software." There are only connections. All connections are in some sense hardwired (in as much as they are physical entities) and all are software (in as much as they can be changed.) Thus, it may very well be that there is a part of the network prewired to deal with this or that processing task. If that task is not relevant in the organism's environment, that part of the network can be used for something else. If that part of the network is damaged, another part can come to play the role "normally" carried out by the damaged portion. These very properties have been noted characteristics of the brain since Hughlings-Jackson 's work in the late 19th century (e.g. , Jackson, 1869/1958) ; Jackson pointed them out as difficulties for the strict local izationist views then popular among students of the brain. Note too that our scheme allows for the organism to be especial ly sensitive to certain relationships (such as the relationship between nausea and eating, for which there might be stronger or more direct prewired

<sup>7</sup>Here agai n. our orga<sup>n</sup> ism oversimplifies a bit. It appears that some parts of the nervous system -particularly lower level. reOexive. or regulatory mechanisms -seem to be prewired and subject only to control by trainable modulatory conn<sup>e</sup>cti<sup>o</sup>nsto higher level. more adaptive mechanisms. rather than to be directly modifiable themselves: for discus-

connections) while at the same time allowing quite arbitrary associa-

tions to be learned.

Finally, it should be mentioned that all of the learning schemes that have been proposed for networks of the sort we have studied are incremental (cf. Chapters 7, 8, 11 , 18, 19, and 25) , and therefore as an organism moves from its primarily genetically predetermined start state to its primarily environmentally determined final state, it will pass through a sequence of more or less intermediate states. There will be a kind of trajectory through the space of possible networks. This trajectory will constitute the developmental sequence for the organism. To the degree that different individuals share the same genetics (start state) and to the degree that their environments are similar, they will pass through similar trajectories. It should also be said that since, in PDP systems, what is learned is a product of both the current state of the organism and the current pattern of inputs, the start state will have an important effect on what is learned and the shape of the network following any given set of experiences. However, the greater the amount of experience, the more independent the system should be from its start state and the more dependent it should be on the struc-

ture of its environment.

Of course, not all connections may be plastic-certainly, many subcortical mechanisms are considerably less plastic than cortical ones. Also, plasticity may not continue throughout life (see Chapter 24) . It would, of course, be a simple matter to suppose that certain connections are not modifiable. This is an issue about which our framework provides no answer. The major point is that there is no inconsistency between prewired, innate knowledge, and mutability and adaptability. We cannot resist making one more point about the nativism/empiricism issue. This is that our PDP account of innate knowledge seems to provide a rather plausible account of how we can come to have innate "knowledge." To the extent that stored knowledge is assumed to be in the form of explicit, inaccessible rules of the kind often postulated by linguists as the basis for linguistic competence (see Chapter 18), it is hard to see how it could "get into the head" of the newborn. It seems to us implausible that the newborn possesses elaborate symbol systems and the systems for interpreting them required to put these explicit, inaccessible rules to use in guiding behavior. On our account, we do not need to attribute such complex machinery. If the innate knowledge is simply the prewired connections, it is encoded from the start in just the right way to be of use by the processing

mechanisms.

Why Are People Smarter Than Rats?

Some have argued that since we claim that human cognition can be explained in terms of PDP networks and that the behavior of lower animals such as rats can also be described in terms of such networks we have no princi pled way of explaining why rats are not as smart as people. Given all of the above, the Question does seem a bit puzzl ing. We are not claiming, in any way, that people and rats and all other organisms start out with the same prewired hardware. People have much more cortex than rats do or even than other primates do; in particular they have very much more prefrontal and parietal cortex-more brain structure not dedicated to input /output -and presumably, this extra cortex is strategically placed in the brain to subserve just those functions that differentiate people from rats or even apes. A case in point is the part of the brain known as the angular gyrus. This part of the brain does not exist even in chimpanzees. It sits at the intersection between the language areas of the temporal lobe and the visual areas of the parietal lobe, and damage to this area produces serious deficits in language and in the mapping of words onto meanings. While it is possible that structures like the angular gyrus possess some special internal wiring that makes them fundamental ly different, somehow, in the kinds of cognitive operations they perform , their cytoarchitecture is not markedly different from that of other parts of the brain (see Chapters 20 and 21). Thus it seems to us quite plausible that some of the differences between rats and people lie in the potentiality for forming connections that can subserve the vital functions of language and thought

that humans exhibit and other animals do not.

But there must be another aspect to the difference between rats and people as wel l. This is that the human environment includes other people and the cultural devices that they have developed to organize their thinking processes. Some thoughts on how we imagine these cultural devices are exploited in higher forms of intelligent behavior are

presented in Chapter 14.

Conscious Knowledge and Explicit Reasoning

There may be cognitive scientists who accept some or al l of what we have said up to this point, but still feel that something is missing, namely, an account of how we guide behavior using explicit, conscious knowledge, how we reason from what we know to new conclusions based on that knowled�'O�9r1gRRW1 'NIBtQR9t a path through a problem

PART II

BASIC MECHANISMS

The chapters of Part II represent explorations into specific architectures and learning mechanisms for PDP models. These explorations proceed through mathematical analysis coupled with results from simulations. The major theme which runs through all of these explorations is a focus on the learning problem. How can PDP networks evolve to perform the kinds of tasks we require of them? Since one of the primary features of PDP models in general is their ability to self-modify, these studies form an important base for the application of these

models to specific psychological and biological phenomena.

In Chapter 5, Rumelhart and Zipser begin with a summary of the history of early work on learning in parallel distributed processing systems. They then study an unsupervised learning procedure called competitive learning. This is a procedure whereby feature detectors capable of discriminating among the members of a set of stimulus input patterns evolve without a specific teacher guiding the learning. The basic idea is to let pool s of potential feature detector units compete among themselves to respond to each stimulus pattern. The winner within each pool-the one whose connections make it respond most strongly to the pattern-then adjusts its connections slightly toward the pattern that it won. Several earlier invest igators have considered variants of the competitive learning idea (e.g., Grossberg, 1976; von der Malsberg, 1973). Rumelhart and Zipser show that when a competitive network is trained through repeated presentations of members of a set of patterns, each unit in a pool CO�}P;d"RPic?�at���P patterns with a particular

attribute or property are presented. If there are two units in a pool, each comes to respond to opposite values of a binary feature which is useful in describing the stimulus set. If there are three units in the pool, each unit comes to respond to a value of a trinary feature, etc. It is shown through simulations and mathematical analysis that the competitive learning system can serve as a basis for the development of

useful pattern descriptions.

Chapters 6 and 7 describe Smolensky's harmony theory and Hinton and Sejnowski's Boltzmann machine, respectively. These approaches were developed at the same time, and they have much in common. Both harmony theory and Boltzmann machines employ binary units whose values are determined probabilistically according to the Boltzmann equation. Each employs simulated annealing in which the temperature of the Boltzmann equation is moved slowly to zero as the system relaxes into its solution state where it finally freezes. Both systems apply mathematical formulations borrowed from physics to their

systems to describe and analyze their behavior.

In spite of these similarities, the two systems were developed from very different perspectives. The similarities arose largely because both systems tapped mathematical physics as a tool for formalizing their ideas. Smolensky's harmony theory grew from an attempt to formalize the notion of schema and the ideas of schema theory. Hinton and Sejnowski's Boltzmann machine is based on the idea that stochastic units can be used as a mechanism of search-for finding globally good states of networks through simulated annealing. It combines insights on simulated annealing from Kirkpatrick, Gelatt, and Vecchi (1983) with the proof by Hopfield (982) that there is a global energy function that can be locally minimized through a process of asynchronously

updating individual units.

Chapter 6 provides a mathematical development of harmony theory and shows how a symbolic level of description can be seen as emerging from interactions among the individual processing units in harmony theory. It shows how harmony theory can be applied to a variety of phenomena, including intuitive problem solving and aspects of perception. It also provides a useful description of the mathematical relationships among harmony theory, Boltzmann machines, and the related

mechanisms studied by S. Geman and D. Geman (1984).

Chapter 7 focuses on the issue of learning in Boltzmann machines. One of the most important contributions of the work on Boltzmann machines is the development of the two phase (wake/sleep) learning procedure. Hinton and Sejnowski show that if a Boltzmann machine runs under the influence of environmental inputs for a while and then runs "freely" -without inputs from the environment-there is a very simple learning rule which will allow the Boltzmann machine to pick up Copyrighted Material

environmental regularities and develop its own internal representations for describing those regularities. The major part of Chapter 7 is an

analysis of this learning procedure.

Chapter 8 is the study of still another learning procedure. In this chapter, Rumelhart, Hinton, and Williams show that it is possible to develop a generalization of the delta rule described in Chapter 2 so that arbitrary multilayered networks of units can be can be trained to do interesting tasks . Using this learning rule, the system can learn to associate arbitrary input/ output pairs and in this way can learn to compute arbitrary input/output functions . The generalized delta rule is shown to provide a method of modifying any weight in any network, based on locally available information, so as to implement a gradient descent process that searches for those weights that minimize the error at the output units. Further, simulation work presented in the chapter shows that the problems of local minima often associated with gradient des-

cent and other hill- climbing methods are suprisingly rare.

In general , the chapters in this section demonstrate that the barriers to progress in understanding learning in networks of simple neuron-like units have begun to crumble. There are still deep problems that remain unsolved, but the learning mechanisms described in these chapters make several inroads into some of the most challenging aspects of the

theory of parallel distributed processing.

CHAPTER S

Feature Discovery by Competitive Learning

D. E. RUMELHART and D. ZIPSER

This chapter reports the results of our studies with an unsupervised learning paradigm that we call competitive learning. We have examined competitive learning using both computer simulation and formal analysis and have found that when it is applied to parallel networks of neuron-like elements, many potentially useful learning tasks can be accomplished. We were attracted to competitive learning because it seems to provide a way to discover the salient, general features which can be used to classify a set of patterns. The basic components of the

competitive learning scheme are:

• Start with a set of units that are all the same except for some randomly distributed parameter which makes each of them

respond slightly differently to a set of input patterns.

• Limit the "strength" of each unit.

• Allow the units to compete in some way for the right to

respond to a given subset of inputs.

The net result of correctly applying these three components to a learning paradigm is that individual units learn to specialize on sets of

similar patterns and thus become "feature detectors" or "pattern classifiers." In addition to Frank Rosenblatt, whose work will be discussed below, several others have exploited competitive learning in one form or another over the years. These include von der Malsburg (973), Grossberg (976), Fukushima (975), and Kohonen (982). Our analyses differ from many of these in that we focus on the development of feature detectors rather than pattern classification. We address

these issues further below.

One of the central issues in the study of the processing capacities of neuron-like elements concerns the limitations inherent in a one-level system and the difficulty of developing learning schemes for multilayered systems. Competitive learning is a scheme in which important features can be discovered at one level that a multilayer system can use to classify pattern sets which cannot be classified with a single level

system.

Thirty-five years of experience have shown that getting neuron-li ke elements to learn some easy things is often quite straightforward, but designing systems with powerful general learning properties is a difficult problem, and the competitive learning paradigm does not change this fact. What we hope to show is that competitive learning is a powerful strategy which, when used in a variety of situations, greatly expedites some difficult tasks. Since the competitive learning paradigm has roots which go back to the very beginnings of the study of artificial learning devices, it seems reasonable to put the whole issue into historical perspective. This is even more to the point, since one of the first simple learning devices, the perceptron, caused great furor and debate, the

reverberations of which are still with us.

In the beginning, thirty-five or forty years ago, it was very hard to see how anything resembling a neural network could learn at all, so any example of learning was immensely interesting. Learning was elevated to a status of great importance in those days because it was somehow uniquely associated with the properties of animal brains. After McCulloch and Pitts (943) showed how neural-like networks could compute, the main problem then facing workers in this area was to

understand how such networks could learn.

The first set of ideas that really got the enterprise going were contained in Donald Hebb's Organization oj Behavior (I 949). Before Hebb's work, it was believed that some physical change must occur in a network to support learning, but it was not clear what this change could be. Hebb proposed that a reasonable and biologically plausible change would be to strengthen the connections between elements of the network only when both the presynaptic and postsynaptic units were active simultaneously. The essence of Hebb's ideas still persists today in many learning paradigms. The details of the rules for changing weight Copyrighted Material

may be different, but the essential notion that the strength of connections between the units must change in response to some function of the cor<sup>r</sup>ela<sup>t</sup>ed activity of the connected units still dominates learning

models.

Hebb's ideas remained untested speculations about the nervous system until it became possible to build some form of simulated network to test learning theories. Probably the first such attempt occurred in 1951 when Dean Edmonds and Marvin Minsky built their learning machine. The flavor of this machine and the milieu in which it operated is captured in Minsky's own words which appeared in <sup>a</sup> wonderful New Yorker profile of him by Jeremy Bernstein (1981):

> In the <sup>s</sup>ummer of 1951 Dean Edmonds and I went up to Harvard and bui lt our machine. It had three hundred tubes and a lot of motors. It needed some automaticelectric clutches, which we machined ourselves. The memory of the machine was stored in the positions of its control knobs, 40 of them, and when the machine was learning, it used the clutches to adjust its own knobs. We used a surplus gyropilot from a B24

bomber to move the clutches. (p. 69)

This machine actually worked and was so fascinating to watch that

Minsky remembers:

We sort of quit science for <sup>a</sup>whil <sup>e</sup>to watch the machine. We were amazed that it could have several activities going on at once in this little nervous system. Because of the random wiring it had a sort of fail <sup>s</sup>afe characteristic. If one of the neurons wasn't working, it wouldn't make much difference and with nearly three hundred tubes, and the thousands of <sup>c</sup>onnections we had soldered there would usual ly be something wrong somewhere. . I don't think we ever debugged our machine completely, but that didn't matter. By having thi<sup>s</sup> crazy random design it w<sup>a</sup>s almost sure to work no matter how

you built it. (p. 69)

In fact , the functioning of this machine apparently stimulated Minsk<sup>y</sup> sufficiently to write his PhD thesis on a proble<sup>m</sup>related to learning (Minsky, 1954). The whole idea must have generated rather wide interest; von Neumann, for example, was on Minsky's PhD committee and gave him encouragem<sup>e</sup>nt. Although Minsky was perhaps the first on the scene with a learning machine, the real beginnings of meaningful neuron-li ke network learning can probably be traced to the work of Frank Rosenblatt, a BE?� �g�1 of Science classmate of

Minsky's. Rosenblatt invented a class of simple neuron-l ike learning networks which he called perceptrons. In his book, Principles of Neurodynamics (1962), Rosenblatt brought together all of his results on perceptrons. In that book he gives a particularly clear description of what

he thought he was doing:

Perceptrons are not intended to serve as detai led copies of any actual nervous system. They're simplified networks, designed to permit the study of lawful relationships between the organization of a nerve net, the organization of its environment, and the "psychological" performances of which it is capable. Perceptrons might actually correspond to parts of more extended networks and biological systems� in this case, the results obtained will be directly appl icable. More li kely they represent extreme simplifications of the central nervous system, in which some properties are exaggerated and others suppressed. In this case, successive perturbations and refinements of the system

may yield a closer approximation.

The main strength of this approach is that it permits meaningful questions to be asked and answered about particular types of organizations, hypothetical memory mechanisms, and neural models. When exact analytical answers are unobtainable, experimental methods, either with digital simulation or hardware models, are employed. The model is not the terminal result, but a starting point for exploratory analysis of its

behavior. (p. 28)

Rosenblatt pioneered two techniques of fundamental importance to the study of learning in neural-like networks: digital computer simulation and formal mathematical analysis, although he was not the first to simulate neural networks that could learn on digital computers (cf.

Farley & Clark, 1954).

Since the paradigm of competitive learning uses concepts that appear in the work of Rosenblatt, it is worthwhile reviewing some of his ideas in this area. His most influential result was the "perceptron learning

theorem" which boldly asserts:

Gi ven an elementary IX -percept ron , a stimulus world W, and any classification C (W) for which a solution exists� let all stimuli in W occur in any sequence, provided that each stimulus must reoccur in finite time� then beginning from an arbitrary initial state, an error correction procedure wi ll always

yield a solution to C (W) in finite time, . " (p. 596)

As it turned out, the real problems arose out of the phrase "for which <sup>a</sup>

<sup>s</sup>olution exists" -more about this later.

Less widely known is Rosenblatt's work on what he called " spontaneous learning." All network learning models require rules which tell how to present the stimuli and change the values of the weights in accordance with the re<sup>s</sup>pons<sup>e</sup>of the model. These rules can be characterized as forming a spectrum, at one end of which is learning with an errorcorrecting teacher, and at the other is completely spontaneous, unsupervised discovery . In between is a continuum of rules that depend on <sup>m</sup>anipulating the content of the input stimulus stream to bring about learning. These intermediate rules are often referred to as "forced learning." Here we are concerned primari ly with attempts to design a perceptron that would discover something interesting without a teacher because this is similar to what happens in the competitive learning case. In fact, Rosenblatt was able to build a perceptron that was able to spontaneously dichotomize a random sequence of input patterns into classes such that the members of a single class were similar to each other, and different from the members of the other class. Rosenblatt realized that any randomly initialized perceptron would have to dichotomize an arbitrary input pattern stream into a "I-set," consisting of those patterns that happened to produce a response of I, and a "O-set," consisting of those that produced a response of O. Of course one of these sets could be empty by chance and neither would be of <sup>m</sup>u<sup>c</sup>h interest in general. He reasoned that if a perceptron could reinforce these sets by an appropriate rule based only on the perceptron's spontaneous response and not on a teacher's error correction, it might eventually end up with a dichotomization in which the members of each set were more like each other than like the members of the opposite set. What was the appropriate rule to use to achieve the desired dicotomization? The first rule he tried for these perceptrons, which he called C -type, was t<sup>o</sup> increment weights on l ines active with patterns in the I-set, and decrement weights on lines active with patterns in the O-set. The idea was to force a dichotomization into sets whose members were similar in the <sup>s</sup>ense that they activated overlapping subsets of lines. The· results were disastrou<sup>s</sup> . Sooner or later all the input patterns were classified in one <sup>s</sup>et. There was no di chotomy but there was stability. Once one of the

sets won, it remained the victor forever.

Not to be daunted, he examined why this undesirable result occurred and realized that the problem lay in the fact that since the weights could grow without limit, the set that initially had a majority of the patterns would receive the majority of the reinforcement. This meant that weights on lines which could be activated by patterns in both sets would grow to infinite magnitudes in favor of the majority set, which in turn would lead to the captuBbPYr1BiHBaw!81�»Sfns by the majority set and

ultimate total victory for the maJorIty. Even where there was initial equality between the sets, inevitable fluctuations in the random presentation of patterns would create a majority set that would then go on to win. Rosenblatt overcame this problem by introducing mechani sms to limit weight growth in such a way that the set that was to be positively reinforced at active lines would compensate the other set by giving up some weight from all its lines. He called the modi fied perceptrons C. An example of a C rule is to lower the magnitude of all weights by a fixed fraction of their current value before specifically incrementing the magni tude of some of the weights on the basis of the response to an input pattern. This type of rule had the desired result of making an equal dichotomy of patterns a stable rather than an unstable state. Patterns in each of the sets were similar to each other in the sense that they depended on simi lar sets of input lines to produce a response. In Rosenblatt's initial experiment, the main feature of similarity was not so much the shape of the patterns involved, but their location on the retina. That is , his system was able to spontaneously learn something about the geometry of its input line arrangement. Later, we will examine this important property of spontaneous geometry learning in considerable detail. Depending on the desired learning task, it can be either a

boon or a nuisance.

Rosenblatt was extremely enthusiastic about his spontaneous learning results. In fact, his response can be described as sheer ecstasy. To see what he thought about his achievements, consider his claim

(Rosenblatt, 1959):

It seems clear that the class C perceptron introduces a new kind of information processing automaton: For the first time, we have a machine which is capable of having original ideas . As an analogue of the biological brain, the perceptron, more precisely, the theory of statistical separabi lity, seems to come closer to meeting the requirements of a functional explanation of the nervous system than any system previously proposed.

(p. 449)

Although Rosenblatt's results were both interesting and significant, the claims implied in the above quote struck his contemporaries as unfounded. What was also significant was that Rosenblatt appeared to be saying that the type of spontaneous learning he had demonstrated was a property of perceptrons, which could not be repl icated by ordinary computers. Consider the fol lowing quote from the same source:

> As a concept, it would seem that the perceptron has established, beyond doubt, the feasibi lity and principle of

non-human systems which may embody human cognitive functions at a level far beyond that which can be achieved through present day automatons. The future of information processing devices which operate on statistical, rather then logical princi-

ples seems to be clearly indicated. (p. 449)

It is this notion of Rosenblatt's-that perceptrons are in some way superior to computers-that ignited a debate in artificial intel ligence that had significant effects on the development of neural-like network models for both learning and other cognitive processes. Elements of the debate are sti ll wi th us today in arguments about what the brain can do that computers can't do. There is no doubt that this was an important issue in Rosenblatt's mind, and almost certai nly contri buted to the acrimonious debate at that time. Consider the following statement by Rosenblatt made at the important conference on Mechanization of

Thought Processes back in 1959:

Computers seem to share two main functions with the brain: (a) Decision making, based on logical rule, and (b) control, again based on logical rules. The human brain performs these functions, together with a third: interpretation of the environment. Why do we hold interpretation of the environment to be so important? The answer, I think, is to be found in the laws of thermodynamics. A system with a completely self contained logic can never spontaneously improve its abi lity to organize, and to draw valid conclusions from information. (Rosenblatt,

1959, p. 423)

Clearly in some sense, Rosenblatt was saying that there were things that the brain and perceptrons, because of their statistical properties, could do which computers could not do. Now this may seem strange since Rosenblatt knew that a computer program could be written that would simulate the behavior of statistical perceptrons to any arbitrary degree of accuracy. Indeed, he was one of the pioneers in the application of digital simulation to this type of problem. What he was actual ly referring to is made clear when we examine the comments of other partici pants at the conference, such as Minsky (I959) and McCarthy (I 959), who were using the symbol manipulating capabilities of the computer to directly simulate the logical processes involved in decision making, theorem proving, and other intellectual activities of this sort. Rosenblatt believed the computer used in this way would be inadequate to mimic the brain's true intellectual powers. This task, he thought, could only be accompl ished if the computer or other electronic devices were used to simulate perceptrons. We can summarize these di vergent Copyrighted Material

points of view by saying that Rosenblatt was concerned not only with what the brain did, but with how it did it, whereas others, such as Minsky and McCarthy, were concerned with simulating what the brain did, and didn't real ly care how it was done. The subsequent history of AI has shown both the successes and failures of the standard AI approach. We sti ll have the problems today, and it's still not clear to what degree computational strategies similar to the ones used by the brain must be

employed in order to simulate its performance.

In addition to producing fertilizer, as all debates do, this one also stimulated the growth of some new results on perceptrons, some of which came from Minsky. Rosenblatt had shown that a two layer perceptron could carry out any of the 22N possible classifications of N binary inputs; that is, a solution to the classification problem had always existed in principle. This result was of no practical value however, because 2N units were required to accomplish the task in the completely general case. Rosenblatt's approach to this problem was to use a much smaller number of units in the first layer with each unit connected to a small subset of the N inputs at random. His hope was that this would give the percept ron a high probability of learning to carry out classifications of interest. Experiments and formal analysis showed that these random devices could learn to recognize patterns to a significant degree but that they had severe limitations. Rosenblatt (I 962)

characterized his random perceptron as follows:

It does not generalize well to similar forms occurring in new positions in the retinal field, and its performance in detection experiments, where a familiar figure appears against an unfamiliar background, is apt to be weak. More sophisticated psychological capabilities, which depend on the recognition of topological properties of the stimulus field, or on abstract relations between the components of a complex image, are lacking.

(pp. 191-192)

Minsky and Papert worked through most of the sixties on a mathematical analysis of the computing powers of perceptrons with the goal of understanding these limitations. The results of their work are available in a book cal led Perceptrons (Minsky & Papert, 1969). The central theme of this work is that parallel recognizing elements, such as perceptrons, are beset by the same problems of scale as serial pattern recognizers. Combinatorial explosion catches you sooner or later, although sometimes in di fferent ways in parallel than in serial. Minsky and Papert's book had a very dampening effect on the study of neuron-l ike networks as computational devices. Minsky has recently

I now bel ieve the book was overkill. . So after being irritated with Rosenblatt for overclaiming and diverting all those people along a false path, I started to real ize that for what you get out of it - the kind of recognition it can do-it is such a simple machine that it would be astonishi ng if nature did not make use

of it somewhere. (Bernstein, 1981, p. 103)

Perhaps the real lesson from all this is that it real ly is worthwhile trying

to put things in perspective.

Once the problem of scale has been understood, networks of neuron-like elements are often very useful in practical problems of recognition and classification. These networks are somewhat analogous to computers, in that they won't do much unless programmed by a clever person; networks, of course, are not so much programmed as designed. The problem of finding networks of practical size to solve a particular problem is challenging because relatively small changes in network design can have very large effects on the scale of a problem. Consider networks of neuron-like units that determine the parity of their N bi nary inputs (see Figure 1). In the simple perceptrons studied by Minsky and Papert, units in the first layer output 1 only if all their inputs are 1 and output 0 otherwise. This takes 2N units in the first layer, and a single linear threshold unit with a fan-in of 2N in the second layer, to determine parity. If the units in the first layer are changed to linear threshold elements, then only N of them are required, but all must have a fan-in of N. If we allow a multilayer network to do the job, then about 3N units are needed, but none needs a fan-in of more than 2. The number of layers is of order log2N. The importance of all this to the competitive learning paradigm, or any other for that matter, is that no network can learn what it is not capable of doing in principle. What any particular network can do is dependent on its structure and the computational properties of its component elements. Unfortunately, there is no canonical way to find the best network or to determine what it will learn, so the whole enterprise still has

much of the flavor of an experimental science.

THE COMPETITIVE LEARNING MECHANISM

Paradigms of Learning

It is possible to classify learning mechanisms in several ways. One useful classification is in. terms of e learning paradigm in which the Gopynghted Malenal

A B

1 -1 cp

c o

=�

FIGURE I. A: Parity network from Minsky and Papert (1969). Each.p unit has an output of I only if all of its inputs are I. 1: is a linear threshold unit with threshold of 0, i.e., like all <sup>t</sup>he other linear threshold units in the figure, it fires only when the sum of its weighted inputs is greater than the threshold. This and all the other networks signal odd parity with a I in the rightmost unit of the network. B: Parity network made from two layers of linear threshold units. C: Three-unit network for determining the parity of <sup>a</sup> pair of inputs. D: Two-layer network using the subnetwork described in (C). In general, the number of P-units is of order N and the number of layers is of order log2N.

model is supposed to work. There are at least four common learning

paradigms in neural-li ke processing systems:

• Auto Associator. In this paradigm a set of patterns are repeatedly presented and the system is supposed to "store" the patterns. Then, later, parts of one of the original patterns or possibly a pattern similar to one of the original patterns is presented, and the task is to "retrieve" the original pattern through a kind of pattern completion procedure. This is an auto-association process in which a pattern is associated with itself so that a degraded version of the original pattern can act

as a retrieval cue.

• Pattern Associator. This paradigm is really a variant on the auto-association paradigm. A set of pairs of patterns are repeatedly presented. The system is to learn that when one member of the pair is presented it is supposed to produce the other. In this paradigm one seeks a mechanism in which an essentially arbitrary set of input patterns can be paired with an arbitrary set

of output patterns.

• Classification Paradigm. The classification paradigm also can be considered as a variant on the previous learning paradigms, although the goals are sufficiently different and it is sufficiently common that it deserves separate mention. In this case, there is a fixed set of categories into which the stimulus patterns are to be classified. There is a training session in which the system is presented with the stimulus patterns along with the categories to which each stimulus belongs. The goal is to learn to correctly classify the stimuli so that in the future when a particular stimulus or a slightly distorted version of one of the stimuli is presented, the system will classify it properly. This is the typical paradigm in which the perceptron is designed to operate and in which the perceptron convergence theorem is

proved.

• Regularity Detector. In this paradigm there is a population of stimulus patterns and each stimulus pattern, Sk' is presented with some probability Pk. The system is supposed to discover statistically salient features of the input population. Unlike the classification paradigm, there is no a priori set of categories into which the patterns are to be classified; rather, the system must develop its own featural representation of the input stimuli which captures the most salient features of the population of

input patterns.

Layer 3 Inhibitory Clusters

Layer 2 Inhibitory Clusters

> Layer 1 Input Units

INPUT PATTERN

Excitatory Connections

Excitatory Connections

FIGURE 2. The architecture of the competitive learning mechanism. Competitive learning takes place in a context of sets of hierarchically layered units. Units are represented in the diagram as dots. Units may be active or inactive. Active units are represented by filled dots, inactive ones by open dots. In general, a unit in a given layer can receive inputs from aJl of the units in the next lower layer and can project outputs 10 all of the units in the next higher layer. Connections between layers are excitatory and connections within layers are inhibitory. Each layer consists of a set of cluster<sup>s</sup>of mutually inhibitory units. The units within a cluster inhibit one another in such a way that only one unit per <sup>c</sup>luster may be active. We think of the configuration of active units on any given layer as representing the input pattern for the next higher level. There can be an arbitrary number of such layers. A given cluster contains a fixed <sup>n</sup>umber of units, but different

• Every element in every cluster receives inputs from the same

lines.

• A unit learns if and only if it wins the competition with other

units in its cluster.

• A stimulus pattern S. consists of a binary pattern in which each element of the patteln is either active or inactive. An active element is assigned the value I and an inactive element is

assigned the value O.

• Each unit has a fixed amount of weight (all weights are positi ve) which is distributed among its input lines. The weight on the line connecting unit i on the lower (or input) layer to unitj on the upper layer, is designated wi}. The fixed total amount of weight for unit j is designated I, wi} = 1. A unit learns by i shifting weight from its inactive to its active input lines. If a unit does not respond to a particular pattern, no learning takes place in that unit. If a unit wins the competition, then each of its input lines give up some proportion g of its weight and that weight is then distributed equal ly among the active input lines. 2

More formally, the learning rule we have studied is:

if unit j loses on stimulus k

if unit j wins on stimulus k

where Cik is equal to 1 if in stimulus pattern Sk, unit i in the lower layer is active and zero otherwise, and nk is the number

of acti ve units in pattern Sk (thus nk = I,Cik).

Figure 3 illustrates a useful geometric analogy to this system. We can consider each stimulus pattern as a vector. If all patterns contain the same number of active lines, then all vectors are the same length and each can be viewed as a point on an N-dimensional hypersphere,

<sup>2</sup>This learning rule was proposed by von der Malsburg (1973). As Grossberg (1976) points out, renormalization of the weights is not necessary. The same result can be obtained by normalizing the input patterns and then assuming that the weights approach the values on the input lines. Normalizing weights is simpler to implement than normalizing pallerns, so we chose that option. For most of our experiments, however, it does not mailer which of these two rules we chose since all pallerns were of the same

Each x in Figure 3A represents a particular pattern. Those patterns that are very simi lar are near one another on the sphere; those that are very different wi ll be far from one another on the sphere. Now note that since there are N input lines to each unit in the upper layer, its weights can also be considered a vector in N-dimensional space. Since all ul'lits have the same total quantity of weight, we have N-dimensional vectors of approximately fi xed length for each unit in the cluster. 3 Thus, properly scaled, the weights themselves form a set of vectors which (approximately) fall on the surface of the same hypersphere. In Figure 38, the o's represent the weights of two units superimposed on the same sphere with the stimulus patterns. Now, whenever a stimulus pattern is presented, the unit which responds most strongly is simply the one whose weight vector is nearest that for the stimulus. The learning rule specifies that whenever a unit wins a competition for a stimulus pattern, it moves a percentage g of the way from its current location toward the location of the stimulus pattern on the hypersphere. Now, suppose that the input patterns fell into some number, M, "natural" groupings. Further, suppose that an inhibitory cluster receiving inputs from these stimuli contained exactly M units (as in Figure 3C). After sufficient training, and assuming that the stimulus groupings are sufficiently distinct, we expect to find one of the vectors for the M units placed roughly in the center of each of the stimulus groupings. In this case, the units have come to detect the grouping to which the input patterns belong. In this sense, they have "discovered" the

structure of the input pattern sets.

Some Features of Competitive Learning

There are several characteristics of a competitive learning mechanism that make it an interesting candidate for further study, for example:

• Each cluster classifies the stimulus set into M groups, one for each unit in the cluster. Each of the units captures roughly an equal number of stimulus patterns. It is possible to consider a cluster as forming an M-ary feature in which every stimulus pattern is classified as having exactly one of the M possible

3 It should be noted that this geometric interpretation is only approximate. We have used the constraint that 1: Wu = 1 rather than the constraint that 1: wJ = 1. This latter I I constraint would ensure that all vectors are in fact the same length. Our assumption only

assures that they will be approximately the same length.

values of this feature. Thus, a cluster containing 2 units acts as a binary feature detector. One element of the cluster responds when a particular feature is present in the stimulus pattern , oth-

erwise the other element responds.

• If there is structure in the stimulus patterns, the units will break up the patterns along structurally relevant lines. Roughly speaking, this means that the system will find clusters if they are there. (A key problem, which we address below, is specifying the nature of the structure that this system discovers.)

• If the stimuli are highly structured, the classi fications are highly stable. If the stimuli are less well-structured, the classifications are more variable, and a given stimulus pattern will be responded to first by one and then by another member of the cluster. In our experiments, we started the weight vectors in random directions and presented the stimuli randomly. In this case, there is rapid movement as the system reaches a relatively stable configuration (such as one with a unit roughly in the center of each cluster of stimulus patterns). These configurations can be more or less stable. For example , if the stimulus points don't actually fall into nice clusters , then the configurations will be relatively unstable, and the presentation of each stimulus will modi fy the pattern of responding so that the system will undergo continual evolution. On the other hand, if the stimulus patterns fall rather nicely into clusters, then the system will become very stable in the sense that the same units

will always respond to the same stimuli.4

• The particular grouping done by a particular cluster depends on the starting value of the weights and the sequence of stimulus patterns actually presented. A large number of clusters , each receiving inputs from the same input lines can, in general, classify the inputs into a large number of different groupings , or alternatively, discover a variety of independent features present in the stimulus population. This can provide a kind of coarse

coding of the stimulus patterns. 5

<sup>4</sup>G<sup>r</sup>ossber<sup>g</sup>(976) <sup>h</sup>as addressed this problem in hi<sup>s</sup>very similar system. He has proved that if the patterns are sufficiently sparse, and/ or when there are enough units in the cluster, <sup>t</sup>hen a system such as this will fi<sup>n</sup>d a perfectly stable classification. He also points out that when these conditions don't hold, the classification can be unstable. Most of our work is with cases in which there is no perfectly stable classification and the number of patterns is much larg.er than the number of units in the inhibitory clusters. (;opyrighted Material

wins, P (li nei = 11 unitj wins) . Thus, if all patterns are of the same size , i.e., nk = n for all k, then the weight wi} becomes proportional to

the probability that line i is active given unit j wins. That is,

wi} - ! P ( Iinei = 11 unitj wins ).

We are now in a position to speci fy the response, at equilibri um, of unit j when stimulus S, is presented. Let Ol jI be the input to unit j in the face of stimulus S, . This is simply the sum of weights on the acti ve

input lines. This can be written

1: Pk Cik VJk k nk LPk vjk k

which implies that at equi librium

Oljl <sup>=</sup>

where 'Ii represents the overlap between stimulus I and stimulus i,

',,' \_\_ � Cki Ck' L.

Thus, at equi librium a unit responds most strongly to

k ni

patterns that overlap other patterns to which the unit responds and responds most weakly to patterns that are far from patterns to which it responds. Finally, it should be noted that there is another set of

restrictions on the value of Vj

k - the probabi lity that unit j responds to

stimulus Sk

' In fact , the competitive learning rule we have studied has

the further restriction that

\_1 1 Oljk > Olik for all i,cj

Vjk - 0 otherwise.

Thus, in general , there are many solutions to the equilibrium equations described above. The competitive learning mechanisms can only reach those equi librium states in which the above-stated relationships

between the Vjk and the Ol jk also hold.

Whenever the system is in a state in which, on average , the weights are not changing, we say that the system has reached an equilib,ium state. In such a state the values of Ol Jk become relati vely stable, and therefore, the values of Vik become stable. When this happens, the system always responds the same way to a particular stimulus pattern. However, it is possible that the weights will be pushed out of Copyrighted Material

equilibrium by an unfortunate sequence of stimuli. In this case, the system can move toward a new equilibrium state (or possibly back to a previous one) . Some equi librium states are more stable than others in the sense that the "ik become very unlikely to change values for long periods of time. In particular, this wi ll happen whenever the largest otjk is much larger than any other otik for all stimulus patterns Sk ' In this case, small movements in the weight vector of one of the units is very unli kely to change which unit responds to which stimulus pattern. Such equilibrium states are said to be highly stable. We should expect, then, that after it has been learning for a period of time, the system wi ll spend most of its time in the most highly stable of the equilibrium states. One good measure of the stability of an equil ibrium state is given by the average amount by which the input to the winning units is greater than the response of all of the other units averaged over all patterns and all units in a cluster. This measure is given by T below:

T = LPkl>jk (otjk - otik ) '

k j ,i

The larger the value of T, the more stable the system can be expected to be and the more time we can expect the system �o spend in that state. Roughly, if we assume that the system moves into states which maximize T, we can show that this amounts to maximizing the overlap among patterns within a group while minimizing the overlap among patterns between groups. In the geometric analogy above, this will occur when the weight vectors point toward maximally compact stimulus regions that are as distant as possible from other such regions.

SOME EXPERIMENTAL RESULTS

Dipole Experiments

The essential structure that a competiti ve learning mechanism can discover is represented in the overlap of stimulus patterns. The simplest stimulus population in which stimulus patterns can overlap with one another is one constructed out of dipoles-stimulus patterns consisting of exactly two active elements and the rest inactive. If we have a total of N input units there are N(N-J)/2 possible dipole stimuli. Of course, if the actual stimulus population consists of al <sup>l</sup>N(N-J) / 2 possibi lities, there is no structure to be discovered. There are no clusters for our units to point at (unless we have one unit for each of the possible stimUl i, in which case we can point a weight vector at each of the Copyrighted Material

FIGURE 5. The architecture of a co<sup>m</sup>peti<sup>t</sup>ive learning system with 16 input units and

one cluster of size two in the second layer.

that Unit 1 was the winner, the narrow line indicates that Unit 2 was the winner. It should be noted, therefore, that two unfilled circles must always be joi ned by a narrow line and two fi lled circles must always be joined by a wide line. The reason for this is that if a particular unit has more weight on both of the active lines then that unit must win the competition. The results clearly show that the weights move from a rather chaotic initial arrangement to an arrangement in which essentially al l of those on one side of the grid are fi lled and all on the other side are unfil led. The border separating the two halves of the grid may be at any orientation , but most often it is oriented vertical ly and hori zontal ly, as shown in the upper two examples. Only rarely is the orientation diagonal, as in the example in the lower right-hand grid. Thus, we have a case in which each unit has chosen a coherent half of the grid to which they respond. It is important to real ize that as far as the competitive learn ing mechanism is concerned the si xteen input Copyrighted Material

A

J

o 400

B

o 400

c

50 400

FIGURE 6. Relative weight values for the two members of the inhibitory cluster. A: The results for one run with the dipole stimuli defined over a two-dimensional grid. The left-hand grid shows the relative values of the weights initially and the right-hand grid shows the relative values of the weights after 400 trials. A filled circle means that Uni<sup>t</sup><sup>1</sup> had the larger weight on the corresponding i<sup>n</sup>pu<sup>t</sup> . An unfilled circle means that Unit <sup>2</sup> had the larger weight. A heavy line connecting two circles means that Unit 1 responded to the stimulus pattern consisting of the activation of the two circles, and a light line means that Unit 2 won the corresponding pattern. In this case the system has divided the grid horizontally. B: The results for a second run under the same conditions. In this case the system has divided the grid horizontally. C: The results for a third run. In this case the left-hand grid represents the state of the system after 50 trials. Here the grid

lines are unordered. The two-dimensional grid-like arrangement exists only in the statistics of the population of stimulus patterns. Thus, the system has discovered the dimensional structure inherent in the stimulus population and has devised binary feature detectors that tell which half of the grid contains the stimulus pattern. Note, each unit responds to roughly half of the stimulus patterns. Note also that while some units break the grid vertically, some break the grid horizontally, and some break it diagonally; a combination of several clusters offers a rather

more precise classification of a stimulus pattern.

In other experiments, we tried clusters of other sizes. For example, Figure 7 shows the results for a cluster of size four. It shows the initial configuration and its sequence of evolution after 100, 200, 400, 800, and after 4000 training trials. Again, initially the regions are chaotic. After training, however, the system settles into a state in which stimuli in compact regions of the grid are responded to by the same units. It can be seen, in this case, that the trend is toward a given unit responding to a maximally compact group of stimuli. In this experiment, three of the units settled on compact square regions while the remaining one settled on two unconnected stimulus regions. It can be shown that the state into which the system settled does not quite maximize the value

T, but does represent a relatively stable equilibrium state.

In the examples discussed thus far, the system, to a first approximation, settled on a highly compact representation of the input patterns in which all patterns in a region are captured by one of the units. The grids discussed above have all been two-dimensional . There is no need to restrict the analysis to a two-dimensional grid. In fact, a two-unit cluster will, essentially, pass a plane through a space of any dimensionality. There is a preference for planes perpendicular to the axes of the spaces. Figure 8 shows a typical result for the system learning a three-dimensional space. In the case of three dimensions , there are three equally good planes which can be passed through the space and, depending on the starting directions of the weight vectors and on the sequence of stimuli, different clusters wi ll choose different ones of these planes. Thus, a system which receives input from a set of such clusters wi ll be gi ven information as to which quadrant of the space in which the pattern appears . It is important to emphasize that the coherence of the space is entirely in the choice of input stimul i, not in the architecture of the competitive learning mechanism. The system discov-

ers the spatial structure in the input lines.

Formal analysis. For the dipole examples described above, it is possible to develop a rather precise characterization of the behavior of the competitive learning system. Recal l our argument that the most stable equi librium state (and therefore the one the system is most likely to Copyrighted Material

o 100

200 400

800 4000

FIGURE 7. The relative weights of each of the four elements of the cluster after 0, 100,

200, 400, 800, and 4000 stimulus presentations.

end up in) is the one that maximizes the function

T = LPk L Vjk (CXjk - CXik )'

k j,i

Now, in the dipole examples, all stimulus patterns of the stimulus population are equally likely (i.e. , Pk = 1/ N) , all stimulus patterns involve two active lines, and for every stimulus pattern in the population of patterns there are a fixed number of other stimulus patterns in Copyrighted Material

FIGURE 8. The relative weights ror a system in which the stimulus patterns were

chosen rrom a three-dimensional grid after 4000 presen tations.

the population which overlap it. 6 This implies that f/kj = R for all j.

k

With these assumptions, it is possible to show that maximizing T is

equivalent to minimizing the function

fli I N;

(see appendix for deri vation) , where N; is the number of patterns on which unit i wins, M is the number of units in the cluster, and B; is the number of cases in which unit i responds to a particular pattern and does not respond to a pattern which overlaps it. This is the number of

border patterns to which unit i responds. Formal ly, we have

B; = tt vi} (I-v;k ) for rjk <sup>&</sup>gt;O.

i k

From this analysis , it is clear that the most stable states are ones in which the size of the border is minimized . Since total border region is minimized when regions are spherical , we can conclude that in a situation in which stimulus pairs are drawn from adjacent points in a

6 Note that this latter condition does not Quite hold ror the examples presented above due to edge effects. It is possible to eliminate edge effects by the use or a toru<sup>s</sup>. We have carried out experiments on tori as well, and the results are essentially the same. Copyrighted Material

high -dimensional hyperspace, our competitive learning mechanism wi ll form essentially spherical regions that partition the space into one such

spherical region for each element of the cluster.

Another result of our simulations which can be explained by these equations is the tendency for each element of the cluster to capture roughly equally sized regions. This results from the interconnectedness of the stimulus population . The result is easiest in the case in which M= 2. In this case, the function we want to minimize is given by

> B <sup>I</sup>B2 NI + N2 '

Now, in the case of M= 2, we have B <sup>1</sup><sup>=</sup>B2, since the two regions <sup>m</sup>ust border on one another. Moreover, we have N 1<sup>+</sup>N 2 = N, since every pattern is either responded to by Unit 1 or Unit 2. Thu<sup>s</sup>, we

want to minimize the function

This function is minimized when NI = N1 2. Thus, there are two pressures which determine the perfo<sup>r</sup>mance of the system in these cases:

• There is a pressure to reduce the number of border stimuli to a

minimum.

• There is a pressure to divide the sti <sup>m</sup>ul <sup>u</sup><sup>s</sup>patterns among the units in a way that depends on the total amount of weight that unit has. If two unitshave the same amount of weigh<sup>t</sup> , they wi ll capture roughly equal numbers of equally l ikely stimulus

patterns .

Learning Words and Letters

It is common practice to handcraft networks to carry out particula<sup>r</sup> tasks. Whenever one creates such a network that performs a task rather successfully, the question arises as to how such a network migh<sup>t</sup> have evolved. The word perception model developed in McCle<sup>l</sup> <sup>l</sup>and

and Rumelhart (

l98l) and Rumelhart and McClelland (I982) is one

such case - <sup>i</sup><sup>n</sup> - <sup>p</sup>oint . That model offers rather detailed accounts of <sup>a</sup> variety of word perceptio<sup>n</sup>experiments, but it was crafted to do its job. Copyrighted Material

possible that one of the units will never win, and therefore never learn . This, of course, takes the competition out of competiti ve learning. This situation is analogous to the situation in the geometric analogy in which all of the stimulus points are relatively close together on the hypersphere, and one of the weight vectors, by chance, points near the cluster while the other one points far from the stimuli. (See Figure 1 0) . It is clear that the more distant vector is not closest to any stimulus and thus can never move toward the collectio<sup>n</sup> . We have investigated two modifications to the system which deal with the problem. One, which we call the leaky learning model, modifies the learning rule to state that both the winning and the losing units move toward the presented stimulus: the close vector simply moves much further.

In symbols this suggests that

Cik g,- - g,w· nk IJ

Cik gw -;;; - gw wij if unit j loses on stimulus k

if unit j wins on stimulus k

where g, is the learning rate for the losing units, gw is the learning rate for the winning unit, and where g,« gw . In our experiments we made

FIGURE 10. A geometric interpretation of changes in stimulus sensitivity. The larger the circle around the head of the weight vector the more sensitive the unit. The decision as to which unit wins is made on the basis of the distance from the circle rather than from the head of the weight vector. In the example, the stimulus pattern indicated by the y is actually closer to the head of one vector 0, but since it is closer to the circle sur-

rounding vector p, unit p wO�jhWftJ?8W�al

g, an order of magnitude smaller than gw . This change has the property that it slowly moves the losing units into the region where the actual stimuli lie, at which point they begin to capture some units and the

ordinary dynamics of competitive learning take over.

The second method is simi lar to that employed by Bienenstock,

Cooper, and Munro (1982)

, in which a unit modulates its own sensi-

tivity so that when it is not receiving enough inputs, it becomes increasingly sensitive. When it is receiving too many inputs , it decreases its sensitivity. This mechanism can be implemented in the present context by assuming that there is a threshold and that the relevant activation is the degree to which the unit exceeds its threshold. If, whenever a unit fails to win it decreases its threshold and whenever it does win it increases its threshold, then this method wi ll also make all of the units eventually respond, thereby engaging the mechanism of competitive learning. This second method can be understood in terms of the geometric analogy that the weight vectors have a circle surrounding the end of the vector. The relevant measure is not the distance to the vector itself but the distance to the circle surrounding the vector. Every time a unit loses , it increases the radius of the circle; every time it wins, it decreases the radius of the circle. Eventually, the circle on the losing unit wi ll be large enough to be closer to some stimulus pat-

tern than the other units.

We have used both of these mechanisms in our experiments and they appear to result in essentially similar behavior. The former, the leaky learning method, does not alter the formal analysis as long as the ratio gd gw is sufficiently small. The varying threshold method is more difficult to analyze and may, under some circumstances, distort the competitive learning process somewhat. After this diversion, we can now return to our experiments on the development of word/ position-

specific letter detectors and other feature detectors.

Position-specific letter detectors. In our first experiment, we presented letter pairs drawn from the set: AA, AB, BA, and BB. We began with clusters of size two. The results were unequivocal . The system developed position-specific letter detectors. In some experimental runs, one of the units responded whenever AA or AB was presented, and the other responded whenever BA or BB was presented. In this case, Unit 1 represents an A detector in position 1 and Unit 2 represents a B detector for position 1. Moreover, as in the word perception model , the letter detectors are, of course, in a mutually inhibitory pool . On other experimental runs , the pattern was reversed. One of the units responded whenever there was an A in the second position and the other unit responded whenever there was a B in the second position. Figure 11 shows the final configuration of weights for one of Copyrighted Material

• • • • • ••••• • • • • • •

•••• • • • • ••• • • • • ••• •

.. ... ,. • . . • · ... . • • · .......

.. ..... • . . • .. .... • •

· ........ Unit 1

Unit 2

FIGURE 11. The final configuration of weights for a system trained on the stimulus

patterns A, B, C, D.

our experimental runs. Note that although the units illustrated here respond only to the letter in the first position, there is sti ll weight on <sup>t</sup>he active' lines in the second position. It is just that the weights on the first position differentiate between A and 8, whereas those on the second position respond equal ly to the two letters. In particular, as suggested by our formal analysis, a<sup>s</sup>ymptotical ly the weights on a given li<sup>n</sup>eare proportionj'! to the probabi lity that that line is active when the unit wins. That i�, wij - <sup>p</sup>(unit; = 11 unitjwins) . Since the lower level units unique to A occur equally as often as those unique to 8, the weights on those lines are roughly equal . The input lines common to the two letters are on twice as often as those unique to either letter, and hence, they have twice as much weight. Those lines that never

come on reach zero weight.

Word detection units. In another experiment, we presented the same stimulus patterns , but increased the elements in thecluster from two to four. In this case, each of the four level-two units came to respond to one of the four input patterns -in short, the system developed word detectors. Thus, if layer two were to consist of <sup>a</sup> number of clusters of various sizes, large clusters with approximately one unit per word pattern wi II develop into word detectors, while smal ler clusters with approximately the number of letters per spatial position wi II develop into position-specific letter detectors. As we shal l see below, i f the number of elements of a cluster is substantially less than the number of letters per position , then the cluster wi ll come to

detect position-specific letter features .

Effects of number of elements per serial position. In another experiment, we varied the number of elements in a cluster and the number of letters per serial P�gW�JMl stimulus patterns drawn

from the set: AA. AB. AC. AD. BA. BB. BC. BD. In this case, we found that with clusters of size two, one unit responded to the patterns beginning with A and the other responded to those beginning with B. In our previous experiment, when we had the same number of letters in each position, we found that the clusters were indifferent as to which serial position they responded. Some responded to position 1 and others to position 2. In this experiment, we found that a two-element cluster always becomes a letter detector specific to serial position in which two letters vary. Similarly, in the case of clusters of size four we found that they always became letter detectors for the position in which four letters varied. Thus, in this case one responded to an A in the second position, one responded to a B in the second position , one responded to a C in the second position, and one responded to a D in the second position. Clearly, there are two natural ways to cluster the stimulus patterns-two levels of structure. If the patterns are to be put in two categories , then the binary feature A or B in the first position is the relevant distinction. On the other hand, if the stimuli are to be grouped into four groups , the four value feature determining the second letter is the relevant distinction. The competitive learning algorithm can discover either of the levels of structure-depending on the

number of elements in a cluster .

Letter similarity effects. In another experiment, we studied the effects of letter similarity to look for units that detect letter features. We presented letter patterns consisting of a letter in the fi rst position only. We chose the patterns so they formed two natural clusters based on the similarity of the letters to one another. We presented the letters A. B. S. and E. The letters were chosen so that they fell natural ly into two classes. In our font, the letters A and E are quite simi lar and the letters B and S are very similar. We used a cluster of size two. Naturally, one of the units responded to the A or the E while the other unit responded to the B or the S. The weights were largest on those features of the stimulus pai rs which were common among each of these similar pai rs. Thus, the system developed subletter-size feature detec-

tors for the features relevant to the discrimination.

Correlated teaching inputs. We carried out one other set of experiments with the word/letter patterns. In this case, we used clusters of size two and presented stimuli drawn from the set: AA. BA. SB. EB. Note that on the left-hand side, we have the same four letters as we had in the previous experiment, but on the right-hand side we have only two patterns; these two patterns are correlated with the letter in the first position. An A in the second position means that the first position contains either an A or a B, whereas a B in the second position Copyrighted Material

means that the first position contains either an S or an E. Note further that those correlations between the first and second positions are in opposition to the "natural" similarity of the letters in the first serial position. In this experiment, we first trained the system on the four stimuli described above. Si nce the second serial position had only two letters in it, the size-two cluster became a position-specific letter detector for the second serial position. One unit responded to the A and one to the B in the second position. Notice that the units are also responding to the letters in the fi rst serial position as wel l. One unit is responding to an A or a B in the first position while the other responds to an E or an S. Figure 12 shows the patterns of weights developed by the two units. After training, the system was then presented patterns containing only the first letter of the pai r and, as expected, the system had learned the "unnatural " classification of the letters in the fi rst position. Here the strong correlation between the fi rst and second position led the competitive learning mechanism to override the strong correlation between the highly similar stimulus patterns in the first serial position. This suggests that even though the competitive learning system is an "unsupervised " learning mechanism, one can control what it learns by controlling the statistical structure of the stimulus patterns being presented to it. In this sense, we can think of the right-hand letter in this experiment as being a kind of teaching stimulus aimed at determining the classification learned for other aspects of the stimulus. It should also be noted that this teaching mechani sm is essentially the same as the so-called errorless learning procedure used by Terrace (963) in training pigeons to peck a certain color key by associating that color with a response situation where their pecking is determined by other factors. As we shall see below, this correlational teaching mechanism is useful in allowing the competitive learning mechanism to

discover features which it otherwise would be unable to discover .

. . . • .. • · ....

· .

• • . .

· ... • • · .. . • .. .. .

• • •

• • Unit 1 ••• ••

• • • • •

• ••• • • • •

••• Unit 2

• • • • • • ••

FIGURE 12. The pattern of weights developed in the correlated learning experiment. Copyrighted Material

Horizontal and Vertical Lines

One of the classical ly difficult problems for a linear threshold device like a perceptron is to distinguish between horizontal and vertical lines. In general , horizontal and vertical lines are not linearly separable and requi re a multi layer perceptron system to distinguish them. One of the goals of the competitive learning device is for it to discover features that, at a higher level of analysis, might be useful for discriminating patterns which might not otherwise be discriminable with a linear threshold-type device. It is therefore of some interest to see what kinds of features the competitive learning mechanism discovers when presented with a set of vertical and horizontal lines. In the following discussion, we chronicle a series of experiments on this problem. Several of the experiments ended in fai lure, but we were able to discover a way i n which competitive learning systems can be put together to build a hierarchical feature detection system capable of discriminating vertical and horizontal lines. We proceed by sketching several of our fai lures as well as our successes because the way in which the system fails is elucidating. It should be noted at the outset that our goal is not so much to present a model of how the human learns to distinguish between vertical and horizontal lines (indeed, such a distinction is probably prewired in the human system) , but rather to show how competitive learning can discover features which allow for the system to learn distinctions with multi ple layers of units that cannot be learned by single-layered systems. Learning to distinguish vertical and horizontal

lines is simply a paradigm case.

In this set of experiments, we represented the lower level of units as if they were on a 6x6 grid. We then had a total of 12 stimulus patterns, each consisting of turning on six Level 1 units in a row on the grid. Figure 13 illust rates the grid and several of the stimulus patterns. Ideally, one might hope that one of the units would respond whenever a vertical line is presented� the other would respond whenever a horizontal line is presented. Unfortunately, a little thought indicates that this is impossible. Since every input unit partici pates in exactly one vertical and one horizontal line, there is no configuration of weights which wi ll distinguish vertical from horizontal. This is exactly why no linear threshold device can distinguish between vertical and horizontal lines in one level. Since that must fai l, we might hope that some clusters in the competitive learning device wi ll respond to vertical lines by assigning weights as illustrated in Figure 14. In this case, one unit of the pai r would respond whenever the first, second, or fourth vertical line was presented, and another would respond whenever the third, fifth, or sixth vertical line was presented� since both units would Copyrighted Material

· ... . .. .. · .. .. ..

· . . . . . Unit 1 Cluster 1

· .. .. . .. · . .. . .. · ...... .

.. . .. .. .. · .. . ..

.. . . .. .. Unit 2 Cluster 1

· . . . . . · . .. .. . . .. . . .. ..

•••••• • •••• •

Unit 1 Cluster 2

• •• •••

• ••••• Unit 2 Cluster 2

• •• ••• • •• • • •

FIGURE 14. A possi<sup>b</sup>le weight configuration which could distinguish vertical rrom

horizontal.

· ...... • • • • • · .....

• ••• •

• ••• • • ••• • Unit 1

Unit 2

FIGURE 15. <sup>A</sup>typical configuration or weights for the vertical/horizontal discrimina-

tion.

ones. It is this simi larity that the competitive learning mechanism has

discovered.

Now, suppose that we change the system somewhat. Suppose that we "teach" the system the difference between vertical and horizontal (as we did in the previous experiments with letter strings) . In this experiment we used a 12 x 6 grid. On the right-hand side of the grid we presented either a vertiebtPyri3hPJ>hiijlOfI);Jjne, as we did before. On

the right-hand grid are identical for the two cluster members. Thus, when the "teacher" is turned off, and only the right-hand figure is presented, the two units respond randomly and show no evidence of

having learned the horizontal I vertical distinction.

Suppose, however, that we have four, rather than two, units in the level-two clusters. We ran this experiment and found that of the four units, two of them divided up the vertical patterns and two of them di vided up the horizontal patterns . Figure 1 7 ill ustrates the weight values for one of our runs. One of the units took three of the vertical line patterns; another unit took three other vertical patterns. A third unit responded to three of the horizontal line patterns , and the last unit responded to the remaining three horizontal lines. Moreover, after we took away the "teaching" pattern, the system continued to classify the vertical and horizontal lines just as it did when the left-hand "teaching"

pattern was present.

• • • • • • Unit 1

Unit 3

Cluster 1

• • • • • • Unit 2

Unit 4

.... .. . . . . . . ••••••

• • • • Unit 1

Unit 3

• •••••

Cluster 2

Unit 2 ...... . . ....

• • • • • • Unit 4

FIGURE 17. The weight values for the two clusters of size f<sup>o</sup>ur for the vertical/ horizontal discrimination experiment wi th a correlated "teaching" stimulus. Copyrighted Material

In one final experiment with vertical and horizontal lines, we developed a three-level system in which we used the same stimulus patterns as in the previous experiment; the only difference was that we had two clusters of four units at the second level and one cluster of two units at the third level . Figure 18 shows the architecture employed. In this case, the two four-element clusters each learned to respond to subsets of the vertical and horizontal lines as in the previous experiment. The two clusters generally responded to different subsets , however. Thus, when the upper horizontal line was presented, Unit 1 of the first cluster responded and Unit 3 of the second cluster responded. When the bottom horizontal line was presented, Unit 1 of the first cl uster responded again, but Unit 4 of the second cluster also responded. Thus, the cluster of size two at the highest level was recei ving a kind of dipole stimulus. It has four inputs and on any trial , two of them are act i ve . As with our analysis of di pole stimuli, we know that stimul i that overlap are always put in the same category. Note that when a vertical line is presented, one of the two units in each of the middle layers of clusters that responds to vertical lines wi ll become active, and that none of the units that respond to horizontal lines will ever be active; thus, this means that there are two units in each middle layer cluster that respond to vertical lines . Whenever a vertical line is presented, one of the units in each cluster wi ll become active. None of the horizontal units will ever be active in the face of a vertical stimulus. Thus, one of the units at the highest level learns to respond whenever a vertical line is presented , and the other unit responds whenever a horizontal line is

Layer 1

Layer 2

•••• • Input Units

presented. Once the system has been trained, this occurs despite the absence of the "teachingh stimulus. Thus , what we have shown is that the competitive learning mechanism can, under certain conditions, develop feature detectors which allow the system to distinguish among patterns that are not differentiable by a simple linear unit in one level.

CONCLUSION

We have shown how a very simple competitive mechanism can discover a set of feature detectors that capture important aspects of the set of stimulus input patterns. We have also shown how these feature detectors can form the basis of a multilayer system that can serve to learn categorizations of stimulus sets that are not linearly separable. We have shown how the use of correlated stimuli can serve as a kind of ''teachingh input to the system to allow the development of feature detectors which would not develop otherwise. Although we find the competitive learning mechanism a very interesting and powerful learning principle, we do not, of course, imagine that it is the only learning principle. Competitive learning is an essentially nonassociative, statistical learning scheme. We certainly imagine that other kinds of learning mechanisms will be involved in the building of associations among patterns of activation in a more complete neural network. We offer this analysis of these competitive learning mechanisms to further our understanding of how simple adaptive networks can discover features important in the description of the stimulus environment in which the

system finds itself.

ACKNOWLEDGMENTS

This research was supported by grants from the System Development Foundation and by Contract NOOOI4-79-C-0323, NR 667-437 with the Personnel and Training Research Programs of the Office of Naval

Research.

APPENDIX

For the case of homogeneous dipole stimulus patterns, it is possible to derive an expression for the most stable equilibrium state of the system. We say that a set of dipole stimulus patterns is homogeneous if (a) they are equally likely and (b) for every input pattern in the set there are a fixed number of other input patterns that overlap them. These conditions were met in our simulations. Our measure of stability

is given by

T = LPkLL Vjk (£rjk - £rik)'

k j i

S

. 1 .

mce Pk <sup>=</sup><sup>N</sup>

' we can WrIte

Summing the first portion of the equation over i and the second over j

we have

T = M

<sup>N</sup> LL Vjk£rjk - <sup>N</sup>

<sup>1</sup>LL£rikLVjk '

j k i k j

Now note that when Pk = 1/ N, we have £r ik = L 'kj vijl L Vkl' Further-

j ,

more, L V,k =1 and L V'k = N" where N, is the number of patterns

I k captured by unit I. Thus, we have

Now, since all stimuli are the same size, we have 'ij = 'j

i' Moreover,

since all stimuli have the same number of neighbors , we have L'/j = L'i} = R, where R is a constant determined by the dimen<sup>s</sup>lonalit/of the stimulus space from which the dipole stimuli are drawn.

Thus, we have LVii

M R"" I T <sup>=</sup>-N LL Vjk£rjk - <sup>N</sup> �-N.. ' j k I <sup>I</sup>

procedure, including languages for precisely expressing them and theoretical machines for realizing them. This body of theory grew out of mathematical logic, and in tum contributed to computer science, physical computing systems, and the theoretical paradigm in cognitive

science often called the (von Neumann) computer metaphor.l

In his paper "Physical Symbol Systems," Allen Newell (1 980) articulated the role of the mathematical theory of symbolic computation in cognitive science and furnished a manifesto for what I will call the symbolic paradigm. The present book offers an alternative paradigm for cognitive science, the subsymbolic paradigm, in which the most powerful level of description of cognitive systems is hypothesized to be lower than the level that is naturally described by symbol manipulation.

The fundamental insights into cognition explored by the subsymbolic paradigm do not involve effective procedures and symbol manipulation. Instead they involve the "spread of activation," relaxation, and statistical correlation. The mathematical language in which these concepts are naturally expressed are probability theory and the theory of dynamical systems. By dynamical systems theory I mean the study of sets of numerical variables (e.g., activation levels) that evolve in time in parallel and interact through differential equations. The classical theory of dynamical systems includes the study of natural physical systems (e.g., mathematical physics) and artificially designed systems (e.g., control theory). Mathematical characterizations of dynamical systems that formalize the insights of the subsymbolic paradigm would be most helpful

in developing the paradigm.

This chapter introduces harmony theory, a mathematical framework for studying a class of dynamical systems that . perform cognitive tasks according to the account of the subsymbolic paradigm. These dynamical systems can serve as models of human cognition or as designs for artificial cognitive systems. The ultimate goal of the enterprise is to develop a body of mathematical results for the theory of information processing that complements the results of the classical theory of (symbolic) computation. These results would serve as the basis for a manifesto for the subsymbolic paradigm comparable to Newell's manifesto for the symbolic paradigm. The promise offered by this goal will, I hope, be suggested by the results of this chapter, despite their very lim-

ited scope.

1 Mathematical logic has recently given rise to another approach to formalizing information: situation semantics (Barwise & Perry, 1983). This is related to Shannon's (1948/1963) measure of information through the work of Dretske (1981). The approach of this chapter is more faithful to the probabilistic formulation of Shannon than is the symbolic approach of situation semantics. (This results from Dretske's move of identify-

It should be noted that harmony theory is a "theoryh in the mathematical sense, not the scientific sense. By a "mathematical theory" -e.g., number theory, group theory, probability theory, the theory of computation-I mean a body of knowledge about a part of the ideal mathematical world� a set of definitions, axioms, theorems, and analytic techniques that are tightly interrelated. Such mathematical theories are distinct from scientific theories, which are of course bodies of knowledge about a part of the "real" world. Mathematical theories provide a language for expressing scientific theories� a given mathematical theory can be used to express a large class of scientific theories. Group theory, for example, provides a language for expressing many competing theories of elementary particles. Similarly, harmony theory can be used to express many alternative theories about various cognitive phenomena. The point is that without the concepts and techniques of the mathematical language of group theory, the formulation of any of the current scientific theories of elementary particles would be essen-

tially impossible.

The goal of harmony theory is to provide a powerful language for expressing cognitive theories in the subsymbolic paradigm, a language that complements the existing languages for symbol manipulation. Since harmony theory is conceived as a language for using the subsymbolic paradigm to describe cognition, it embodies the fundamental scientific claims of that paradigm. But on many important issues, such as how knowledge is represented in detail for particular cases, harmony theory does not itself make commitments. Rather, it provides <sup>a</sup> language for stating alternative hypotheses and techniques for studying

their consequences.

A Top-Down Theoretical Strategy

How can mathematical analysis be used to study the processing

mechanisms underlying the performance of some cognitive task?

One strategy, often associated with David Marr (I 982), is to characterize the task in a way that allows mathematical derivation of mechanisms that perform it. This top-down theoretical strategy is pursued in harmony theory. My claim is not that the strategy leads to descriptions that are necessarily applicable to all cognitive systems, but rather that the strategy leads to new insights, mathematical results, computer architectures, and computer models that fill in the relatively unexplored conceptual world of parallel, massively distributed systems that perform cognitive tasks. Filling in this conceptual world is a necessary subtask, I believe, for understanding how brains and minds are capable of intelligence and for assessing whether computers with novel architectures

might share this capabili�opyrighted Material

introduced by thisexpository organization will be repaid by greater

accessibility.

Section 1 is a top-down presentation of how the perceptual perspective on cognition leads to the basic feature<sup>s</sup>of harmony theory. This presentation starts with a particular perceptual model, the letterperception model of McClelland and Rumelhart (I 98 1), a<sup>n</sup>d abstracts from it general features that can apply to modeling of higher cognitive processes. Crucial to the development is a particular formulation of

<sup>a</sup>spectsof schema theory, along the lines of Rumelhart (I 980) .

Section 2, the majority of the chapter, is a bottom-up presentation of harmony theory that starts with the primitives of the knowledge representation. Theorems are informally described that provide a competence theory for a cognitive system that performs thecompletion task, a machine that realizes this theory, and a learning procedure through which the machine can absorb the necessary information from its environment. Then an appl ication of the general theory is described: a model of intuitive, qualitative problem-solving in elementary electric circuits. This model illustrates several points about the relation between symbolic and subsymbolic descriptions of cognitive phenomena; for example, it furnishes a sharp contrast between the description at these two levels of the nature and acquisition of

expertise.

The final part of the chapter is an Appendix containing a concise but

self-contained formal presentation of the definitions and theorems.

SECTION 1: SCHEMA THEORY AND SELF -CONSISTENCY

THE LOGICAL STRUCTURE OF HARMONY THEORY

The logical structure of harmony theory is shown schematically in Figure 1. The box labeled Mathematical Theory represents the use of mathematical analysis and computer simulation for drawing out the implications of the fundamental principles. These principles comprise a mathematical characterization of computational requirements of a cognitive system that perforCJljip}lnyRftJlJJM8mffctask. From these principles

schemata Inference

� harmony probabIlity CONCEPTUAL FOUNDATIONS

descriptive characterization of computational requirements

- mathematical

�- characterization

Of computation raqulrements

\�

machine implementation " (simulation): PROCESS

,, '

MATHEMATICAL THEORY

------, <sup>7</sup>)

•

formalization derivation

FIGURE \. The logical structure of harmony theory.

characterization of performance

, " ,

> --- � Simulation

it is possible to mathematically analyze aspects of the resulting performance as well as rigorously derive the rules for a machine implementing the computational requirements. The rules defining this machine have a different status from those defining most other computer models of cognition: They are not ad hoc, or post hoc; rather they are logically <sup>d</sup>erived from a set of computational requirements. This is one sense in

which harmony theory has a top-down theoretical development.

Where do the " mathematically characterized computational requirements" of Figure I come from? They are a formalization of a descriptive characterization of cognitive processing, a simple form of schema theory. In Section 1 of this chapter, I will give a description of this form of schema theory and show how to transform the descriptive characterization into a mathematical one-how to get from the conceptual box of Figure 1 into the mathematical box. Once we are in the formal world, mathematical analysis and computer simulation can be put to

work.

Throughout Section 1, the main points of the development will be

explicitly enumerated.

Point I. The mathematics of harmony theory is founded on familiar concepts Q{ cognitive science: inference through activation q{ schemata. Copyrighted Material

DYNAMIC CONSTRUCTION OF SCHEMATA

The basic problem can be posed a la Schank (980), While eating at a fancy restaurant, you get a headache. Without effort, you ask the waitress if she could possibly get you an aspirin. How is this plan created? You have never had a headache in a restaurant before. Ordinarily, when you get a headache your plan is to go to your medicine cabinet and get yourself some aspirin. In the current situation, this plan must be modified by the knowledge that in good restaurants, the management is wi lling to expend effort to please its customers, and that

the waitress is a liaison to that management.

The cognitive demands of this situation are schematically illustrated in Figure 2. Ordinari ly, the restaurant context calls for a "restaurant script" which supports the planning and inferencing required to reach the usual goal of getting a meal. Ordinarily, the headache context calls for a "headache script" which supports the planning required to get aspirin in the usual context of home. The completely novel context of a headache in a restaurant calls for a special-purpose script integrating the

knowledge that ordinarily manifests itself in two separate scripts.

What kind of cognitive system is capable of this degree of flexi bility? Suppose that the knowledge base of the system does not consist of a set of scripts like the restaurant script and the headache script. Suppose

restaurant context

headache contexl

restaurant ---"'... & headache --v-"'"

context

Headache in a Restaurant

restaurant script

headache script

specialpurpose script

... Inferences, goals

... inferences, goals

... 'ask waitress for aspirin'

However, the model also accounts for the observed facilitation of letter perception within orthographically regular nonwords or pseudowords like MAVE. When the model processes this stimulus, several word units become and stay quite acti ve, including MAKE, WA VE, HA VE, and other words orthographically similar to MA VE. In this case, the perception of a letter in the stimulus is the result of an inference process that is supported by the collection of acti vated units. This

collection is a dynamically created pseudo word schema.

When an orthographically irregular non word is processed by the model, letter perception is slowest. As in the case of pseudowords, many word units become active. However, none become very active, and very many are equally active, and these words have very little similarity to each other, so they do not support inference about the letters effectively. Thus the knowledge base is incapable of creating schemata

for irregular nonwords.

Point 3. Schemata are coherent assemblies of knowledge atoms; only

these can support inference.

Note that schemata are created simply by activating the appropriate atoms. This brings us to what was labeled in Figure 1 the "descriptively

characterized computational requirements" for harmony theory:

Point 4: The harmony principle. The cognitive system is an engine for activating coherent assemblies of atoms and drawing inferences that are

consistent with the knowled,?e represented by the activated atoms.

Subassemblies of activated atums that tend to recur exactly or approxi-

mately are the schemata.

This principle focuses attention on the notion of coherency or consistency. This concept will be formalized under the name of harmony,

and its centrality is acknowledged by the name of the theory.

MICRO· AND MACROLEVELS

It is important to realize that harmony theory, like al <sup>l</sup> subsymbolic accounts of cognition, exists on two distinc<sup>t</sup>levels of description: a microlevel involving knowledge atoms and a mac<sup>r</sup>olevel involving schemata (see Chapter 14). These levels of description are c<sup>o</sup>mpletely analogous to other micro- and macrotheories, for example, in physics. The microtheory, quantum physics, is as<sup>s</sup>umed to be universally valid. Part of its job as a theory is to explain why the approximate macrotheory classical phvsics..2 works when it does and why it breaks , COPYflghted Matenal

TABLE I A PROCEDURE FOR PERFORMING THE COMPLETION TASK

Input: <sup>A</sup><sup>c</sup>tivat <sup>i</sup>on: Inference:

Assign values to some features in the representation Activate atoms that are consistent with the representation Assign values to unknown features of representation that

are consistent with the active knowledge

fil led in, and how the representation is fi lled in affects which atoms are activated. The acti vation and inference processes mutually constrain each other; these processes must run in paral lel . Note also that all the

decisions come out of a stri ving for consistency.

Point 6. Assembly of schemata (activation of atoms) and iriference (completing missing parts of the representation) are both achieved by finding maximally self-consistent states Qf the system that are also con-

sistent with the input.

The completion of the stimulus shown in Figure 5 is shown in Figure 6. The consistency is high because wherever an active atom is

o \_ active;

on

inactive; off

above, and reports the results of some particular studies. The most for-

mal matters are treated in the Appendix.

SECTION 2: HARMONY THEORY

. .. the privileged unconscious phenomena, those susceptible of becoming conscious, are those which . . . affect most profoundly our emotional sensibility . .. Now, what are the mathematic entities to which we attribute this character of beauty and elegance . .. ? They are those whose elements are harmoniously disposed so that the mind without effort can embrace their totality while realizing the details. This harmony is at once a satisfaction of our esthetic needs and an aid to the mind, sustaining and guiding. . .. Figure the future elements of our combinations as something like the unhooked atoms of Epicurus. . .. They flash in every direction through the space . .. like the molecules of a gas in the kinematic theory of gases. Then their mutual impacts may produce new combinations.

> Henri Poincare (1913) Mathematical Creation \0

In Section 1, a top-down analysis led from the demands of the completion task and a probabilistic formulation of schema theory to p<sup>e</sup>rceptual features, knowledge atoms, the central notion of harmony, and the role of harmony in estimating probabi lities of environmental states. In Section 2, the presentation will be bottom-up, starting from the

primitives .

KNOWLEDGE REPRESENTATION

Representation Vector

At the center of any harmony theoretic model of a particular cogni-

tive process is a set of representational features rj, r2, ..

. These

Knowledge Atoms

Representational Features

> k W,A2 kM A = , 2 kA K = 2 3

(- (- (0

+ + 0 + + 0 0 0 + +

FIGURE 8. The graphical representation of a particular harmony model .

0) 0) -)

would them be indicated above the atom in the drawing. (For the com-

pletely general case, see Figure 13.)

Hierarchies and the Architecture of Harmony Networks

One of the characteristics that distinguishes harmony models from other paral lel network models is that the graph always contains two layers of nodes, with rather different semantics. As in many networks, the nodes in the upper layer correspond to patterns of values in the lower layer. In the letter-perception model of McClelland and Rumelhart, for example, the word nodes correspond to patterns over the letter nodes, and the letter nodes in turn correspond to patterns over the line-segment nodes. The letter-perception model is typical in its hierarchical structure: The nodes are stratified into a sequence of several layers, with nodes in one layer being connected only to nodes in

adjacent layers. Harmony models use only two layers.

The formal ism could be extended to many layers, but the use of two layers has a principled foundation in the semantics of these layers. The nodes in the representation layer support representations of the environment at all levels of abstractness. In the case of written words, this layer could support representation at the levels of line segments, letters, and words, as shown schematically in Figure 9. The upper, knowledge, layer encodes the patterns among these representations. If information is gi ven about line sef!l1p9}fgh;&Y?AcNUfI5l of the knowledge atoms

The harmony function has as parameters the set of knowledge vectors and their strengths: {(ka,fT a)} ; I wi ll cal l this the knowledge base K. The basic requirement on the harmony function H is that it be additive under decompositions of the system. IJ This means that if a network can be partitioned into two unconnected networks. as in Figure 12. the harmony of the whole network is the sum of the harmonies of the

parts:

H(r, a ) = H (r h a l ) + H(r2, a 2) .

In this case, the knowledge and representational feature nodes can each be broken into two subsets so that the knowledge atoms in subset 1 all have 0 connections with the representational features in subset 2, and vice versa. Corresponding to this partition of nodes there is a decom-

position of the vectors r and a into the pieces rh r2 and aJ ,a2' The harmony function I have studied (recall Figure 7) is

(1)

a

Here, hI( ( r, ka) is the harmony contributed by acti vating atom a,

given the current representation r. I have taken this to be

r"k

hl« r. ka) =Ikai -I( .

1--10---------- r ------------1 .. 1

FIGURE 12. A decomposable harmony network.

val

ues of the unknown rj and those values of the <sup>Q</sup>

athat together max-

imize the harmony H ( r, a) and thereby maximize the estimated proba-

bi lity p(r ,a).

This discussion is summarized in the fol lowing theorem:

Theorem 1: Competence. Suppose a cognitive system can observe

the frequency of the patterns Ika

l in its environment. The probabi l-

ity distribution with the most Shannon missing information that is

consistent with the observations is

7T ( r) ex: e<sup>V</sup> (

x)

with U defined as above. The maximum-likelihood completions of

this distribution are the same as those of

p (r ,8) ex: <sup>e</sup> H(r ,<sup>a</sup> )

with the harmony function defined above.

This theorem describes how a cognitive system should perform completions, according to some mathematical principles for statistical extrapolation and inference, In this sense, it is a competence theorem. The obvious next question is: Can we design a system that wi ll really compute completions according to the specifications of the competence

theorem?

The " Physics Analogy"

It turns out that designing a machine to do the required computations is a relatively straightforward appl ication of a computational technique from statistical physics. It is therefore an appropriate time to discuss the "analogy" to physics that is exploited in harmony theory.

Why is the relation between probabi lity and harmony expressed in the competence theorem the same as the relation between probabi lity and energy in statistical physics? The mapping between statistical physics and inference that is being exploited is one that has been known

for a long time.

The second law of thermodynamics states that as physical systems evolve in tim<sup>e</sup> , they wi ll approach conditions that maximize randomness or entropy, subject to the constraint that a few conserved quantities like the systems' energy must always remain unchanged. One of the triumphs of statistical mechanics was the understanding that this law is the macroscopic manifesfJU(K{i�t�I\1ft(ftff4Ying microscopic descri ption

functions. Physicists at IBM independently applied the technique, under the name simulated annealing, to both practical computer design problems and classical maximization problems (Kirkpatrick, Gelatt, & Vecchi , 1983) . Benchmarks of simulated anneal ing against other search procedures have produced mi xed results (Aragon, Johnson, &

McGeoch , 1985) .

The contribution of harmony theory is not so much the search procedure for finding maxima of H, but rather the function H itself. Theorem 2 is important: It describes a statistical dynamical system that performs completions; it gives an implementation-level descri ption of a kind of completion machine. But Theorem 1 is more central : It gi ves a high , functional-level characterization of the performance of the system-says what the machine does-and introduces the concept of harmony. More central to the theory also is Theorem 3 , which says

how the harmony function can be tuned with experience.

The Learnability Theorem

Performing the completion task in different environments cal ls for different knowledge. In the formal ism of Theorem 1, a gi ven cognitive system is assumed to be capable of observing the frequency in its environment of a predetermi ned set of feature patterns. What varies for a given cognitive system across environments is the frequencies of the patterns; this manifests itself in the variation across environments of the strengths of the knowledge atoms representing those patterns.

Theorem 3: Learnability. Suppose states of the environment are selected according to the probabi lity distri bution defining that environment, and each state is presented to a cogn itive system. Then there is a procedure for gradual ly modifying the strengths of the knowledge atoms that will converge to the val ues requi <sup>r</sup>ed by

Theorem 1.

The basic idea of the learning procedure is simple. Whenever one of the patterns the cognitive system can observe is present in a stimulus from the environment, the parameter associated with that pattern is incremented. In harmonium, this means that whenever a knowledge atom matches a stimulus, its strength increases by a smal l amount D.rr . In the simulation machine, this means that the A parameter on al l the connections corresponding to that atom must be incremented by D.A =D.rr (1 - K). In t8b�1nBNiedW/a?�Wl}1 corresponds to a memory

Instruction, and Ginsburg, 1983.) Even such simple problems as that of Figure 16 have important instructional implications (Riley, 1984) .

The model I wi ll describe was studied in col laboration with Mary S. Riley (Riley & Smolensky, 1984) and Peter DeMarzo (1984) . This model provides answers, without any symbolic mani pulation of rules, to qual itative questions about the particular circuit of Figure 16. It should not be assumed that we imagine that a different harmony network like the one I wi ll describe is created for every different circuit that is analyzed. Rather we assume that experts contain a small number of fixed networks like the one we propose, that these networks represent the effects of much cumulated experience with many different circuits, that they form the "chunks" with which the expert's intuition represents the circuit domain, and that complex problem sol ving somehow employs these networks to direct the problem solving as a whole through intuitions about chunks of the problem. At this early stage we cannot say much about the coordination of activity in complex problem solving. But we do claim that by gi ving an explicit example of a nonsymbol ic account of problem solving, our model offers insights into expertise that complement nicely those of traditional production-system models. The model also serves to render concrete many of the general

features of harmony theory that have been described above.

Representational features. The first step in developing a harmony model is to select features for representing the environment. Here the environment is the set of qualitative changes in the electric circuit of Figure 16 that obey the laws of physics. What must obviously be represented are the changes in the physical components: whether R <sup>1</sup> goes up, goes down , or stays the same, and simi larly for R 2 and the battery's voltage Vrola/' We also hypothesize that experts represent deeper features of this environment, like the current I, the voltage drops VI and V2 across the two resistors, and the effective resistance Rlola/ of the circuit. We claim that experts "see " these deeper features; that perceiving the problem of Figure 16 for experts invol ves fil l ing in the deeper features just as for all sighted people-experts in vision perceiving a scene involves fill ing in the features describing objects in three-dimensional space. Many studies of expertise in the psychological literature show that experts percei ve their domain differently from novices: Their representations are much richer; they possess additional representational features that are specially developed for capt uring the structure of the particular environment. (See, for example, Chase &

Simon, 1973; Larkin, 1983.)

So the representational features in our model encode the qual itative changes in the seven circuit variables: R It R <sup>2</sup> , R'o(ail V It V 2, Vrola/' and

although there seems to be no natural formal definition for the concept. To study the properties of macrodecisions, it is appropriate to look at how the average values of the stochastic node variables change during the computation. For each of the unknown variables, the node val ues were averaged over 30 runs of the completion problem of Figure 16, separately for each time during the computation . The resulting graphs are shown in Figure 20. The plots hover around 0 initially, indicating that values + and - are equal ly likely at high temperatures-lots of

fl

 ickering. As the system cools, the average values of the representation variables drift toward the values they have in the correct sol ution

to the problem (RlolaJ = up, I = down, VI = down, V2 = up) .

Emergent seriality. To better see the macrodecisions, in Figure 21 the graphs have been superimposed and the " indecisive" band around 0 has been removed. The stri king result is that out of the statistical din

of paral lel microdecisions emerges a sequence of macrodecisions.

Propagation of givens. The result is even more interesting when it is observed that in symbolic forward-chaining reasoning about this problem, the decisions are made in the order R, I, VI, V2. Thus not on ly ic; the ('(JmnPlpncp of the mnrlel neHt ly clec;cri hahl e <;ym hol ical ly. hut even the performance, when descri bed at Ihe macrolevel , cou ld be modeled by the sequen tial fi ri ng of prod uct ions that chai <sup>n</sup>Ih rough the inferences. Of course, macrodecisions emerge first about those variables that are most directly constrained by the given inputs, but not because rules are being used that have conditions that only al low them to apply when all but one of the variables is known. Rather it is because the variables given in the input are fIXed and do not fluctuate: They provide the information that is the most consistent over time, and therefore the knowledge consistent with the input is most consistently acti vated, allowing those variables involved in this knowledge to be more consistently completed than other variables. As the temperature is lowered, those variables "near" the input (with respect to the connections provided by the knowledge) stop fluctuating first, and their relative constancy of value over time makes them function somewhat like the original input to support the next wave of completion. In this sense, the stability of variables "spreads out " through the network, starting at the inputs and propagating with the help of cool ing. Unlike the simple feedforward "spread of activation " through a standard activation network, this process is a spread of feedback-mediated coherency through a decision-making network. Like the growth of droplets or crystals, this amounts to the expansion of pockets of order into a sea of

20

15

u

10

5

o \_\_ � \_\_\_\_ � \_\_\_\_ � \_\_ �L-\_\_ � \_\_\_\_ � \_\_\_\_ � \_\_ �

o 100 200

Time

300 400

FIGURE 22. The specific heat of the circuit analysis model th rough the course of t he

computation.

/ / /

/ / \_ /

o 100

/ I -I I / /

// R

200 Time 300 400

FIGURE 23. There is a peak in the specific heat at the time when the R and I decisions

are being made.

The analysis of decision making in this model considered the limit as the number of features and atoms goes to infinity-for only in this "thermodynamic limit" can we see real phase transitions. In this limit, the set of possible values for the averages that define the aggregate variables comes closer and closer to a continuum. The central l imit theorem constrains these averages to deviate less and less from their means; statistical fluctuations become less and less significant; the

model 's behavior becomes more and more deterministic.

Thus, just as the statistical behavior of matter disappears into the deterministic laws of thermodynamics as systems become macroscopic in size, so the statistical behavior of individual features and atoms in harmony models becomes more and more closely approximated by the higher level descri ption in terms of schemata as the number of constituents aggregated into the schemata increases. However there are two important differences between harmony theory and statistical physics relevant here. First, the number of constituents aggregated into schemata is nowhere near the number- I0<sup>23</sup> -of particles aggregated into bulk matter. Schemata provide a useful but significantly limited description of real cognitive processing. And second, the process of aggregation in harmony theory is much more complex than in physics. This point can be brought out by passing from the grossly oversimplified two-choice decision model just considered to a more realistic cogni-

tive domain.

Schemata for rooms. In a real istical ly complicated and large network, the schema approximation would go something like this. The knowledge atoms encode clusters of values for features that occur in the environment . Commonly recurring clusters would show up in many atoms that differ slightly from each other. (In a different language, the many exemplars of a schema would correspond to knowledge atoms that differ slightly but share many common features.) These atoms can be aggregated into a schema , and their average activation at any moment defines the activation of the schema. Now among the atoms in the cluster corresponding to a schema for a living-room, for example, might be a subcluster corresponding to the schema for sofa/ coffee-table. These atoms comprise a subschema and the average of their activations would be the activation variable for this subschema. The many atoms comprising the schema for kitchen share a set of connections to representational features relating to cooking devices. It is convenient to group together these connections into a cooking-device slot, Smoking ' Different atoms for different instances of kitchen encode various patterns of values over these representational features, corresponding to instances of stove, con ventional oven, microwave oven ,

Development Foundation, the Alfred P. Sloan Foundation, National Institute of Mental Health Grant PHS MH 14268 to the Center for Human Information Processing, and Personnel and Training Research Programs of the Office of Naval Research Contract N00014-79-C-0323,

NR 667-437.

Transducer

Mental Space: M knowledge A

FIGURE 26. A schematic representation of the theoretical framework .

representation are taken to be binary. The prediction problem is to take some features of an environmental state as input and make best guesses about the unknown features. This amounts to extrapolating from some observed statistics of the environment to an entire probability distribution over all possible feature combinations. This extrapolation proceeds by constructing the distribution that adds minimal infor-

mation (in Shannon's sense) to what is observed.

Notation. B = (- 1, + I) , the default binary values. R =the real

numbers. x

n

<sup>=</sup>XxXx ... xX (n times) , where x is the cartesian

product. If X, y E Xn

, then x·y = r.�\_lxmYm and Ix i = L�=llxml .

<sup>2</sup>x is the set of all subsets of X. IXI is the number of elements of X. Bn is called a binary hypercube. The i <sup>t</sup>h coordinate junction of Bn (; = 1, . . . ,<sup>n</sup> ) gives for any point (i.e., vector) in Bn its ith B-valued

coordinate (i.e. , component) .

De/. A distal environment Edistol = (E, P) is a set E of environmental

events and a probability distribution P on E.

De/. A representational space R is a cartesian product Rex x Ren of two binary hypercubes. Each of the N (Nex ; Nen ) binary-valued coordinate functions ri of R (Rex ; Ren ) is cal led an (exogenous; endogenous)

feature.

De/. A transduction map T from an environment Edistol to a representational space R = RexxRen is a map T: E --+ Rex. T induces a proba-

De! Let R be a representational space. Associated with this space

is the input space I = {- I, 0, + II Na.

De! A point r in R is called a completion of a point t in I if every nonzero feature of , agrees with the corresponding feature of r. This relationship will be designated r ::> t. A completion junction c is a map from I to 2R (the subsets of R) for which r E dd implies r ::> to The features of L with val ue 0 are the "unknowns" that must be filled in by

the completion function.

De! Let p be a probabi lity distribution on a space X = Rex xA . The maximum-likelihood completion junction determined by p,

c p : I -2R , is defined by

dd = { r E R I for some a E A, and all (a' ,r') E R x A

such that r' ::> , : p (r , a ) � p (r

, a') I

(A wi ll be either empty or the set of possi ble knowledge atom activa-

tion vectors.)

De! A basic event (X has the form

a : [r· = bl] & [r· = <sup>b</sup> 2] & ... & [r. = b,,] '1 '2 '{J ,..

where {ril' ri 2 • ... . ri

B

I is a collection of exogenous features and

(b I> b 2 ••••• <sup>b</sup>�) E B� . a can be characterized by the function

Xor : R -( 0, 1) defined by

Xa( r) <sup>=</sup>Ii: 'h I r; (r)+bJ 1' = 1 I'

which is I if the features al l have the correct values, and 0 otherwise.

A convenient specification of a is as the knowledge vector

ka = (0, O •...• 0, bi

l

, O •...• 0, b'2

' O •...• 0, b;{J ' <sup>O</sup>•... • 0)

E {- I, O,+ II <sup>N</sup>

in which the i

l'

th element is bl' and the remaining elements are all

zero.

De! A set 0 of observables is a col lection of basic events.

De! Let p be an environment and 0 be a set of observables. The observable statistics of p is the set of probabi lities of all the events in 0:

{p (a )}a E o·

some arbitrary initial distribution, pr(x (O) = x). Given the initial state x, the new state at Time 1, x(1), is constructed as follows. One of the <sup>n</sup>coordinates of M is selected (with uniform distribution) for updating. All the other n- <sup>l</sup>coordinates of x (1) will be the same as those of x (0) = x. The updated coordinate can retain its previous value, leading to x (1) = x, or it can flip to the other binary value, leading to a new state that wi ll be denoted x' . The selection of the value of the updated coordinate for x (1) is stochastically chosen according to the

likelihood ratio:

pr(x (1) = x' )

=

Po(x' )

pr(x (1) = x) Po(x )

(where Po is the probabil ity distribution for t = 0 in the given sequence (p, } ;",\_ 0) . This process-randomly select a coordinate to update and stochastically select a binary value for that coordinate-is iterated indefinitely, producing states x (t) for all times <sup>t</sup>= 1, 2, . . . . At each time t, the likelihood ratio of values for the

stochastic choice is determined by the distribution p, .

De/. Let P be a probability distribution. Define the one-parameter

family of distributions Pr by Pr = Ni l <sup>p</sup>

v r

where the normalization constants are

Nr = I. P (x) <sup>v</sup> <sup>r</sup> . x E x

<sup>T</sup>is cal led the temperature parameter. An annealing schedule T is a sequence of positive values (T,},':o that converge to zero. The annealing process determined by P and T is the heat bath stochastic process

determined by the sequence of distributions, Pr

. If P is the Gibbs dis-

I

tribution determined by V, then

where

Pr (x ) = Zi l eV(x )/r

Zr = I. eV ( <sup>x</sup> )/T. <sup>x</sup>EX

This is the same (except for the sign of the exponent) as the relationship that holds in classical statistical mechanics between the probabi lity P ( x) of a microscopic state x, its energy V (x) , and the temperature T. This is the basis for the name� "temperature" and "anneal ing schedule." In the anneal ing processCfefJ(fli'iJ18tM�ution PH of Theorem 1 on

which states that in an ensemble of systems with states distributed according to p, the number of transitions from x' to x is equal to the

number from x to x

. Detai led balance holds because, for the non-

trivial case in which x' and x differ in the single coordinate v, the tran-

sition matrix W determined by the distribution p is

w. = p p (x ) x I v

p (x ) + p (x' )

where Pv is the probabi lity of selecting for update the coordinate v.

Now we have

pr(x (t+l) = x ) = L p ( x') WX' I = L p (x ) WI I·

x' E X. x' E X.

= p ( x ) <sup>L</sup>Wxx' = p(x ). y' E X.

The last equal ity follows from

<sup>L</sup>Wxx' = 1

x' E X.

which simply states that the probability of a transition from x to some state x· is 1. The conclusion is that the probabi <sup>l</sup> ity distribution at time

t+ 1 remains p , which is therefore a stationary distribution.

Part B: Part A assures us that with infinite patience we can arbitrarily well approximate the distribution Pr at any finite temperature T. It seems intuitively clear that with still further patience we could sequential ly approximate in one long stochastic process a series of distributions Pr with temperatures T, monotonical ly decreasing to zero. This pro-

I

cess would presumably converge to the zero-temperature distribution that corresponds to the maximum-likel ihood completion function. A

proof that this is true, provided

T,>C/lnt

for suitable C, can be found in S. Geman and D. Geman (1 984) .

Theorem 3. We now pick up the analysis from the end of the proof

of Theorem 1.

Lemma. (S. Geman, personal communication , 1 984 ,) The values of the Lagrange multipliers A. = !A-ala E 0 defining the function U of

Theorem 2 are those that minimize the convex function: Copyrighted Material

I rA.a [)Ca (r ) - Pa 1/

F(A) = In Zv (A ) = In L e

aE 0

r E R

Proof of Lemma: Note that

pu (r) = pv(r) = ZV (}.. )- l <sup>e</sup> V( <sup>r</sup>

)

where

V(r) <sup>=</sup>L A"lx,,( r)- p,,] = U( r)- <sup>L</sup>A-" p". " E O "E O

From this it follows that the gradient of F is

of

(lA-a =<sup>&</sup>lt;x'" > Pu - p",

The constraint that A enforces is precisely that this vanish for all a; then Pu = 1f'. Thus the correct A is a critical point of F. To see that in fact the correct A is a minimum of F, we show that F has a positive-definite matrix of second-partial deri vatives and is therefore

convex. It is straightforward to verify that the quadratic form

<sup>a</sup> <sup>2</sup>F L q q,,'

",,,,' EO

" OA-" (lA-,,'

is the variance

< (Q - <sup>&</sup>lt; Q>p)

2 >Pu

of the random variable Q defined by Q ( r) = L q" x'" ( r). This

"E O

variance is clearly nonnegative definite. That Q cannot vanish is assured by the assumption that the Xa are linearly independent. Since a Gibbs distribution Pu is nowhere zero, this means that the variance of

Q is positive, so the Lemma is proved.

Proof of Theorem 3: Since F is convex, we can find its minimum, A- , by gradient descent from any starting point. The process of learning the correct A, then, can proceed in time according to the gradient des-

cent equation

dA" � ex:

of - aA"

= - « X,,>pu - p",

) = <X"'

>P - <X,,

>pu

where it is understood that the function U changes as A. changes. The two phases of the trace learning procedure generate the two terms in this equation. In the e�tttalA6IImJjallttion phase, the increment

<Xa> p is estimated� in the environmental simulation phase, the decrement <Xa >pu is estimated (following Theorem 2) . By hypothesis, these estimates are accurate. (That is, this theorem treats the ideal case of perfect samples, with sample means equal to the true population means.) Thus A wi ll converge to the correct val ue. The proportional relation between CT and A was derived in the proof of Theorem 1 .

Theorem 4. The proof of Theorem 3 shows that the trace learning procedure does gradient descent in the function F. The Boltzmann

learning procedure does gradient descent in the function G: Pu

(r) G(A) <sup>=</sup>-<sup>L</sup>

r

P(r) In--P (r)

where, as always, the function U impl icitly depends on A. Theorem 4 wi ll be proved by showing that in fact F and G differ by a constant independent of A, and therefore they define the same gradient descent

trajectories. From the above definition of V, we have

V( r) = U(r) - L Aa<Xa> = <sup>U</sup> (r) - <U>

aEO

where, here and henceforth, < > denotes expectation values with

respect to the environmental distribution p. This implies

i.e. ,

Zv <sup>=</sup>Zu e-<u>.

By the definition of F,

<sup>F</sup>= In Zv = InZu - < U> = < In Zu - U > .

To evaluate the last quantity in angle brackets, note that

pu (r) = ZiJ l eU (r)

implies

In p u

(r) = -ln Zu + U (r)

so that the preceding equation for F becomes

<sup>F</sup>= < InZu - U> = - <sup>&</sup>lt;Inpu > = - LP(r) InPu(r).

Now,

G <sup>=</sup>- LP (r) In pu(r) + !:p (r) Inp(r),

so we have

G (). ) = F (). ) - S (P).

Thus, as claimed, G is just F minus a constant that is independent of

).: the entropy of the environment.

constraints of this kind as well as possible, and the human visual system stores enough plausible constraints and is good enough at applying them that it can arrive at the correct interpretation of most normal

images.

The computation may be performed by an iterative search which starts with a poor interpretation and progressively improves it by reducing a cost function that measures the extent to which the current interpretation violates the plausible constraints. Suppose, for example, that each unit stands for a small three-dimensional surface fragment, and the state of the unit indicates the current bet about whether that surface fragment is part of the best three-dimensional interpretation. Plausible constraints about the nature of surfaces can then be encoded by the pairwise interactions between processing elements. For example, two units that stand for neighboring surface fragments of similar depth and surface orientation can be mutually excitatory to encode the constraints that each of these hypotheses tends to support

the other (because objects tend to have continuous surfaces).

RELAXATION SEARCHES

The general idea of using parallel networks to perform relaxation searches that simultaneously satisfy mUltiple constraints is appealing. It might even provide a successor to telephone exchanges, holograms, or communities of agents as a metaphor for the style of computation in cerebral cortex. But some tough technical questions have to be answered before this style of computation can be accepted as either

efficient or plausible:

• Wi1\ the network settle down or will it osci1\ate or wander aim-

lessly?

• What does the network compute by settling down? We need some characterization of the computation that the network performs other than the network itself. Ideally we would like to be able to say what ought to be computed (Marr, 1982) and

then to show that a network can be made to compute it.

• How long does the network take to settle on a solution? If thousands of iterations are required the method becomes implausible as a model of how the cortex solves constraint-

This means that it is bound to give a significant number of errors in modeling environments where very similar vectors have very different probabilities. Better performance can be achieved by annealing the network to a lower final temperature (which is equivalent to making all the weights larger) , but this will make the learning worse for two separate reasons. First, with less errors there is less to drive the learning because it relies on the difference between the phase+ and phasestatistics. Second, it will be harder to reach thermal equil ibrium at this lower temperature and so the co-occurrence statistics will be unreliable. One way of getting good statistics to drive the learning and also getting very few overt errors is to measure the co-occurrence statistics at a

temperature higher than the final one.

Another way of ensuring that the network approaches equilibrium is to el iminate deep, narrow minima that are often not found by the annealing process. Derthick (1984) has shown that this can be done using a longer gentler anneal ing schedule in phase-. This means that the network is more likely to occupy the hard-to-find minima in phasethan in phase+ , and so these minima will get filled in because the learning rule raises the energies of states that are occupied more in phase-

than in phase+ .

AN EXAMPLE OF HARD LEARNING

A simple example which can only be solved by capturing the higher order statistical structure in the ensemble of input vectors is the "shifter" problem. The visible units are divided into three groups. Group VI is a one-di mensional array of 8 units, each of which is clamped on or off at random with a probability of 0. 3 of being on. Group V2 also contains 8 units and their states are determined by shifting and copying the states of the units in group VI' The only shifts allowed are one to the left, one to the right, or no shift. Wrap-around is used so that when there is a right shift, the state of the right-most unit in VI determines the state of the left-most unit in V2• The three possible shifts are chosen at random with equal probabi liti es. Group V3 contains three units to represent the three possible shifts, so at any one

time one of them is clamped on and the others are clamped off.

The problem is to learn the structure that relates the states of the three groups. One facet of this problem is to "recognize" the shifti.e., to complete a partial input vector in which the states of VI and V2 are clamped but the units in V3 are left free. It is fairly easy to see why this problem cannot possibly be solved by just adding together a lot of pairwise interactions between units in Vb V2, and V3• If you know Copyrighted Material

FIGURE 3. Theweights of the 24 hidden units in the shifter network . Each large regi<sup>o</sup><sup>n</sup> corresponds to a unit. Within this region the black rectangles represent negati ve weights and the white rectangles represent positive ones. The size of a rectangle represents t h<sup>e</sup> magnitude of the weight. Thetwo rows of weights at the bottom of each uni t are its connections to the two groups of input units, VI and V2 . These weights the refore represent the"receptive field" of the hidde<sup>n</sup>unit. The three weights in the middle of the top row of each unit are its connections to the thre<sup>e</sup>output units that represent shift - l<sup>e</sup> ft , noshift, and shift-right. The solitary weight at the top left of each unit is its threshold . Each hidden unit is directly connected to all 16 input units and all 3 <sup>o</sup>utput units. In this example, the hidden units are not connected to e<sup>a</sup>ch other. The top-left unit has weight<sup>s</sup> that are easy to understand: Its optimal stimulusis act ivity in thefourth unit of VI an<sup>d</sup> the fifth unit of V2 , and it votes for shift-right. It has negative wei ghts to make it less likely to <sup>c</sup>ome on when there is an alternati ve explanation for why its two f<sup>a</sup>voriteinput

units are act ive.

different clamped vectors and the co-occurrence statistics were averaged over all 20 runs to yield an estimate, for each connection , of Pi) in Equation 7. In phase- , none of the units were clamped and the network was annealed in the same way. The network was then run for <sup>a</sup> further 10 iterations and the co-occurrence statistics were collected for all connected pairs of units. This was repeated 20 times and the co-

The entire set of 40 annealings that were used to estimate Pit and Pi} was called a sweep. After each sweep, every weight was incremented by 5 (Pit - Pin . In addition, every weight had its absolute magnitude decreased by 0.0005 times its absolute magnitude. This weight decay prevented the weights from becoming too large and it also helped to resuscitate hidden units which had predominantly negative or predominantly positive weights. Such units spend all their time in the same state and therefore convey no information. The phase+ and phasestatistics are identical for these units, and so the weight decay gradually erodes their weights until they come back to life (units with all zero

weights come on half the time) .

The Annealing Schedule

The annealing schedule spent the following number of iterations at the following temperatures : 2 at 40, 2 at 35, 2 at 30, 2 at 25, 2 at 20, 2 at 15, 2 at 1 <sup>2</sup> , 2 at 10. One iteration is defined as the number of random probes required so that each unit is probed one time on average. When it is probed, a unit uses its energy gap to decide which of its two states to adopt using the stochastic decision rule in Equation 3. Since each unit gets to see the most recent states of all the other units, an iteration cannot be regarded as a single parallel step. An truly parallel asynchronous system must tolerate time delays. Units must decide on their new states without being aware of very recent changes in the states of other units. It can be shown (Sejnowski , Hinton , Kienker, & Schumacher, 1985) that first-order time delays act like added tempera-

ture and can therefore be tolerated by networks of this kind.

The Performance of the Shifter Network

The shifter network is encouraging because it is a clear example of the kind of learning of higher order structure that was beyond the capability of perceptrons, but it also il lustrates several weaknesses in the

current approach.

• The learning was very slow. It required 9000 learning sweeps, each of which invol ved reaching equili brium 20 times in phase+ with vectors clamped on VI> V2, and V3, and 20 times in phase- with no units clamped. Even for low-level perceptual

magnitude decreased by 1. For each weight, the probability of this hap-

pening was 0.0005 times the absolute magnitude of the weight.

We found that the network performed better if there was a preliminary learning stage which just involved the sememe units. In this stage, the intermediate units were not yet connected. During phase+ the required patterns were clamped on the sememe units an<sup>d</sup>Pit wa<sup>s</sup>measured (annealing was not required because al l the units involved were clamped) . During phase- no units were clamped and the network was allowed to reach equi librium 20 times using the annealing schedule given above. After anneal <sup>i</sup>n<sup>g</sup> , pi; was estimated from the cooccurrences as before, except that only 20 phase- anneal ings were used instead of 40. There were 300 sweeps of this learning stage and they resulted in weights between pairs of sememe units that were sufficient to give the sememe group an energy landscape with 20 strong minima corresponding to the 20 possible "word meanings." This hel ped subsequent learning considerably, because it reduced the tendency for the intermediate units to be recruited for the job of modeling the structure among the sememe units. They were therefore free to model the structure between the grapheme units and the sememe units.9 The results described here were obtained using the preliminary learning stage and so they correspond to learning to associate grapheme strings with

"meanings" that are already fami liar.

The Performance of the Network

Using the same annealing schedule as was used during learning, the network can be tested by clamping a grapheme string and looking at the resulting activities of the sememe units. After 5000 learning sweeps, it gets the semantic features exactly correct 99.3% of the time. A performance level of 99.9% can be achieved by using a "careful" annealing schedule that spends twice as long at each temperature and

goes down to half the final temperature.

The Effect of Local Damage

The learning procedure generates weights which cause each of the units in the intermediate layer to be used for many different words.

9 There was no need to have a similar stage for learning the structure among the grapheme units because in the main stage of learning the grapheme units are always clamped and so there is no tendency fr(;ttp�StJtMlltf!HitIfOdel the structure among them.

yv <sup>B</sup>

V\J A B

increase weights that help B ...

FIGURE 5. One cross-section of a ravine in weight-space. Each point in weight space <sup>c</sup>orresponds to a whole energy landscape. To indicate this, we show how a very simple landscape changes as the weights are changed. Moveme<sup>n</sup>t to the right along the K-aKis corresponds to increasing the weights between pairs of units that are both on in state B and not both on in state A. This increases the depth of A. If the task requires that <sup>A</sup> and B have about the same depth, an imbalance between them will lower the

performance and thus raise G.

between the various minima without significantly affecting the gross topography of the energy landscape. Relearning can then restore most of the performance by restoring the balance between the existing

minima.

The simulation behaved as predicted. The mean absolute value of the weights connecting the intermediate units to the other two groups was 21. 5. These weights were first perturbed by adding uniform random noise in the range - <sup>2</sup>to + 2. This had surprisingly little effect, reducing the performance using the normal annealing schedule from 99.3% to 98.0%. This shows that the network is robust against slight noise in the weights. To cause significant deterioration, uniform random noise between - <sup>22</sup>and + <sup>22</sup>was added. On average, this perturbs each weight by about half its magnitude which was enough to reduce normal performance to 64.3% correct. Figure 6 shows the course of the relearning and compares it with the speed of the original learning when performance was at this level. It also shows that other kinds of damage

learning in which the network constructs efficient internal codes for communicating information across narrow bandwidth channels. At present, the learning algorithm is too slow to be tested properly on large networks and future progress hinges on being able to speed it up.

ACKNOWLEDGMENTS

This research was supported by grants from the System Development Foundation. We thank David Ackley, Peter Brown, Francis Crick, Mark Derthick, Scott Fahlman, Stuart Geman, John Hopfield, Paul Kienker , Jay McClelland, Barak Pearlmutter, David Rumelhart, Tim

Shallice, and Paul Smolensky for helpful discussions.

APPENDIX:

DERIV ATION OF THE LEARNING ALGORITHM

When a network is free-running at equilibrium the probability distrib-

ution over the visible units is given by

P-( Va) == LP-( VaA Hp) p

(8)

where Va is a vector of the states of the visible units, HI3 is a vector of states of the hidden units , and Eal3 is the energy of the system in state

Va AHp

Hence,

Ea/3 - L wij sri3 st13 , i <j

n -EafJ IT <sup>e</sup>\_

1 aQ al3 -EafJ

l T

- -so "'s · <sup>e</sup>nUw-IJ . T I }

Differentiating (8) then yields

This derivative is used to compute the gradient of the G -measure

where P + (Va) is the clamped probability distribution over the visible

units and is independent of Wij ' So

8G = \_ L P:(Va) ap -(va) aWl} aP ( Va) aWij

Now,

and

P+(Va AHf,i ) =P+(Hf,i1 Va )P <sup>+</sup> ( Va),

P-( Va AHp ) =P-(Hp IVa)P-( Va ) ,

(9)

Equation 9 holds because the probability of a hidden state given some visible state must be the same in equilibrium whether the visible units were clamped in that state or arrived there by free-running. Hence,

Also,

where

and

\_ P+ (Va) + P ( Va AHp ) P- (Va ) = P ( VaA Hp ) .

a

Pij= !.P+ (VaA Hp )srP slP

ap

Pij=!.P - (V>, A HjJ)S/ILS/IL.

>'IL

The Boltzmann machine learning algorithm can also be formulated as an input-output model. The visible units are divided into an input set f and an output set 0, and an environment specifies a set of conditional probabi lities of the form P + (Op Ifa)' During phase+ the environment Copyrighted Material

clamps both the input and output units, and the Pit <sup>s</sup>are estimated. During phase- the input units are clamped and the output units and hidden units free-run , and the PijS are estimated. The appropriate G

measure in this case is

<sup>G</sup>=l:P <sup>+</sup> (

Ja <sup>A</sup> O/3 ) lnP+ (O/3l /a )

.

a/3 p- (O/3 l/a )

Similar <sup>m</sup>athematics apply in this formulation and aG/a wij is the same

as before.

CHAPTER S

Learning Internal Representations

by Error Propagation

D. E. RUMELHART, G. E. HINTON, and R. 1. WILLIAMS

THE PROBLEM

We now have a rather good understanding of simple two-layer associative networks in which a set of input patterns arriving at an input layer are mapped directly to a set of output patterns at an output layer. Such networks have no hidden units. They involve only input and output units. In these cases there is no internal representation. The coding provided by the external world must suffice. These networks have proved useful in a wide variety of appl ications (cf. Chapters 2, 17, and 18). Perhaps the essential character of such networks is that they map similar input patterns to similar output patterns. This is what al lows these networks to make reasonable generalizations and perform reasonably on patterns that have never before been presented. The similarity of patterns in a PDP system is determi ned by their overlap. The overlap in such networks is determined outside the learning system itself-by

whatever produces the patterns.

The constraint that similar input patterns lead to similar outputs can lead to an inability of the system to learn certain mappings from input to output. Whenever the representation provided by the outside world is such that the similarity structure of the input and output patterns are very different, a net\\C),\*Jy�ijfltetttMatt8i8h1 representations (Le. , a

network without hidden units) will be unable to perform the necessary mappings. A classic example of this case is the exclusive-or (XOR) problem illustrated in Table 1. Here we see that those patterns which overlap least are supposed to generate identical output values. This problem and many others like it cannot be performed by networks without hidden units with which to create their own internal representations of the input patterns. It is interesting to note that had the input patterns contained a third input taking the value 1 whenever the first two have value 1 as shown in Table 2 , a two-layer system would be able

to solve the problem.

Minsky and Papert (I969) have provided a very careful analysis of conditions under which such systems are capable of carrying out the required mappings . They show that in a large number of interesting cases , networks of this kind are incapable of solving the problems. On the other hand, as Minsky and Papert also pointed out, if there is a layer of simple perceptron-like hidden units, as shown in Figure 1, with which the original input pattern can be augmented, there is always a recoding (i.e., an internal representation) of the input patterns in the hidden units in which the similarity of the patterns among the hidden units can support any required mapping from the input to the output units. Thus, if we have the right connections from the input units to a large enough set of hidden units, we can always find a representation that wi ll perform any mapping from input to output through these hidden units. In the case of the XOR problem, the addition of a feature that detects the conjunction of the input units changes the similarity

> Input Patterns

TABLE <sup>1</sup>

Output Patterns

TABLE <sup>2</sup>

o 1 I o

Output Patterns

000 0 <sup>010</sup>1 100 1

111

0

Output Patterns

Input Patterns

Internal

Representation

Units

FIGURE I. A multilayer network. In this case the information coming to the input units is reroded into an internal representation and the outputs are generated by the inter· nal representation rather than by the original pattern. Input patterns can always be encoded, if there are enough hidden units, in a form so tha<sup>t</sup>the <sup>a</sup>ppropriate output pat·

tern can be generated from any input pattern.

structure of the patterns sufficiently to al low the solution to be learned. As illustrated in Figure 2, this can be done with a single hidden unit. The numbers on the arrows represent the strengths of the connections among the units. The numbers written in the circles represent the thresholds of the units. The value of + 1. 5 for the threshold of the hidden unit insures that it wi ll be turned on only when both input units are on. The value 0.5 for the output unit insures that it will tu rn on only when it receives a net positive input greater than 0.5. The weight of - 2 from the hidden unit to the output unit insures that the output unit wi ll not come on when both input units are on. Note that from the point of view of the output unit, the hidden unit is treated as simply another input unit. It is as jf the jOP'ut . p'atterns consi sted of three

Copynghted Matenal

& Anandan, 1985). In this chapter we present another alternative that works with deterministic units, that involves only local computations, and that is a clear generalization of the delta rule. We call this the generalized delta rule. From other considerations, Parker (1985) has independently derived a simi lar generalization, which he calls learninglogic. Le Cun (1985) has also studied a roughly similar learning scheme. In the remainder of this chapter we first derive the generalized delta rule, then we il lustrate its use by providing some results of our simulations , and finally we indicate some further generalizations of

the basic idea.

THE GENERALIZED DELTA RULE

The learning procedure we propose involves the presentation of a set of pairs of input and output patterns. The system first uses the input vector to produce its own output vector and then compares this with the desired output, or target vector. If there is no difference, no learning takes place. Otherwise the weights are changed to reduce the difference. In this case, with no hidden units, this generates the standard delta rule as described in Chapters 2 and 11. The rule for changing

weights following presentation of input/ output pair p is given by

(1)

where tpj is the target input for jth component of the output pattern for pattern p, Opj is the jth element of the actual output pattern produced by the presentation of input pattern p, ip; is the value of the ith element of the input pattern, 8pi = <sup>t</sup>pi - 0pi' and flp wi} is the change to be made to the weight from the ith to the jth unit following presentation

of pattern p .

The delta rule and gradient descent. There are many ways of deriving this rule. For present purposes, it is useful to see that for linear units it minimizes the squares of the differences between the actual and the desired output values summed over the output units and all pairs of input/ output vectors. One way to show this is to show that the derivative of the error measure with respect to each weight is proportional to the weight change dictated by the delta rule, with negative constant of proportionality. This corresponds to performing steepest descent on a surface in weight space whose height at any point in weight space is equal to the error measure. (Note that some of the following sections Copyrighted Material

are written in italics. These sections constitute informal derivations of the claims made in the surrounding text and can be omitted by the

reader who finds such derivations tedious')

To be more specific, then, let

<sup>1</sup>Ep <sup>=</sup>2'1;: «(pj - Opj)2

I

be our measure of the error on input/output pattern p and let E = LEp be our overall measure of the error. We wish to show that the delta rule implements a gradient descent in E when the units are linear. We will proceed by simply showing

that

which is proportional to Lip Wj; as prescribed by the delta rule. When there are no hidden units it is straightforward to compute the relevant derivative. For this purpose we use the chain rule to write the derivative as the product of two parts: the derivative of the error with respect to the output of the unit times the derivative of the out-

put with respect to the weight.

aEp aEp aOpj aWj; = aOpj aWj; • (])

The first part tells how the error changes with the output of the j th unit and the second part tells how much changing Wj; changes that output. Now, the derivatives

are easy to compute. First, from Equation 2

aEp !l

= - (tpj - op) = - apj'

uOpj

(4)

Not surprisingly, the contribution of unit Uj to the error is simply proportional to a pj .

Moreover, since we have linear units,

Opj = LWjlip;. i

from which we conclude that aOpj . !l = '

p i' u Wj;

Thus, substituting back into Equation 3, we see that

aEp . - -!l- <sup>=</sup>apj/PI UWji

as desired. Now, combining this with the observation that

aE = 1: aEp aWj; p aWj;

should lead us to conclude that the net change in Wj; after one complete cycle of pattern presentations is proportional to this derivative and hence that the delta rule implements a gradient descent in E. In fact, this is strictly true only if the values of the weights are not changed during this cycle. By changing the weights after each pattern is presented we depart to some extent from a true gradient descent in E. Nevertheless, provided the learning rate (i.e., the constant of proportionality) is sufficiently small, this departure will be negligible and the delta rule will implement a very close approximation to gradient descent in sum-squared error. In particular, with small enough learning rate, the delta rule will find a set of weights minimizing this

error function.

The delta rule for semilinear activation functions in feedforward networks. We have shown how the standard delta rule essentially implements gradient descent in sum-squared error for linear activation functions. In this case, without hidden units, the error surface is shaped like a bowl with only one minimum, so gradient descent is guaranteed to find the best set of weights . With hidden units, however, it is not so obvious how to compute the derivatives , and the error surface is not concave upwards , so there is the danger of getting stuck in local minima. The main theoretical contribution of this chapter is to show that there is an efficient way of computing the deri vatives. The main empirical contribution is to show that the apparently fatal problem of

local minima is irrelevant in a wide variety of learning tasks.

At the end of the chapter we show how the general ized delta rule can be appl ied to arbitrary networks , but, to begin With , we confine ourselves to layered feedforward networks . In these networks , the input units are the bottom layer and the output units are the top layer. There can be many layers of hidden units in between , but every unit must send its output to higher layers than its own and must receive its input from lower layers than its own. Given an input vector, the output vector is computed by a forward pass which computes the activity levels of each layer in turn using the already computed activity levels in the ear-

lier layers.

Since we are primari ly interested in extending this result to the case with hidden units and since, for reasons outlined in Chapter 2, hidden units with linear activation functions provide no advantage, we begin by general izing our analysis to the set of nonlinear activation functions which we cal <sup>l</sup>semilinear (see Chapter 2). A semilinear activation function is one in which the output of a unit is a nondecreasi ng and dif-

ferentiable function of t�d>MfJl8tjal

where 0; = i; if unit i is an input unit. Thus, a semilinear acti vation

function is one in which

(8)

and f is differentiable and nondecreasi ng. The gene <sup>r</sup>al i <sup>z</sup>ed delta rule works if the network consists of units having semilinear activation functions. Notice that li near threshold units do not sati sfy the requirement because their derivative is infinite at the threshold and zero elsewhere.

To get the correct generalization of the delta rule. we must set

aEp

ap wji ex: - -!\-,

V wji

where E is the same sum-squared error function defined earlier. As in the standard delta rule it is again useful to see thiS derivative as resulting from the product of two parts: one part reflecting the change in error as a function of the change in the net input to the unit and one part representing the effect of changing a particular weight

on the net input. Thus we can write

aEp aEp anetpj aWji = ane/pj aWji .

By Eq<sup>u</sup>ation 7 we see that the second factor is

Now let us define

aEp

0·

PJ <sup>=</sup> ---anel .. PJ

(9)

(JO)

(By <sup>c</sup>omparing this to Equation 4. note that this is consistent with the definition of o pj used in the original delta rule jor linear units since Opj = netpj when unit Uj is

linear. ) Equation 9 thus has the equi valent form

aEp --!\- <sup>=</sup>OpjOp;. VWji

This says that to implement gradient descent in E we should make our weight

changes according to

just as in the standard delta rule. The trick is to figure out what 8p

j should be for

each unit U) in the network. The interesting result, which we now derive, is that there is a simple recursive computation 0/ these 8 's which can be implemented by

propagating error signals backward through the network. aE

To compute 8 p) = - �, we apply the chain rule to write this partial deriva-

onetp)

tive as the product 0/ two factors, one factor reflecting the change in error as a /unction 0/ the output 0/ the unit and one reflecting the change in the output as a /unc-

tion 0/ changes in the input. Thus, we have

aEp aEp aop) 8p) = --- = ------, anetpj aOpj anetpj

Let us compute the second factor. By Equation 8 we see that

aOpj \_ I �-- - I ; (netpj),

onetpj

(1)

which is simply the derivative 0/ the squashing /unction Ij for the j th unit, evaluated at the net input netp) to that unit. To compute the first factor, we consider two cases. First, assume that unit Uj is an output unit 0/ the network. In this

case, it /ollows /rom the definition 0/ Ep that

aEp

� = - (tpj - Opj),

Op)

which is the same result as we obtained with the standard delta rule. Substituting

for the two factors in Equation 11, we get

(J3)

for any output unit U). /fUj is not an output unit we use the chain rule to write

12 flEp anetplc = 12 aEp a-Lwk;op

;= 12 aE

p Wkj=-L8p1cWkj'

k anetpk aOpj k anetpk aOpj ; k anetpk k

In this case, substituting /or the two factors in Equation 12 yields

8 pj = I 'j (netpj ) 128 pic Wkj

(J4)

k

whenever u) is not an output unit. Equations J3 and 14 give a recursive procedure for computing the 8 's for all units in the network, which are then used to compute the weight changes in the network according to Equation 11. This procedure constitutes the generalized delta rule for a /eed/orward network 0/ semilinear units.

These results can be summarized in three equations. First, the generalized delta rule has exactly the same form as the standard delta rule of Equation 1. The weight on each line should be changed by an amount proportional to ctbp� MfJitftialrror signal, 8, available to

the unit receiving input along that line and the output of the unit send-

ing activation along that line. In symbols,

The other two equations specify the error signal. Essentially, the determination of the error signal is a recursive process which starts with the output units. If a unit is an output unit, its error signal is very simi lar

to the standard delta rule. It is given by

S pj = (tp

j - Opj)/ j (netp

j)

where / j (netpj) is the derivative of the semi linear activation function which maps the total input to the unit to an output value. Finally, the error signal for hidden units for which there is no specified target is determined recursively in terms of the error signals of the units to which it directly connects and the weights of those connections. That is,

S pj = / j (netp

j ) I,s pk Wkj k

whenever the unit is not an output unit.

The application of the general ized delta rule, thus, involves two phases: During the first phase the input is presented and propagated forward through the network to compute the output value Opj for each unit. This output is then compared with the targets, resulting in an

error signal Sp

j for each output unit. The second phase involves a

backward pass through the network (analogous to the initial forward pass) during which the error signal is passed to each unit in the network and the appropriate weight changes are made. This second, backward pass allows the recursive computation of 8 as indicated above. The first step is to compute S for each of the output units. This is simply the difference between the actual and desired output values times the derivative of the squashing function. We can then compute weight changes for all connections that feed into the final layer. After this is done, then compute 8 's for all units in the penultimate layer. This propagates the errors back one layer, and the same process can be repeated for every layer. The backward pass has the same computational complexity as the forward pass , and so it is not unduly expensive. We have now generated a gradient descent method for finding weights in any feedforward network with semi linear units. Before reporting our results with these networks , it is useful to note some further observations. It is interesting that not all weights need be variable. Any number of weights in the network can be fixed. In this case, error is still propag��.YIfghrJW'M�"GOxed weights are simply not

modified. It should also be noted that there is no reason why some output units might not recei ve inputs from other output units in earl ier layers. In this case, those units receive two di fferent ki nds of error: that from the direct comparison with the target and that passed through the other output units whose acti vation it affects. In this case , the correct procedure is to simply add the weight changes dictated by the direct comparison to that propagated back from the other output units.

SIMULATION RESULTS

We now have a learning procedure which could, in principle, evolve a set of weights to produce an arbitrary mapping from input to output. However, the procedure we have produced is a gradient descent procedure and, as such, is bound by all of the problems of any hill climbing procedure-namely, the problem of local maxima or (in our case) minima. Moreover, there is a question of how long it might take a system to learn. Even if we could guarantee that it would eventually find a solution , there is the question of whether our procedure could learn in a reasonable period of time. It is interesting to ask what hidden units the system actually develops in the solution of particular problems. This is the question of what kinds of internal representations the system actually creates. We do not yet have definitive answers to these questions. However, we have carried out many simulations which lead us to be optimistic about the local minima and time questions and to be surprised by the kinds of representations our learning mechanism discovers. Before proceeding with our results, we must describe our simulation system in more detail. In particular, we must specify an activation function and show how the system can compute the derivative of

this function.

A useful activation function. In our above deri vations the derivative of the acti vation function of unit u), r j (net), always played a role. This implies that we need an acti vation function for which a deri vative exists. It is in teresting to note that the linear threshold function, on which the percept ron is based, is discontinuous and hence will not suffice for the generalized delta rule. Simi larly, since a linear system achieves no advantage from hidden uni ts, a linear activation function will not suffice either. Thus, we need a contin uous, nonlinear activation function. In most of our experi ments we have used the logistic

acti vation function in w��Jyrighted Material

(15)

where () j is a bias similar in function to a threshold. 1 In order to apply our learning rule, we need to know the derivative of this function with

respect to its total input, netpj, where netpj <sup>=</sup>L, wJ

; op; + () <sup>J</sup>

. It is easy to

show that this derivative is given by

aOpj

-!l-- = Opj 0- op).

unetpj

Thus, for the logistic acti vation function, the error signal, Bpj, for an

output unit is given by

Bpj = (tP) - Opj )op} (1 - op}),

and the error for an arbitrary hidden Uj is given by

Spj = op}O - op})L,SpkWk}'

k

It should be noted that the derivative, Opj (1 - op), reaches its maximum for Opj = 0.5 and, since 0::::; Opj::::; 1, approaches its minimum as Opj approaches zero or one. Since the amount of change in a given weight is proportional to this deri vati ve, weights will be changed most for those units that are near their midrange and, in some sense, not yel committed to being either on or off. This feature, we believe, contri-

butes to the stability of the learning of the system.

One other feature of this activation function should be noted. The system can not actual ly reach its extreme values of 1 or 0 without infinitely large weights. Therefore, in a practical learni <sup>n</sup>g situation in which the desired outputs are binary (O, 1), the system can never actually achieve these values. Therefore, we typical <sup>l</sup>y use the val ues of 0.1 and 0.9 as the targets, even though we wi ll talk as if values of (0, I} are

sought.

The learning rate. Our learning procedure requires only that the change in weight be proportional to aEp/aw. True gradient descent requires that infinitesimal steps be taken. The constant of proportionality is the learning rate in our procedure . The larger this constant, the larger the changes in the weights. For practical purposes we choose a

learning rate that is as large as possi ble without leading to osci llation. This offers the most rapid learning. One way to increase the learning rate without leading to oscillation is to modify the generalized delta rule to include a momentum term. This can be accomplished by the follow-

ing rule:

(16)

where the subscript n indexes the presentation number, 'T/ is the learning rate, and a is a constant which determines the effect of past weight changes on the current direction of movement in weight space. This provides a kind of momentum in weight space that effectively filters out high-frequency variations of the error-surface in the weight space. This is useful in spaces containing long ravines that are characterized by sharp curvature across the ravine and a gently sloping floor. The sharp curvature tends to cause divergent osci llations across the ravine. To prevent these it is necessary to take very small steps, but this causes very slow progress along the ravine. The momentum fi lters out the high curvature and thus allows the effective weight steps to be bigger. In most of our simulations a was about 0.9. Our experience has been that we get the same solutions by setting a = 0 and reducing the size of 'T/, but the system learns much faster overall with larger values of <sup>a</sup>

and TJ.

Symmetry breaking. Our learning procedure has one more problem that can be readily overcome and this is the problem of symmetry breaking. If all weights start out with equal values and if the solution requires that unequal weights be developed, the system can never learn. This is because error is propagated back through the weights in proportion to the values of the weights. This means that all hidden units connected directly to the output inputs will get identical error signals, and, since the weight changes depend on the error signals , the weights from those units to the output units must always be the same. The system is starting out at a kind of local maximum. which keeps the weights equal , but it is a maximum of the error function, so once it escapes it will never return. We counteract this problem by starting the system with small random weights. Under these conditions symmetry problems of

this kind do not arise.

The XOR Problem

It is useful to begin with the exclusive-or problem since it is the classic problem requiring hidden units and since many other difficult Copyrighted Material

problems involve an XOR as a subproblem. We have run the XOR problem many times and with a couple of exceptions discussed below, the system has always solved the problem. Figure 3 shows one of the solutions to the problem. This solution was reached after 558 sweeps through the four stimulus patterns with a learning rate of." = 0.5. In this case, both the hidden unit and the output unit have positive biases so they are on unless turned off. The hidden unit turns on if neither input unit is on. When it is on, it turns off the output unit. The connections from input to output units arranged themselves so that they turn off the output unit whenever both inputs are on. In this case, the network has settled to a solution which is a sort of mirror image of the

one illustrated in Figure 2.

We have taught the system to solve the XOR problem hundreds of times. Sometimes we have used a single hidden unit and direct connections to the output unit as il lustrated here, and other times we have allowed two hidden units and set the connections from the input units to the outputs to be zero, as shown in Figure 4. In only two cases has the system encountered a local minimum and thus been unable to solve the problem. Both cases inv<sup>o</sup>lved the two hidden units version of the

Output Unit

-4.2 I 1 I

\-42 \

I \ -9. 41

I

I � <sup>I</sup>\_\_ X \ Hidden Unit

\ \

---6.4

Input Units

FIGURE 4. A simple architecture for solving XOR with two hidden units and no direct

connections from input to output.

problem and both ended up in the same local mInimum. Figure 5 shows the weights for the local minimum. In this case, the system correctly responds to two of the patterns-namely, the patterns 00 and 10. In the cases of the other two patterns 11 and 01, the output unit gets a net input of zero. This leads to an output value of 0.5 for both of these patterns. This state was reached after 6,587 presentations of each pattern with 'T/=O.25. 2 Although many problems require more presentations for learning to occur, further trials on this problem merely increase the magnitude of the weights but do not lead to any improvement in performance. We do not know the frequency of such local minima, but our experience with this and other problems is that they are quite rare. We have found only one other situation in which a local minimum has occurred in many hundreds of problems of various

sorts. We wi ll discuss this case below.

The XOR problem has proved a useful test case for a number of other studies. Using the architecture illustrated in Figure 4, a student in our laboratory, Yves Chauvin, has studied the effect of varying the

FIGURE 5. A network at a local minimum for the exclusive-or problem. The dotted lines indicate negative weights. Note that whenever the right most input unit is on it turns on both hidden units. The weights connecting the hidden units to the output are arranged so that when both hidden units are on, the output unit gets a net input of zero. This leads to an output value of 0.5. In the other cases the network provides the correct

answer.

number of hidden units and varying the learning rate on time to solve the problem. Using as a learning criterion an error of 0.01 per pattern, Yves found that the average number of presentations to solve the problem with '1/ = 0.25 varied from about 245 for the case with two hidden units to about 120 presentations for 32 hidden units. The results can be summarized by P = 280 - 3310g2H, where P is the required number of presentations and H is the number of hidden units employed. Thus, the time to solve XOR is reduced linearly with the logarithm of the number of hidden units. This result holds for values of H up to about 40 in the case of XOR. The general result that the time to solution is reduced by increasing the number of hidden units has been observed in virtually all of our simulations. Yves also studied the time to solution as a function of learning rate for the case of eight hid· den units. He found an average of about 450 presentations with '1/ = 0.1 to about 68 presentations with '1/ = 0.75. He also found that Copyrighted Material

themselves so that they count the number of inputs. In the diagram, the one at the far left comes on if one or more input units are on , the next comes on if two or more are on, etc. All of the hidden units come on if all of the input lines are on. The first m hidden units come on whenever m bits are on in the input pattern. The hidden units then connect with alternately positive and negative weights . In this way the net input from the hidden units is zero for even numbers and + 1 for odd numbers. Table 3 shows the actual �ol ution attained for one of our simulations with four input lines and four hidden units. This solution was reached after 2,825 presentations of each of the si xteen patterns with "fI = 0.5. Note that the solution is roughly a mirror image of that shown in Figure 6 in that the number of hidden units turned on is equal to the number of zero input values rather than the number of ones. Beyond that the princi ple is that shown above. It should be noted that the internal representation created by the learning rule is to arrange that the number of hidden units that come on is equal to the number of zeros in the input and that the particular hidden units that come on depend only on the number, not on which input units are on. This is exactly the sort of recoding required by parity. It is not the kind of representation readily discovered by unsupervi sed learning schemes

such as competitive learning.

The Encoding Problem

Ackley, Hinton, and Sejnowski (1 985) have posed a problem in which a set of orthogonal input patterns are mapped to a set of orthogonal output patterns through a small set of hidden units. In such cases the internal representations of the patterns on the hidden units must be rather efficient. Suppose that we attempt to map N input patterns onto N output patterns. Suppose further that log2N hidden units are provided. In this case, we expect that the system will learn to use the

> Number of On Input Units

TABLE 3

Hidden Unit Patterns

o <sup>1111</sup> 1 1011 2 1010 3 0010 4 Copynghted lIfINerial Output Value o 1 o 1 o

output as in the input. Table 5 shows the mapping generated by our learning system on this example. It is of some interest that the system employed its ability to use intermediate values in solving this problem. It could, of course, have found a solution in which the hidden units took on only the values of zero and one. Often it does just that, but in this instance, and many others, there are solutions that use the intermediate values, and the learning system finds them even though it has a bias toward extreme values . It is possible to set up problems that require the system to make use of intermediate values in order to solve

a problem. We now turn to such a case.

Table 6 shows a very simple problem in which we have to convert from a distributed representation over two units into a local representation over four units. The similarity structure of the distributed input pat-

terns is simply not preserved in the local output representation.

We presented this problem to our learning system with a number of constraints which made it especially difficult. The two input units were only allowed to connect to a single hidden unit which, in turn, was allowed to connect to four more hidden units. Only these four hidden units were allowed to connect to the four output units. To solve this problem, then, the system must first convert the distributed

TABLE 5

Input Hidden Unit Output Patterns Patterns Patterns <sup>10000000</sup>.5 0 0 10000000 01000000 0 1 0 01000000 00100000 1 1 0 00100000 <sup>00010000</sup>1 1 1 00010000 00001 000 0 1 1 00001000 <sup>00000100</sup>.5 0 1 00000100 <sup>00000010</sup>1 0 .5 <sup>00000010</sup> 00000001 0 0 .5 <sup>00000001</sup>

TABLE 6

Input Patterns Output Patterns

representation of the input patterns into various intermediate values of the singleton hidden unit in which different activation values correspond to the different input patterns. These continuous values must then be converted back through the next layer of hidden unitsfirst to another distributed representation and then, finally, to a local representation. This problem was presented to the system and it reached a solution after 5,226 presentations with", = 0.05.3 Table 7 shows the sequence of representations the system actually developed in order to transform the patterns and solve the problem. Note each of the four input patterns was mapped onto a particular activation value of the singleton hidden unit. These values were then mapped onto distributed patterns at the next layer of hidden units which were finally mapped into the required local representation at the output level . In princi ple, this trick of mapping patterns into activation values and then converting those activation values back into patterns could be done for any number of patterns, but it becomes increasingly difficult for the system to make the necessary distinctions as ever smal ler differences among activation values must be distingui shed. Figure 8 shows the network the system developed to do this job. The connection weights from the hidden units to the output units have been suppressed for clarity. (The sign of the connection, however, is indicated by the form of the connection -e.g., dashed lines mean inhibitory connections) . The four different activation values were generated by having relatively large weights of opposite sign . One input line turns the hidden unit full on, one turns it full off. The two differ by a relatively small amount so that when both turn on, the unit attains a value intermediate between 0 and 0.5. When neither turns on, the near zero bias causes the unit to attain a value slightly over 0.5. The connections to the second layer of hidden units is likewise interesting. When the hidden unit is full on,

TABLE 7

Input Singleton Remaining Output Patterns Hidden Unit Hidden Units Patterns

10 0 1 1 1 0 0010 11 .2 1 1 0 0 0001 <sup>00</sup>.6 .5 0 0 .3 1000 01 1 0 0 0 1 <sup>0100</sup>

Output Un its

Hidden Units

Input Un its

FIGURE 8. The network illustrating the use of intermediate values in sol ving a problem.

See text for explanation.

the right-most of these hidden units is turned on and all others turned off. When the hidden unit is turned off, the other three of these hidden units are on and the left-most unit off. The other connections from the singleton hidden unit to the other hidden units are graded so that a distinct pattern is turned on for its other two values. Here we

have an example of the flexibility of the learning system.

Our experience is that there is a propensity for the hidden units to take on extreme values, but, whenever the learning problem calls for it, they can learn to take on graded values. It is likely that the propensity to take on extreme values follows from the fact that the logistic is a sigmoid so that increasing magnitudes of its inputs push it toward zero or one. This means that in a problem in which intermediate values are required, the incoming weights must remain of moderate size. It is interesting that the derivation of the generalized delta rule does not depend on all of the units having identical activation functions. Thus, it would be possible for some units, those required to encode information in a graded fashion, to be linear while others might be logistic. The linear unit would have a much wider dynamic range and could encode more different values. This would be a useful role for a linear

unit in a network with h�8p9r\_dMaterial

Symmetry

Another interesting problem we studied is that of classifying inpu<sup>t</sup> strings as to whether or not they are symmetric about their center. We used patterns of various lengths with various numbers of hidden units. To our surprise, we discovered that the problem can always be solved with only two hidden units. To understand the derived representation, consider one of the solutions generated by our system for strings of length six. This solution was arrived at after 1,208 presentations of each six-bit pattern with 'T1 = 0. 1. The final network is shown in Figure 9. For simplicity we have shown the six input units in the center of the diagram with one hidden unit above and one below. The output unit, which signals whether or not the string is symmetric about its center, is shown at the far right. The key point to see about this solution is that for a given hidden unit, weights that are symmetric about the middle are equal in magnitude and opposite in sign. That means that if a symmetric pattern is on, both hidden units will receive a net input of zero from the input units, and, since the hidden units have a negative bias, both will be off. In this case, the output unit, having a positive bias,

."

./

..... .....

..... ........ 9. 44 .....

> ./ ./

.....

+ 8.3t /.3.1 7 ./ ./ ./ ./ .,. .,. .,.

.,.. .,. .,.. ·9A4

.,. .,;

..... ..... .....

.,. .,.. .,.

..... .....

'Qoutput Y Unit .,;

.,; .,;

FIGURE 9. Network for solving the symmetry problem. The six open ci rcles represent the input units. There are two hidden units, one shown above and one below the input

will be on. The next most important thing to note about the solution is that the weights on each side of the midpoint of the string are in the ratio of 1: 2:4. This insures that each of the eight patterns that can occur on each side of the midpoint sends a unique activation sum to the hidden unit. This assures that there is no pattern on the left that wi ll exactly balance a non-mirror-image pattern on the right. Finally, the two hidden units have identical patterns of weights from the input units except for sign. This insures that for every nonsymmetric pattern , at least one of the two hidden units wi ll come on and turn on the output unit. To summarize, the network is arranged so that both hidden units wi ll receive exactly zero activation from the input units when the pattern is symmetric, and at least one of them wi ll receive positive

input for every nonsymmetric pattern.

This problem was interesting to us because the learning system developed a much more elegant solution to the problem than we had previously considered. This problem was not the only one in which this happened. The parity solution di scovered by the learning procedure was also one that we had not discovered prior to testing the problem with our learning procedure. Indeed, we frequently discover these more elegant solutions by giving the system more hidden units than it needs and observing that it does not make use of some of those provided. Some analysis of the actual solutions discovered often leads us to the discovery of a better solution involving fewer hidden units.

Addition

Another interesting problem on which we have tested our learning algorithm is the simple binary addition problem. This problem is interesting because there is a very elegant sol ution to it, because it is the one problem we have found where we can rel iably find local minima and because the way of avoiding these local minima gi ves us some insight into the conditions under which local minima may be found and avoided. Figure 10 illustrates the basic problem and a minimal solution to it. There are four input units, three output units, and two hidden units. The output patterns can be viewed as the binary representation of the sum of two two-bit binary numbers represented by the input patterns. The second and fourth input units in the diagram correspond to the low-order bits of the two binary numbers and the first and third units correspond to the two higher order bits. The hidden units correspond to the carry bits in the summation. Thus the hidden unit on the far right comes on when both of the lower order bits in the input patter�yrig/Dtedf �t8li8J the one on the left comes

difficult. Note that the middle bit should come on whenever an odd number of the set containing the two higher order input bits and the lower order carry bit is turned on . Observation wil l confirm that the network shown performs that task. The left-most hidden unit receives inputs from the two higher order bits and from the carry bit. Its bias is such that it will come on whenever two or more of its in puts are turned on . The middle output unit receives positive inputs from the same three units and a negative input of - 2 from the second hidden unit. This insures that whenever just one of the three are turned on, the second hidden unit wi ll remain off and the output bit wil l come on. Whenever exactly two of the three are on, the hidden unit wi ll turn on and counteract the two units exciting the output bit, so it wil l stay off. Final ly, when all three are turned on, the output bit will receive - 2 from its carry bit and + 3 from its other three inputs. The net is positive, so the middle unit wil l be on. Finally, the third output bit should turn on whenever the second hidden unit is on - that is, whenever there is a carry from the second bit. Here then we have a minimal network to carry out the job at hand. Moreover, it should be noted that the concept behind this network is general izable to an arbitrary number of input and output bits. In general , for adding two m bit binary numbers we wil l require 2m input units, m hidden units, and m+ I out-

put units.

Unfortunately, this is the one problem we have found that reliably leads the system into local minima. At the start in our learn ing trials on this problem we al low any input unit to connect to any output unit and to any hidden unit . We allow any hidden unit to connect to any output unit, and we allow one of the hidden units to connect to the other hidden unit, but, since we can have no loops, the connection in the opposite direction is disallowed. Sometimes the system wil l discover essentially the same network shown in the figure. 4 Often , however, the system ends up in a local minimum. The problem arises when the XOR problem on the low-order bits is not sol ved in the way shown in the diagram. One way it can fail is when the "higher" of the two hidden units is "selected " to sol ve the XOR problem. This is a problem because then the other hidden unit cannot "see " the carry bit and therefore cannot finally sol ve the problem. This problem seems to stem from the fact that the learning of the second output bit is always dependent on learning the first (because information about the carry is necessary to learn the second bit) and therefore lags behind the learn ing of the first bit and has no influence on the selection of a hidden unit to

<sup>4</sup>The network is the same except for the highest order bit. The highest order bit is always on whenever three or more o f t h e input units are on. This is always learned first

solve the first XOR problem. Thus, about half of the time (in this problem) the wrong unit is chosen and the problem cannot be solved. In this case , the system finds a solution for all of the sums except the 11 + 11 - 11 0 (3+3 = 6) case in which it misses the carry into the middle bit and gets 11 + 11 - 100 instead. This problem di ffers from others we have sol ved in as much as the hidden units are not "equipotential " here. In most of our other problems the hidden units have

been equipotential, and this problem has not arisen.

It should be noted, however, that there is a relati vely simple way out of the problem -namely, add some extra hidden units. In this case we can afford to make a mistake on one or more selections and the system can still solve the problems. For the problem of adding two-bit numbers we have found that the system always solves the problem with one extra hidden uni t. With larger numbers it may require two or three more. For purposes of illustration , we show the results of one of our runs with three rather than the minimum two hidden units. Figure 11 shows the state reached by the network after 3,020 presentations of each input pattern and with a learning rate of "Y/ = 0.5. For convenience, we show the network in four parts. In Figure 11 A we show the connections to and among the hidden units. This figure shows the internal representation generated for this problem. The "lowest" hidden unit turns off whenever either of the low-order bits are on. In other words it detects the case in which no low-order bit is turn on. The "highest" hidden unit is arranged so that it comes on whenever the sum is less than two. The conditions under which the middle hidden unit comes on are more complex. Table 8 shows the patterns of hidden units which occur to each of the si xteen input patterns. Figure lI B shows the connections to the lowest order output unit. Noting that the relevant hidden unit comes on when neither low-order input unit is on, it is clear how the system computes XOR. When both low-order inputs are off, the output unit is turned off by the hidden unit. When both low-order input units are on , the output is turned off directly by the two input units. If just one is on, the positive bias on the output unit keeps it on . Figure ll C gives the connections to the middle output unit, and in Figure 11 D we show those connections to the left-most, highest order output unit. It is somewhat difficult to see how these connections always lead to the correct output answer, but, as can be verified from the figures, the network is balanced so that this works . It should be pointed out that most of the problems described thus far have invol ved hidden units with quite simple interpretations. It is much more often the case, especially when the number of hidden units exceeds the mini mum number required for the task, that the hidden units are not readily interpreted. This fol lows from the fact that there is very little tendency for loealist representations to develop. Typically Copyrighted Material

Output Un its 000

Input Unit<sup>s</sup>

A

Output Units

Hidden Units

Hidden Units

o

Output Un its

OO� / , <sup>I</sup>\ '· '0

<sup>I</sup>\ \ Hidden

·5

/ \ .• \

/ ' , Units

0

/ \ \ <sup>I</sup>\ \ / \ 0 I \ '-J

I \ <sup>I</sup>\ / \

0

6 0 6

o

Input Units

8

Output Units

··

00

/1\\, 111\\ "1\ \ I I \ \ \ <sup>I</sup>1.,1\ \

., Hidden

Units

--

 .:..' £":\ <sup>0</sup> II� \ I + 11 \_� -2

I \ ,-- \_\_ I I \ \ - - '3

Input Units

c

I, \' I I \' +2/ "2 '2\ \+2

66 66

Input Units

D

FIGURE 11 . Network found for Ihe summation problem. A: The conneclions from th<sup>e</sup> input units to the three hidden units and the connections among the hidden units. B: The connections from the input and hidden units to the lowest order output unit. C: The connections from the input and hidden units to the middle output unit. D: The connec-

TABLE 8

Input Hidden Unit Output Patterns Patterns Patterns

00+ <sup>00</sup>III <sup>000</sup> 00+ 01 11 0 001 00+ 10 01 1 010 00+ 11 010 011 <sup>01</sup>+ 00 110 001 <sup>01</sup>+ 01 010 010 01 + 10 010 011 01 + 11 000 100 10 + 00 01 1 010 10 + 01 010 01 1 10 + 10 001 100 10 + II 000 101 <sup>11</sup>+ 00 010 01 <sup>1</sup> <sup>11</sup>+ 01 000 100 11 + 10 000 101 11 + II 000 11 0

the internal representations are distributed and it is the pattern of activity over the hidden units, not the meaning of any particular hidden

unit that is important.

The Negation Problem

Consider a situation in which the input to a system consi sts of patterns of n+ 1 binary values and an output of n values. Suppose further that the general rule is that n of the input units should be mapped directly to the output patterns. One of the input bits, however, is special . It is a negation bit. When that bit is off, the rest of the pattern is supposed to map straight through , but when it is on , the complement of the pattern is to be mapped to the output. Table 9 shows the appropriate mapping. In this case the left element of the input pattern is the negation bit, but the system has no way of knowing this and must learn which bit is the negation bit. In this case, weights were allowed from any input unit to any hidden or output unit and from any hidden unit to any output unit. The system learned to set all of the weights to zero except those shown in Figure 12 . The basic structure of the problem and of the solution is evident in the figure. Clearly the problem was reduced to a set of three XORs between the negation bit Copyrighted Material

The T -C Problem

Most of the problems discussed so far (except the symmetry problem) are rather abstract mathematical problems . We now turn to a more geometric problem-that of discriminating between a T and a C- independent of translation and rotation. Figure 13 shows the stimulus patterns used in these experiments . Note, these patterns are each made of five squares and differ from one another by a single square. Moreover, as Minsky and Papert (969) point out, when considering the set of patterns over all possible translations and rotations (of 90°, 180°, and 270°) , the patterns do not differ in the set of distances among their pairs of squares. To see a difference between the sets of patterns one must look, at least, at configurations of triplets of squares. Thus Minsky and Papert call this a problem of order three. 5 In order to facilitate the learning, a rather different architecture was employed for this problem. Figure 14 shows the basic structure of the network we employed. Input patterns were now conceptualized as twodimensional patterns superimposed on a rectangular grid. Rather than al lowing each input unit to connect to each hidden unit, the hidden units themselves were organized into a two-dimensional grid with each unit receiving input from a square 3 x 3 region of the input space. In this sense, the overlapping square regions constitute the predefined receptive field of the hidden units. Each of the hidden units, over the entire field, feeds into a single output unit which is to take on the value

FIGURE 13. The stimulus set for the T-C problem. The set consists of a block T and a block C in each of four orientations. One of the eight patterns is presented on each trial.

o o o

o o Output Unit

Hidden Un its

Input Units

FIGURE 14. The network for solving the T-C problem. See text for explanation.

1 if the input is a T (at any location or orientation) and 0 if the input is a C. Further, in order that the learning that occurred be independent of where on the field the pattern appeared, we constrained all of the units to learn exactly the same pattern of weights. [n this way each unit was constrained to compute exactly the same function over its receptive field-the receptive fields were constrained to all have the same shape. This guarantees translation independence and avoids any possible "edge effects " in the learning. The learning can readily be extended to arbitrarily large fields of input units. This constraint was accompl ished by simply adding together the weight changes dictated by the delta rule for

each unit and then chamzing all weil!hts

e�actly the same amount. In

this way, the whole field of hidden units consists simply of replications of a single feature detector centered on different regions of the input space, and the learning that occurs in one part of the field is automati-

cally generalized to the rest of the field. 6

We have run this problem in this way a number of times. As a result, we have found a number of sol utions. Perhaps the simplest way to understand the system is by looking at the form of the receptive field for the hidden units. Figure 15 shows several of the receptive fields we have seen. 7 Figure 15A shows the most local representation developed. This on-center-off-surround detector turns out to be an excellent T detector. Since, as illustrated, a T can extend into the oncenter and achieve a net input of + 1, this detector wi ll be turned on for <sup>a</sup>T at any orientation. On the other hand, any C extending into the center must cover at least two inhi bitory cells. With this detector the bias can be set so that only one of the whole field of inhibitory units wi ll come on whenever a T is presented and none of the hidden units wi ll be turned on by any C. This is a kind of protrusion detector which differentiates between a T and C by detecting the protrusion of the T. The receptive field shown in Figure 15B is again a kind of T detector. Every T activates one of the hidden units by an amount + 2 and none of the hidden units receives more than + 1 from any of the C's. As shown in the figure, T's at 90° and 270° send a total of + 2 to the hidden units on which the crossbar lines up. The T's at the other two orientations receive +2 from the way it detects the vertical protrusions of those two characters. Figure 15C shows a more distributed representation. As illustrated in the figure, each T activates five different hidden units whereas each C excites only three hidden units. In this case the system again is differentiating between the characters on the basis

of the protruding end of the T which is not shared by the C.

Finally, the receptive field shown in Figure 150 is even more interesting. In this case every hidden unit has a positive bias so that it is on unless turned off. The strength of the inhibitory weights are such that if a character overlaps the receptive field of a hidden unit, that unit turns off. The system works because a C is more compact than a T and therefore the T turns off more units that the C. The T turns off 21 hidden units, and the C turns off only 20. This is a truly distributed

6 A similar procedure has been employed by Fukushima (980) in his neocognitron <sup>a</sup>n<sup>d</sup>

by Kienker, Sejnowski , Hinton , and Schumacher (985) .

A

c

8. LEARNING INTERNAL REPRESENTATIONS 351

m[

:: ��r:

i-1 -1 -1

-1

�1+ 2:- 1

0 -2 -2 -2 -2 -2 -2

B

<sup>I</sup>::pf -1 1- 1

.. · .. ,· ,Jf�

��:: :: ·· ··

-2 -2 -2

FIGURE <sup>15</sup> . Receptive fields found in different runs of the T-C problem. A: An oncenter-off-surround recept ive field for detecting T's. B: A vertical bar detector which responds to T's more strongly than C's. C: A diagonal bar detector. A T act ivates five such detectors whereas a <sup>C</sup>activates only th ree such detectors. D: A compact ness detector . This inhibitory receptive field turns off whenever an input covers any region of i ts receptive field . Si nce the <sup>C</sup>is more compact than <sup>t</sup>he T it turns off 20 such detectors

whereas the T turns off 21 of them.

representation. In each case, the solution was reached in from about

5,000 to 10,000 presentations of the set of eight patterns. <sup>8</sup>

It is interesting that the inhibitory type of receptive field shown in Figure 150 was the most common and that there is a predominance of inhibitory connections in this and indeed all of our simulations. This can be understood by considering the traject ory through which the learning typically moves. At first, when the system is presented with a

<sup>8</sup>Si nce translation independence was bui l<sup>t</sup>into the learning procedure, it makes no difference where the input occurs: the same thing will be learned wherever the pattern is presented . Thus, there are on�t5�pr@HiOO MWtIDfaP be presented to the system .

difficult problem, the initial random connections are as likely to mislead as to give the correct answer. In this case, it is best for the output units to take on a value of 0. 5 than to take on a more extreme value. This follows from the form of the error function given in Equation 2. The output unit can achieve a constant output of 0.5 by turning off those units feeding into it. Thus, the first thing that happens in virtually every difficult problem is that the hidden units are turned off. One way to achieve this is to have the input units inhibit the hidden units. As the system begins to sort things out and to learn the appropriate function some of the connections will typical ly go positive, but the majority of the connections will remain negative. This bias for solutions involving inhibitory inputs can often lead to nonintuitive results in which hidden units are often on unless turned off by the input.

More Simulation Results

We have offered a sample of our results in this section. In addition to having studied our learning system on the problems discussed here, we have employed back propagation for learning to multiply binary digits, to play tic-tac-toe, to distinguish between vertical and horizontal lines, to perform sequences of actions , to recognize characters, to associate random vectors , and a host of other applications. In all of these applications we have found that the generalized delta rule was capable of generating the kinds of internal representations required for the problems in question. We have found local minima to be very rare and that the system learns in a reasonable period of time. Sti ll more studies of this type will be required to understand precisely the conditions under which the system wi ll be plagued by local minima. Suffice it to say that the problem has not been serious to date. We now turn to a

pointer to some future developments.

SOME FURTHER GENERALIZATIONS

We have intensively studied the learning characteristics of the general ized delta rule on feed forward networks and semi linear activations functions. Interestingly these are not the most general cases to which the learning procedure is applicable. As yet we have only studied a few examples of the more fully generalized system, but it is relatively easy to apply the same learning rule to sigma-pi units and to recurrent net-

works . We wi ll simply s�qg� here.

The Generalized Delta Rule and Sigma-Pi Units

It wi ll be recalled from Chapter 2 that i n the case of sigma-pi units

we have

(I7)

where i varies over the set of conjuncts feeding into unit j and k varies over the elements of the conjuncts. For simplicity of exposition , we restrict ourselves to the case in which no conjuncts involve more than two elements. In this case we can notate the weight from the conjunction of units i and j to unit k by wkij ' T<sup>h</sup>e weight on the di rect connection from unit i to unit j would, thus, be wji; , and since the relation

is multi plicative, Wkij = Wkj

;' We can now rewrite Equation 17as

OJ = Ij (L Wjh; 0h 0; ). ;.h

We now set

Taking the deri vative and simplifying, we get a rule for sigma-pi units

strictly analogous to the rule for semilinear activation functions:

!l.p wkij = <sup>8</sup> k <sup>o</sup>; oj .

We can see the correct form of the error signal , 8, for this case by inspecting Figure 16. Consider the appropriate value of 8; for unit U; in the figure. As before, the correct val ue of 8; is given by the sum of the 8's for all of the units into which U; feeds , weighted by the amount of effect due to the activation of U; times the derivative of the activation function. In the case of semilinear functions, the measure of a unit's effect on another unit is given simply by the weight W connecting the fi rst unit to the second. In this case, the u, 's effect on Uk

depends not only on Wk;j

, but also on the value of Uj . Thus, we have

8; = I'j (net; ) L8 k Wk;j OJ j ,k if u; is not an output unit and, as before,

8; = .1 '; (net; )(t;-o;)

if it is an output unit. Copyrighted Material

&. &. J <sup>t</sup>

FIGURE 16. The generalized delta rule for s <sup>i</sup>gma - pi units. The products of activation values of individual units activate output units. See text for explanation of how the 8

values are computed in this cas<sup>e</sup> .

Recurrent Nets

We have thus far restricted ourselves to /eed/orward nets. This may seem like a substantial restriction, but as Minsky and Papert point out, there is, for every recurrent network, a feed forward network with identical behavior (over a finite period of ti me) . We wi ll now indicate how this construction can proceed and thereby show the correct form of the learning rule for the recurrent network. Consider the si mple recurrent network shown in Figure 17 A. The same network in a feedforward architecture is shown in Figure 17B. The behavior of a recurrent network can be achieved in a feedforward network at the cost of duplicating the hardware many times over for the feedforward version of the network . 9 We have distinct units and distinct weights for each point in time. For naming convenience, we subscri pt each unit with its unit number in the corresponding recurrent network and the time it represents. As long as we constrain the weights at each level of the feedforward network to be the same, we have a feedforward network which performs identically with the recurrent network of Figure 17 A.

<sup>9</sup>Note that in this discussion, and indeed in <sup>o</sup>ur entire development here, we have assumed a discrete time system with synchronous update and with each connection

involving a unit delay.

A

B

· · ·

· · Time

t+1

o

FIGURE 17 , A comparison of a recu rrent network and a feedforward network with identical behavior, A: A completely connected recurrent network with two units, B: A feedforward network which behaves the same as the recurrent network , In this case, we have a separate unit for each time step and we requ ire that the weights connecting each layer of units to the next be the same for all layers, Moreover, they must be the same as

the analogous weights in the recurrent case,

The appropriate method for maintaining the constraint that all weights be equal is simply to keep track of the changes dictated for each weight at each level and then change each of the weights according to the sum of these individual ly prescribed changes. Now, the general rule for determining the change prescribed for a weight in the system for a particular time is simply CWf,lPighted � of an appropriate error

measure 8 and the input along the relevant line both for the appropriate times. Thus, the problem of specifying the correct learning rule for recurrent networks is simply one of determining the appropriate value of 8 for each time. In a feedforward network we determine 8 by multiplying the derivative of the activation function by the sum of the 8's for those units it feeds into weighted by the connection strengths. The same process works for the recurrent network-except in this case, the value of 8 associated with a particular unit changes in time as a unit passes error back, sometimes to itself. After each iteration, as error is being passed back through the network, the change in weight for that iteration must be added to the weight changes specified by the preceding iterations and the sum stored. This process of passing error through the network should continue for a number of iterations equal to the number of iterations through which the activation was originally passed. At this point, the appropriate changes to all of the weights can

be made.

In general , the procedure for a recurrent network is that an input (generally a sequence) is presented to the system while it runs for some number of iterations. At certain specified times during the operation of the system, the output of certain units are compared to the target for that unit at that time and error signals are generated. Each such error signal is then passed back through the network for a number of iterations equal to the number of iterations used in the forward pass. Weight changes are computed at each iteration and a sum of all the weight changes dictated for a particular weight is saved. Finally, after all such error signals have been propagated through the system, the weights are changed. The major problem with this procedure is the memory required. Not only does the system have to hold its summed weight changes while the error is being propagated, but each unit must somehow record the sequence of activation values through which it was driven during the original processing. This follows from the fact that during each iteration while the error is passed back through the system, the current 8 is relevant to a point earl ier in time and the required weight changes depend on the activation levels of the units at that time. It is not entirely clear how such a mechanism could be implemented in the brain. Nevertheless , it is tantal izing to realize that such a procedure is potentially very powerful, since the problem it is attempting to solve amounts to that of finding a sequential program (like that for a digital computer) that produces specified input-sequence/ output-sequence pairs. Furthermore , the interaction of the teacher with the system can be quite flexible, so that, for example, should the system get stuck in a local minimum, the teacher could introduce "hints" in the form of desired output values for intermediate stages of processing. Our experience with recurrent net�Hmi�tePMII we have carried out some

experiments. We turn first to a very simple problem in which the sys-

tem is induced to invent a shift register to solve the problem.

Learning to be a shift register. Perhaps the simplest class of recurrent problems we have studied is one in which the input and output units are one and the same and there are no hidden units. We simply present a pattern and let the system process it for a period of time. The state of the system is then compared to some target state. If it hasn't reached the target state at the designated time, error is injected into the system and it modifies its weights. Then it is shown a new input pattern and restarted. In these cases , there is no constraint on the connections in the system. Any unit can connect to any other unit. The simplest such problem we have studied is what we call the shift register problem. In this problem, the units are conceptualized as a circular shift register. An arbitrary bit pattern is first established on the units. They are then allowed to process for two time-steps. The target state, after those two time-steps, is the original pattern shifted two spaces to the left. The interesting question here concerns the state of the units between the presentation of the start state and the time at which the target state is presented. One solution to the problem is for the system to become a shift regi ster and shift the pattern exactly one unit to the left during each time period. If the system did this then it would surely be shifted two places to the left after two time units. We have tried this problem with groups of three or five units and, if we constrain the biases on al l of the units to be negative (so the units are off unless turned on) , the system always learns to be a shift register of

this sort.

IO Thus, even though in principle any unit can connect to any

other unit, the system actually learns to set all weights to zero except the ones connecting a unit to its left neighbor. Since the target states were determined on the assumption of a circular register, the left-most unit developed a strong connection to the right-most unit. The system learns this relatively quickly. With T/ = 0.25 it learns perfectly in fewer than 200 sweeps through the set of possible patterns with either three-

or five-unit systems.

The tasks we have described so far are exceptional ly simple, but they do illustrate how the algorithm works with unrestricted networks. We have attempted a few more difficult problems with recurrent networks.

10 If the constraint that biases be negative is not imposed, other solutions are possible. These solutions can involve the units passing through the complements of the shifted pattern or even through more complicated intermediate states. These trajectories are interesting in that they match a simple shift register on all even numbers of shifts, but do

not match following an odd number of shifts.

One of the more interesting involves learning to complete sequences of

patterns. Our final example comes from this domain.

Learning to complete sequences. Table 10 shows a set of 25 sequences which were chosen so that the first two items of a sequence uniquely determine the remaining four. We used this set of sequences to test out the learning abilities of a recurrent network. The network consisted of five input units (A, B, C, D, E) , 30 hidden units, and three output units (I, 2, 3) . At Time 1 , the input unit corresponding to the first item of the sequence is turned on and the other input units are turned off. At Time 2, the input unit for the second item in the sequence is turned on and the others are al l turned off. Then al l the input units are turned off and kept off for the remaining four steps of the forward iteration. The network must learn to make the output units adopt states that represent the rest of the sequence. Unlike simple feedforward networks (or their iterative equivalents) , the errors are not only assessed at the final layer or time. The output units must adopt the appropriate states during the forward iteration, and so during the back-propagation phase, errors are injected at each time-step by comparing the remembered actual states of the output units with their

desired states .

The learning procedure for recurrent nets places no constraints on the allowable connectivity structure. 11 For the sequence completion problem, we used one-way connections from the input units to the hidden units and from the hidden units to the output units . Every hidden unit had a one-way connection to every other hidden unit and to itself,

> TABLE 10 25 SEQUENCES TO BE LEARNED AA 1212 AB1 223 AC 1231 ADl221 AEI 213 BA23 12 BB2323 BC2331 B02321 BE23 13 CA3112 CB3 123 CC3131 C03121 CE311 3 OA2112 OB2 123 OC2131 002121 OE2ll3 EA 13I2 EB 1323 EC 1331 EDl321 EEI3 13

1 1 The constraint in feed forward networks is that it must be possible to arrange the units into layers such that units do not influence units in the same or lower layers. In recurrent networks this amounts to the constraint that during the forward iteration,

and every output unit was also connected to every other output unit and to itself. All the connections started with small random weights uniformly distributed between -0.3 and +0.3. All the hidden and output units started with an activity level of 0.2 at the beginning of each

sequence.

We used a version of the learning procedure in which the gradient of the error with respect to each weight is computed for a whole set of examples before the weights are changed. This means that each connection must accumulate the sum of the gradients for all the examples and for all the time steps involved in each example. During training, we used a particular set of 20 examples , and after these were learned almost perfectly we tested the network on the remaining examples to see if it had picked up on the obvious regularity that relates the first two items of a sequence to the subsequent four. The results are shown in Table 11. For four out of the five test sequences , the output units all have the correct val ues at all times (assuming we treat val ues above 0.5 as 1 and values below 0.5 as 0) . The network has clearly captured the rule that the fi rst item of a sequence determines the third and fourth, and the second determines the fi fth and si xth. We repeated the simulation with a different set of random initial weights, and it got all

five test sequences correct.

The learning required 260 sweeps through all 20 training sequences. The errors in the output units were computed as follows: For a unit that should be on, there was no error if its activity level was above 0.8, otherwise the derivati ve of the error was the amount below 0.8. Similarly, for output units that should be off, the deri vative of the error was the amount above 0. 2. After each sweep, each weight was decremented by .02 times the total gradient accumulated on that sweep plus 0.9

times the previous weight change.

We have shown that the learning procedure can be used to create a network with interesting sequential behavior, but the particular problem we used can be solved by simply using the hidden units to create "delay lines" which hold information for a fi xed length of time before allowing it to influence the output. A harder problem that cannot be solved with delay lines of fixed duration is shown in Table 12 . The output is the same as before, but the two input items can arri ve at variable times so that the item arriving at time 2, for example, could be either the first or the second item and could therefore determine the states of the output units at either the fifth and sixth or the seventh and eighth times. The new task is equi valent to requiring a buffer that receives two input "words " at variable times and outputs their "phonemic realizations " one after the other. This problem was solved successfully by a network similar to the one above except that it had 60 hidden units and hal f of their possi ble iItM1iynnmetf>Ma� omitted at random. The

TABLE 11 PERFORMANCE OF THE NETWORK ON FIVE NOVEL TEST SEQUENCES

Input Sequence A D

Desi red Outputs 2 2

Actual States of:

Output Unit 1 0.2 0. 12 0.90 0.22 0. 11 0.83 Output Unit 2 0.2 0. 16 0. 13 0.82 0.88 0.03 Output Unit 3 0. 2 0.07 0.08 0.03 0.01 0.22

Input Sequence B E

Desired Outputs 2 3 3

Actual States of:

Output Unit 1 0.2 0. 12 0.20 0.25 0.48 0.26 Output Unit 2 0.2 0. 16 0.80 0.05 0.04 0.09 Output Unit 3 0.2 0.07 0.02 0.79 0.48 0.53

Input Sequence C A

Desi red Outputs 3 2

Actual States of:

Output Unit 1 0.2 0. 12 0. 19 0.80 0.87 0. 11 Output Unit 2 0.2 0. 16 0. 19 0.00 0. 13 0.70 Output Unit 3 0.2 0.07 0.80 0. 13 0.01 0.25

Input Sequence D B

Desi red Outputs 2 2 3

Actual States of:

Output Unit 1 0.2 0. 12 0. 16 0.79 0.07 0. 11 Output Unit 2 0.2 0. 16 0.80 0. 15 0.87 0.05 Output Unit 3 0.2 0.07 0.20 0.01 0. 13 0.96

Input Sequence E C

Desired Outputs 3 3

Actual States of:

Output Unit 1 0.2 0.12 0.80 0.09 0.27 0.78 Output Unit 2 0.2 0. 16 0.20 0. 13 0.01 0.02 Output Unit 3 0.2 0.07 0.07 0.94 0.76 0. 13

learning was much slower, requiring thousands of sweeps through all 136 training examples . There were also a few more errors on the 1 4 test examples, but the general ization was still good with most of the

test sequences being coll!RJp)9tgRfd'lieM»terial

TABLE 12 SIX VARIATIONS OF THE SEQUENCE EA \312 PRODUCED BY PRESENTING THE FIRST TWO ITEMS AT VARI ABLE TIMES

EA-- 1312 -EA- <sup>1312</sup> E-A- 1J12 -E-A I312

E- -AI3l2 --EA\312

Note: With these temporal variations, the 25 sequences shown in Table 10 can be used to generate 150 different sequences.

CONCLUSION

In their pessimistic discussion of perceptrons, Minsky and Papert (1 969) finally discuss multilayer machines near the end of their book.

They state:

The perceptron has shown itself worthy of study despite (and even because of!) its severe limitations. It has many features that attract attention: its linearity; its intriguing learning theorem; its clear paradigmatic simplicity as a kind of paral lel computation . There is no reason to suppose that any of these virtues carry over to the many-layered version, Nevertheless, we consider it to be an important research problem to elucidate (or reject) our intuitive judgement that the extension is sterile. Perhaps some powerful convergence theorem wi ll be di scovered, or some profound reason for the failure to produce an interesting "learning theorem " for the multi layered machine

wi ll be found. (pp. 231-232)

Although our learning results do not guarantee that we can find a sol ution for all solvable problems, our analyses and results have shown that as a practical matter, the error propagation scheme leads to solutions i n virtually every case, In short, we bel ieve that we have answered Minsky and Papert's chal lenge and have found a learning result sufficiently powerful to demonstrate that their pessimism about learning in mul-

tilayer machines was misplaced.

One way to view the procedure we have been describing is as a parallel computer that, having been shown the appropriate input/ output exemplars specifying some function, programs itself to compute that function in general . Parallel computers are notoriously difficult to program. Here we have a mechanism whereby we do not actually have to know how to write the program in order to get the system to do it.

Parker (1985) has emph�fflJIigHt-eB�rerial

On many occasions we have been surprised to learn of new methods of computing interesting functions by observing the behavior of our learning algorithm. This also raised the question of generalization. In most of the cases presented above, we have presented the system with the entire set of exemplars. It is interesting to ask what would happen if we presented only a subset of the exemplars at training time and then watched the system generalize to remaining exemplars. In small problems such as those presented here, the system sometimes finds solutions to the problems which do not properly general ize. However, preli minary results on larger problems are very encouraging in this regard. This research is still in progress and cannot be reported here. This is

currently a very active interest of ours.

Final ly, we should say that this work is not yet in a finished form. We have only begun our study of recurrent networks and sigma-pi units. We have not yet appl ied our learning procedure to many very complex problems . However, the results to date are encouragi ng and

we are continuing our work .

PART III

FORMAL ANALYSES

Part III is focused on the formal tools employed in the study of PDP models and their application in the analysis of several specific aspects of

PDP mechanisms.

In Chapter 9, Jordan provides a very accessible introduction to linear algebra and its applications to the analysis of PDP models. This chapter is designed to be a tutorial for those who are not familiar with the basics of linear algebra. Most of the book can be read and understood without knowledge of linear algebra, but an understanding of this important tool will greatly enhance a reader's understanding of the mathematical discussions that can be found in several chapters of the

book.

In Chapter 10, Williams provides a useful analysis of activation functions. Throughout the book we employ several different activation functions. There is a question as to whether we will need to look for more complex activation functions to carry out some of the more complex computations. Williams shows that we will never have to consider

activation functions more complex than the sigma pi function.

In Chapter 11, Stone provides a useful analysis of the delta rule, which plays an important role throughout the book. Stone shows how <sup>a</sup> change of basis can be employed to reveal the internal workings of the delta rule. He shows that when there is no deterministic relationship between inputs and targets, the delta rule leads to a system whose outputs match the central tendencies of the target patterns. Finally, he

In Chapter 12, McClelland analyzes the capacity limitations of two kinds of networks. He studies the effects of limitations of fan-in and fan-out on the capacities of standard pattern-association networks, and he explores the costs in units and connections of using programmable

networks of the kind outlined in Chapter 16.

Finally, in Chapter 13, Zipser and Rabin describe a computer simulation system, called P3, for building computer simulations of PDP models. P3 provides both a language for describing networks and an interface for interacting with these networks and observing their behavior. Chapter 13 gives as an example a description of how the

competitive learning algorithm can be built in P3.

In general, the chapters in this section are useful for two reasons. First, they describe several useful basic results-results that lie behind many of the simulation models described in other sections of the book. Second, and perhaps more importantly, they indicate some of the formal tools that are available for analyzing parallel networks, and show through example how these tools can be used to produce useful results.

A

Carol [�� I 1 21

8

FIGURE 1.

37 72

Joe 1 75

8 1946 Mary [�?I

Brad [�� I 155

to only three components, however. If, for example, we also wanted to keep track of Joe's shoe size and year of birth, then we would simply

make a vector with five components, as in Figure lB.

One important reason for the great utility of linear algebra lies in the simplicity of its notation. We will use bold, lower-case letters such as v to stand for vectors. With this notation, an arbitrarily long list of infor-

mation can be designated by a single symbol.

When a vector has no more than three components, it can be represented graphically by a point or an arrow in three-dimensional space. An example with three components is given in Figure 2 for the vector corresponding to Mary. Each axis in the figure corresponds to

one of the three components of the vector.

It will prove helpful to try and visualize vectors as points or arrows in two- and three-dimensional space in proceeding through this chapter in order to develop geometric intuition for the operations on vectors. Notice, however, that there is no fundamental distinction between such vectors and vectors with more than three components. All of the operations upon vectors described in later sections apply equally well to

vectors with any finite number of components.

In a parallel distributed processing model, many quantities are best represented by vectors. The pattern of numbers representing the activations of many processing units is one example. Other examples are the s�t of weights on the input lines to a particular processing unit,

4

3

2

o�--�--�--�--�--��- 2 3 4 5

FIGURE 3.

Addition of Vectors

Two or more vectors can be added by adding their components. The vectors must have the same number of components to be added; other-

wise the operation is undefined.

Examples:

Vector addition is associative (the vectors can be grouped in any manner) and commutative (the order of addition is unimportant) just like addition in ordinary algebra. This is true because if we consider one component at a time, vector addition is just addition in ordinary

algebra.

How can vector addition be represented graphically? Consider Figure 4, where lhe vectors vI - [ � 1 and. 2 - [ : 1 are being added. It can be

seen that the sum v 1 + V 2 is a vector [ � 1 which lies between v 1 and v 2'

Forming the parallelogram with sides v 1 and v 2, we see that the sum of Copyrighted Material

3

2

2

FIGURE 4.

4 5

the two vectors is the diagonal of this parallelogram. In two and three dimensions this is easy to visualize, but not when the vectors have more than three components. Nevertheless, it will be useful to imagine vector addition as forming the diagonal of a parallelogram. One implication of this view, which we will find useful , is that the sum of two vectors is a vector that lies in the same plane as the vectors being

added.

Example: Calculating averages. We can demonstrate the use of the two operations thus far defined in calculating the average vector. Suppose we want to find the average age, height, and weight of the four individuals in Figure lA. Clearly this involves summing the com-

ponents separately and then dividing each sum by 4. Using vectors , this corresponds to adding the four vectors and then multiplying the resulting sum by the scalar 1/4. Using u to denote the average vector,

<sup>u</sup>= ! II ��l + I !�l + I ��l + I :�ll = I �::�l ·

175 61 121 155 128

Using vector notation, if we denote the four vectors by v I, V 2, V 3, and

v 4, then we can write the averaging operation as

The vector u, then, is a vector whose components are the averages of the components of the four individual vectors. Notice that the same result is obtained if each vector is first multiplied by 1/4, and the resulting vectors are added. This shows that multiplication by scalars and vector addition obey a distributive law, as in ordinary algebra.

LINEAR COMBINATIONS AND LINEAR

INDEPENDENCE

Linear Combinations of Vectors

The average vector calculated in the last section is an example of a linear combination of vectors. In this section, we pursue this idea

further.

Consider the vectors v, = [ �], V 2 = [ �], and u = [ 1 �]. Can u be written as the sum of scalar multiples of v, and v 2? That is, can scalars

c, and C2 be found such that u can be written in the form

U = CIVI+C2V2?

If so, then u is said to be a linear combination of the vectors v I and v 2. The reader can verify that c, = 3 and C2 = 2 witt work, and thus u is a

linear combination of v, and v 2.

This can also be seen directly in Figure 5, where these vectors are plotted. Remembering that multiplication by a scalar shortens or

10

5

o�----�------�------�-- 5 10 15

lengthens a vector and that vector addition corresponds to forming the diagonal of a parallelogram, it seems clear that we can find scalars to adjust v 1 and v 2 to form a parallelogram that yields u. This is indicated in the figure. It also seems clear that, using positive scalars, any vector in the shaded area of the figure can be generated this way. By using both negative and positive scalars, any vector in the plane can be written as a linear combination of v 1 and v 2. This is true because multiplication by a negative scalar reverses the direction of a vector as well as shortening or lengthening it. The vectors v 1 and v 2 are said to span the plane, because any vector in the plane can be generated from these two

vectors.

In general , given a set v I> v 2 ••••• V II of vectors, a vector vis said to be a linear combination of the Vi if scalars el>e2 •...• ell can be found

such that

(1)

The set of all linear combinations of the v, is called the set spanned by

the v,.

Exampk. The � vecto� [�I, [! I Md [ � I s� an ru thre�

dimensional space since any vector <sup>v</sup>- [!I can be written as a linear

combination <sup>v</sup>- <sup>a</sup>[i I +b [! 1+ c m The vecto� are ref erroo to

as the standard basis for three-dimensional space (more on the idea of a

basis in the next section) .

Linear Independence

To say that a set of vectors span a space is to say that all vectors in the space can be generated from the original set by linear combination. We have shown examples in which two vectors span two-dimensional space and three vectors span three-dimensional space. We might be led to expect that, in general, n vectors suffice to span n-dimensional space. In fact, we have been using the term "dimension" without defining what it means; it would seem that a good definition of n-

To make this definition work, we would require that the same size space be generated by any set of n vectors. However, this is not the

case, as can be easily shown. Consider any pair of collinear vectors, for example. Such vectors lie along a single line, thus any linear combination of the vectors will lie along the same line. The space spanned by these two vectors is therefore only a one-dimensional set. The collinear vectors [ � I and [ ; I are a good example. Any linear combina-

tion of these vectors will have equal components, thus they do not span

the plane.

Another example is a set of three vectors that lie on a plane in three-dimensional space. Any parallelograms that we form will be in the same plane, thus all linear combinations will remain in the plane

and we can't span all of three-dimensional space.

The general rule arising from these examples is that of a set of n vectors, if at least one can be written as a linear combination of the others, then the vectors span something less than a full n-dimensional space. We call such a set of vectors linearly dependent. If, on the other hand, none of the vectors can be written as a linear combmation of the others, then the set is called linearly independent. We now revise the definition of dimensionality as follows: n -dimensional space is the set of vectors spanned by a set of n linearly independent vectors. The n

vectors are referred to as a basis for the space.

Examples:

1. [ �

I and [ ; I are linearly dependent. They span only a one-

dimensional space.

2. I: I and 1 � I are linearly independent. Thus they span the

plane, a two-dimensional space.

3. 1 � I, 1 � I, and 1-� I are linearly dependent since 7 times the

first vector minus 4 times the second vector is equal to the third

vector.

4. [ i J.[ � I

, an+ � I are linearly dependent. Clearly they =not

span all of three-dimensional space, because no vector with a

nonzero third component can be generated from this set. Copyrighted Material

VECTOR SPACES

Let us pause to reflect for a moment upon what a vector is. I have implied that a vector is a list of numbers, and I have also used the term to refer to a point or an arrow in space. Are both of these objects vectors, or is one just a heuristic representation for the other? Are there other objects that should be called vectors? Just what is a vector?

As is often the case in mathematics, these kinds of questions are solved by being avoided. Consider the following definition of an

abstract vector space, and try to decide what a vector is.

<sup>A</sup>vector space is a set Vof elements, called vectors , with the follow-

ing properties:

• To every pair, u and v, of vectors in V. there corresponds a vector u + v also in V, called the sum of u and v, in such a way

that addition is commutative and associative.

• For any scalar c and any vector v in V, there is a vector cv in V, called the product of c and v, in such a way that multiplication by scalars is associative and distributive with respect to

vector addition.l

The answer to the question is that a vector is an undefined object in linear algebra, much like a line in geometry. The definition of a vector space simply lists the properties that vectors must have, without specifying what a vector must be. Thus, any set of objects that obey these properties can be called a vector space. Lists of numbers are vectors when addition is defined as adding components separately and scalar multiplication is defined as multiplying all the components by the scalar, because these operations fill all the requirements of a vector space. Arrows or points in space are also vectors when addition is defined geometrically as taking the diagonal of a parallelogram and scalar multiplication is defined as lengthening or shortening the arrow, because again, these operations fi ll the requirements of a vector space. A seemingly unrelated example of a vector space is the set of polynomials of order n, with addition and scalar multiplication defined in the

obvious way.

This sort of abstraction is common in mathematics. It is useful because any theorem that is true about a general vector space must be

1 <sup>I</sup>have left out certain technicalities usually included as axioms for a vector space. These include the axiom that there must be a zero vector, and for every vector, there is

true about any instantiation of a vector space. We can therefore discuss general properties of vector spaces without being committed to choosing a particular representation such as a list of numbers. Much of the discussion about linear combinations and linear independence was of

this nature.

When we do choose numbers to represent vectors, we use the following scheme. First we choose a basis for the space. Since every vector in the space can be written as a linear combination of the basis vectors, each vector has a set of coefficients c J, c 2 ••••• cn which are the

coefficients in the linear combination. These coefficients are the numbers used as the components of the vector. As was shown in the previous section, the coefficients of a given vector are unique because

basis vectors are linearly independent.

There is a certain arbitrariness in assigning the numbers, since there are infinitely many sets of basis vectors, and each vector in th� space has a different description depending on which basis is used. That is, the coefficients, which are referred to as coordinates, are different for

different choices of basis. The implications of this fact are discussed further in a later section where I also di scuss how to relate the coordi-

nates of a vector in one basis to the coordinates of the vector in another basis . Chapter 22 contains a lengthy discussion of several

issues relating to the choice of basis.

INNER PRODUCTS

As of yet, we have no way to speak of the length of a vector or of the similarity between two vectors . This will be rectified with the

notion of an inner product.

The inner product of two vectors is the sum of the products of the vector components. The notation for the inner product of vectors <sup>v</sup>and w is v . w. As with vector addition, the inner product is defined

only if the vectors have the same number of components.

Example:

<sup>v</sup>. w = (3 . 1) + (- 1 . 2) + (2 . 1) = 3.

The inner product is a kind of mUltiplication between vectors, although somewhat of a strange sort of multiplication, since it produces a single number from a pair of vectors. What does this single number

"measure" ?

Length

As a special case, consider taking the inner product of a vector with itself. An example is the vector v = [ ! [ in Figure 7. The inner pro-

duct of v with itself is

v . v = <sup>32</sup>+ <sup>42</sup>= 25.

Consider the right triangle in Figure 7 with sides corresponding to the components of v. and hypotenuse v itself. The Pythagorean theorem tells us that the square of the length of v is equal to the sum of the squares of the sides. Since this is exactly what is calculated by the inner product v . v, it appears that a reasonable definition of the length of a vector is the square root of the inner product of the vector with

itself. Thus we define the length of a vector v, denoted by I

Ivl l , as

IIvll = (v' v)'h.

Although the definition was motivated by an example in two dimensions, it can be applied to any vector. Notice that many of the

properties we intuitively associate with length are included in this definition. For example, if a vector has larger components than another vector, it will be longer, because the squared components will contribute to a larger inner product. Multiplying a vector by a scalar produces a new vector whose length is the absolute value of the scalar

times the length of the old vector:

lIevll = !clllvll·

This is a property that can be easily proved. Somewhat harder to prove is the so-called triangle inequality, which states that the length of the sum of two vectors is less than or equal to the sum of the lengths of

the two vectors:

Geometrically, the triangle inequality corresponds to the statement that one side of a triangle is no longer than the sum of the lengths of the

other two sides.

Thus, in the special case where the operands are the same vector, the inner product is closely related to the idea of length. What if the

operands are different vectors?

Angle

The angle between two vectors v and w is defined in terms of the

inner product by the following definition:

v 'w

cos (J <sup>=</sup>IIvll I

Iwil

(2)

where (J is the angle between v and w. Note that all of the quantities on

the right hand side of the equation are easily calculated for ndimensional vectors. At the end of this section, I will show geometrically why this formula is correct in two-dimensional space, using the

ordinary geometrical definition of angle.

Example. Find the angle (J between the vectors v I = [ � 1 and

<sup>v</sup>2 = [ � I. First, we calculate the necessary inner product and lengths:

must remember to divide the inner product by the lengths of the vec-

tors involved to make such comparative statements.

An important special case occurs when the inner product is zero. In

this case, the two vectors are said to be orthogonal. Plugging zero into

the right side of Equation 2 gives cos () == o.

which implies that the angle between the vectors is 90°. Thus, orthog-

onal vectors are vectors which lie at right angles to one another.

We will often speak of a set of orthogonal vectors. This means that every vector in the set is orthogonal to every other vector in the set. That is, every vector lies at a right angle to every other vector. A good example in three-dimensional space is the standard basis referred to earlier. Although we will skip the proof, it is probably clear that any orthogonal set is linearly independent. Indeed, orthogonality is

stronger than linear independence: whereas every orthogonal set is linearly independent, there are very many linearly independent sets of vectors that are not orthogonal. An example in two-dimensional space

is the pair [ : 1 and [ � I· When we choose a basis for a space, we typi·

cally choose an orthogonal basis. In fact, in much of classical physics and mathematics, there is not the slightest hint that a basis should be

anything but orthogonal .

Projections

A further application of the inner product, closely related to the ideas of length and angle, is the notion of a projection of one vector onto another. An example is given in Figure 8. The distance x is the pro-

jection of v on w, In two dimensions, we readily know how to calculate

the projection. It is

<sup>x</sup>.. IIvll cos () (3)

where () is the angle between v and w, This formula generalizes , and

for any vectors v and w, the projection of v on w is given by Equation 3. It is a scalar which can be thought of as indicating how much v is

pointing in the direction of w .

"

FIGURE 8.

There is a close relationship between the inner product and the projection. Using Equation 2, we can rewrite the formula for the projec-

tion:

x = \

Iv II cos 9

v 'w

<sup>=</sup>IIv\l \lvll IIwll

Thus, the projection is the inner product divided by the length of w. In particular, if w has length one, then IIwll = I, and the projection of v

on wand the inner product of v and ware the same thing. This way of thinking about the inner product is consistent with our earlier comments. That is, if we hold the lengths of v and w constant, then we know that the inner product gets larger as v moves toward w. From the picture, we see that the projection gets larger as well. When the two vectors are orthogonal , the projection as well as the inner product are

zero.

Inner Products in Two Dimensions

Equation 2 can be shown to be correct in two-dimensional space with the help of some simple geometry. Let v and w be two vectors in the plane, and 9 be the angle between them, as shown in Figure 9. Denote

the x and y coordinates of v �nd w by V..J � Vy and wx, w

y

, respectively.

COPYrighted Material

FIGURE 9.

v

w

Let I denote the projection of v on w. We have I = Ilv II cosO from geometry. We can break I into two pieces Ix and Iy as shown in the figure. I, can be computed from the diagram by noticing that triangles

OAD and COB, in Figure 10, are similar triangles. Thus, the ratio of

corresponding sides is constant:

giving

Iy Wy

�= Ilwll'

numbers. In what follows, cand Cj will be any scalars, and the v and <sup>w</sup>

will be n -dimensional vectors.

v·w=

w·v

<sup>C</sup>(v . w) = (e v) . w = <sup>v</sup>. (e w)

<sup>w</sup>. (v I+ v 2) = w . V I + w . V <sup>2</sup>

(4)

(5)

(6)

The first theorem says simply that order is unimportant; the inner product is commutative. The second and third theorems show that the inner product is a linear function, as we will discuss at length in a later section. We can combine these two equations to get <sup>w</sup>. (eIVI + C2V2) = c) (w . VI) <sup>+</sup>C2 (w . V2). It is also well worth our while to use mathematical induction to generalize this formula, giving

us

(7)

This important result tells us how to calculate the inner product of <sup>w</sup>

and a linear combination of vectors. Another useful theorem is

Iv . wi � IIvllllwll (8)

This is known as the Cauchy-Schwartz inequality. It gives an upper

bound on the inner product.

ONE UNIT IN A PARALLEL DISTRIBUTED

PROCESSING SYSTEM

In this section, we show how some of the concepts we have introduced can be used in analyzing a very simple PDP model . Consider the processing unit in Figure 12 which receives inputs from the n units below. Associated with each of the n + 1 units there is a scalar activation value. We shall use the scalar u to denote the activation of the output unit and the vector v to denote the activations of the ninput units. That is , the ith component of v is the activation of the ith input unit.

Associated with each link between the input uni ts and the output unit, there is a scalar wemwr;�& �g�� can think of the set of <sup>n</sup>

FIGURE 12.

FIGURE 13.

weights as an n -dimensional vector w. This is the weight vector

corresponding to the output unit. Later we will discuss a model with many output units, each of which will have its own weight vector.

Another way to draw the same model is shown in Figure 13. Here we have drawn the n input units at the top with the output unit on the right. The components of the weight vector are stored at the junctions where the vertical input lines meet the horizontal output line. Which diagram is to be preferred (Figure 12 or Figure 13 ) is mostly a matter of taste, although we will see that the diagram in Figure 13 generalizes

better to the case of many output units.

Now to the operation of the model : Let us assume that the activation of each input unit is multiplied by the weight on its link, and that these products are added up to give the activation of the output unit. Using the definition of the inner product, we translate that statement

into mathematics as follows:

u =W ·v.

9. INTRODUCTION TO LINEAR ALGEBRA 385

The geometric properties of the inner product give us the following picture to help in understanding what the model is computing. We imagine that the set of possible inputs to the model is a vector space. It is an n -<sup>d</sup>imensional space, where n is the number of input lines.

The weight vector also has n components, thus we can plot the weight vector in the input space. The advantage of doing this is that we can now state how the system will respond to the various inputs. As we have seen, the inner product gives an indication of how close two vec<sup>t</sup>ors are. Thus, in this simple PDP model, the output activation gives

an indication or measurement of how close the input vector is to the stored weight vector. The inputs lying close to the weight vector will yield a large positive response, those lying near 900 will yield a zero response, and those pointing in the opposite direction will yield a large negative response. If we present a succession of input vectors of constant length, the output unit will respond most strongly to that input vector which is closest to its weight vector, and will drop off in

response as the input vectors move away from the weight vector.

One way to describe the functioning of the processing unit is to say that it splits the input space into two parts, the part where the response is negative and the part where the response is positive. We can easily imagine augmenting the unit in the following way: if the inner product is positive, output a 1; if the inner product is negative, output a O.

This unit, referred to as a linear threshold unit, explicitly computes

which part of the space the input lies in.

In some models, the weight vector is assumed to be normalized, that is, Ilwll = 1. As we have seen, in this case, the activation of the output

unit is simply the projection of the input vector on the weight vector.

MATRICES AND LINEAR SYSTEMS

The first section introduced the concepts of a vector space and the inner product. We have seen that vectors may be added together and multiplied by scalars. Vectors also have a length, and there is an angle between any pair of vectors. Thus, we have good ways of describing

the structure of a set of vectors .

The usefulness of vectors can be broadened considerably by introducing the concept of a matrix. From an abstract point of view, matrices are a kind of "operator" �tftlitl{>ing from one vector space

Addition of Matrices

Matrices are added together by adding corresponding elements. Only matrices that have the same number of rows and columns can be added

together.

Example:

[345] [-1 0 2 ] M + <sup>N</sup>= 1 0 1 <sup>+</sup>4 1 -1 = [

Notice that there is a close relationship between these definitions and the corresponding definitions for vectors. In fact, for fixed integers <sup>m</sup>and n, the set of all <sup>m</sup>x n matrices is another example of a vector space. However, we will not exploit this fact, rather, we will think about matrices in another way, in terms of functions from one vector

space to another. This is the subject of the next section.

Multiplication of a Vector by a Matrix

We now link up vectors and matrices by showing how a vector can be multiplied by a matrix to produce a new vector. Consider the matrix <sup>W</sup>-[ � � :] and the vector v = [ � I. We wish to define a vector <sup>u</sup>

which is the product of <sup>W</sup> <sup>a</sup>nd v . and denoted

u = <sup>W</sup><sup>v</sup>-[ � � ;] m

To define this operation, first imagine breaking the matrix into its rows. Each row of the matrix is a list of three numbers. We can think of the

row as a three-dimensional vector and speak of the row vectors of the matrix. There are two such row vectors. Now consider forming the inner products of each of these row vectors with the vector v. This will yield two numbers. These two numbers can be thought of as a two-

dimensional vector u, which is defined to be the product W v. Copyrighted Material

components of the vector must be the same as the number of columns

of the matrix.

<sup>W</sup>(av) = aWv W (u + v) = Wu + Wv

(9)

(0)

These equations are the counterparts to Equations 5 and 6. As in that section, they can be combined and generalized to general linear combi-

nations:

(1 1)

In the next theorem, the matrices M and N must have the same

number of rows and columns.

Mv + Nv = (M + N)v

ONE LAYER OF A PARALLEL DISTRIBUTED

PROCESSING SYSTEM

(12)

I now generalize the simple model presented earlier to show how

matrices can be used in analyzing PDP models . Consider Figure 17, which is the generalization of Figure 12 to the case of many output units. Suppose that there are m output units, each one connected to all

of the n input units. Denote the activation of the output units by U b U 2, ...• Urn· Each output unit has its own weight vector Wi' separate from the other output units. As before, the activation rule

Outputs:

Inputs:

u � W � v

says that the activation of an output unit is given by the inner product

of its weight vector with the input vector, thus,

Ui = Wi ' v.

If we form a matrix W whose row vectors are the Wi' then we can use

the rule for matrix-vector multiplication to write all of the computations at once. Let u be the vector whose components are the Ui' Then

u=Wv.

This is a very succinct expression of the computation performed by the network. It says that for each input vector v. the network produces an output vector u whose components are the activations of the output

units.

Another way to draw the network is shown in Figure 18. which is the generalization of Figure 13 to the case of many output units. At each

junction in the diagram there is a weight connecting an input unit with an output unit. 3 The weight vectors associated with each output unit appear on the horizontal lines. When drawn this way, it is clear why a matrix appears in the equation linking the output vector to the input

vector: The array of junctions in the diagram is exactly the weight

matrixW.

Now let us attempt to understand geometrically what is being computed by the model. Each output unit is computing the inner product

FIGURE 18.

V 1

w ml

W 12

W 21

W m2

Inputs

appear in a multilayer system and thus have both an incoming weight vector and an outgoing weight vector, as shown in Figure 19. In this case, both views of matrix-vector multiplication can be useful : The unit can be thought of as matching its incoming weight vector to the current input using the inner product, and sending the result of this match

multiplied by the outgoing weight vector to the next level.

LINEARITY

A distinction is often made between a linear system and a nonlinear system. In general, linear systems are relatively easy to analyze and understand, whereas nonlinear systems can be difficult. In this section, I will characterize linear systems. Nonlinear systems are defined simply as everything else. In a later section, I will give some specific examples

of nonlinear systems.

Suppose that there is a function f which represents a system in that

for each input x to the system, the output y is given by

y = j(x).

The x and y might be scalars or they might be vectors, depending on the particular system. The function j is said to be linear if for any inputs X l and x2, and any real number c, the following two equations

hold:

f (ex) = c f (x).

f (X l + X2) = f (X I) + f (X2

)'

(13)

(14)

o

o

FIGURE 22.

This equation can be easily generalized to give the strength of the con-

nection between the jth element of v and the ith element of u:

This formula calculates the inner product between the ith row of M and the jth column of N, which shows that P is equal to the product

MN.

This result can be extended to systems with more than two layers by induction. For example, in a three-layer system, the first two layers

can be replaced with a matrix (as we have just seen) , and then that matrix can be multiplied by the matrix of the remaining layer to get a single matrix for the whole system. In general , the cascaded matrices of any n -layer linear system can be replaced by a single matrix which is

the product of the n matrices.

As a final comment, the definition of matrix multiplication may

seem somewhat odd, especially since it would seem more straightforward to define it by analogy with matrix addition as the element-wise product. In fact, it would be perfectly acceptable to define multiplication as the element-wise product, and then to use another name for the operation we have discussed in this section. However, element-wise multiplication has never found much of an application in linear algebra. Therefore, the term multiplication has been reserved for the operation described in this section, which proves to be a useful definition, as the

mappings of all other vectors in the domain are determined (cf. Equa-

tion 15 ).

In this section, let us special ize to the case of square matrices, that is, matrices with the same number of rows as columns. In this case,

the domain and the range will have the same number of dimensions (because the vectors v and u must have the same number of components) , and the vectors in the domain and the range can be plotted in

the same space. This is done in Figure 23 , where we have shown two

vectors before and after multiplication by a matrix.

In general , vectors in this space wi ll change direction as well as length when multipl ied by a matrix. However, as demonstrated by one of the vectors in Figure 23, there wi ll be some vectors that will change only in length, not direction. In other words, for these vectors, multiplication by the matrix is no different than multiplication by a simple scalar. Such vectors are known as eigenvectors. Each eigenvector v of a

matrix obeys the equation

Wv = AV (20)

where A is a scalar. A is called an eigenvalue, and indicates how much v

is shortened or lengthened after multiplication by W.

Example:

eigenvalues, then the n associated eigenvectors are linearly independent. Although the conditions under which a matrix has a full set of distinct eigenvalues are beyond the scope of this chapter, it is quite possible to have matrices with fewer than n eigenvalues, as in the case of the iden-

tity matrix.

I will not discuss how to find eigenvectors and eigenvalues for a particular matrix, but refer the reader to the books on linear algebra listed at the end of the chapter. There are several methods, all of which can be computationally expensive for large matrices. In a later section I will discuss how to construct a certain class of matrices given a set of

desired eigenvectors.

The goal now is to show how eigenvectors can be used. To do so, let us begin by assuming that we are dealing with the most favorable case: an n x n matrix W with n distinct eigenvalues }q , <sup>A</sup> <sup>2</sup>•... • An .

Denote the associated linearly independent eigenvectors by <sup>v</sup>\I V 2 ••••• <sup>V</sup>II. Recall that if we have a set of basis vectors for the domain of a matrix, and if we know the vectors in the range associated

with each basis vector, then the mapping of all other vectors in the domain are determined. The eigenvectors of W form such a basis. This is because there are n eigenvectors, and they are linearly independent. Furthermore, we know the vectors in the range associated with

each eigenvector Vi; they are simply the scalar multiples given by

Wv = AV.

To show how to take advantage of these observations, pick an arbitrary vector v in the domain of W. It can be written as a linear combi-

nation of the eigenvectors, because the eigenvectors form a basis:

We can now write:

u =Wv

Using linearity,

If we next substitute for each of the quantities Wv i, using Equation 20:

U = CIA lv l + C2A 2v 2 + ... + cn AnV n .

(21)

Algebraic Properties of the Transpose

(W T) <sup>T</sup>= w

(CW )T = cW T

(M + N) <sup>T</sup>= MT <sup>+</sup>NT

(MN )T = NTMT

If a matrix is its own transpose, that i s i f W T = W. then the matrix is

symmetric.

Outer Products

Before discussing outer products, let me attempt to ward off what could be a confusing aspect of the notation we are using. Consider, for example, the entity below. Is it a matrix with one column or is it a

vector?

The answer is that it could be either-there is no way of distinguishing one from the other based on the notation. There is nothing wrong with this failure to distinguish between vectors and n x 1 matrices for the following reason. In equations involving vectors and matrices , the same results will be obtained whether entities such as the one above are treated as vectors or as matrices . This is true because the algebra for vectors and matrices is exactly the same, as a review of the relevant earlier sections will show. Thus, as long as we are simply interested in calculating values and manipulating equations, there is no need to distinguish between vectors and n x 1 matrices. Rather, by treating them

as the same thing, we have a uniform set of procedures for dealing with

all equations involving vectors and matrices.

Nevertheless, on the conceptual level , it is important to distinguish between vectors and matrices. The way we are using the terms, a vector is an element in a vector space, whereas a matrix can be used to define a linear mapping from one vector space to another. These are

With this caveat in mind, we will conti{lue to take advantage of the uniformity of notation, bt.�fii'lWdMft{nOObn between a vector and an

in the previous section, the i ,j t,h element of W is equal to the product u, vi ' which is the product of the activation of the jth input unit and the ith output unit. Both of these quantities are available in a physically circumscribed area on the link joining these two units. Thus, the weight on that link can be changed by autonomous local processes. The Hebb rule is often referred to as a local learning rule for this reason. To summarize, we have established a procedure for finding a matrix W which will associate any particular pair of input and output vectors. Clearly for every pair of vectors, we can find a different weight matrix to perform the association. What is less obvious is that the same matrix can be used for several pairs of associations. Let us assume that

we are given n n-dimensional output vectors Ub U 2 ,

.

.. , Un which

we want to associate with n n -dimensional input vectors

vI, V 2 ' .

. . , V n. In other words, for each <sup>i</sup> , we wish to have

Let us further assume that the vectors v, form a mutually orthogonal

set and that each vector v, is of unit length. That is, we assume

<sup>T</sup>11 if <sup>i</sup>=.i <sup>v</sup>j Y j = 0 otherwise.

We now form a set of matrices W, using the learning scheme

developed above:

Finally, we form a composite weight matrix W which is the sum of

the W, :

W=W 1+ ... +W,+ . . . +Wn•

We already know that, for example, W \ above will associate V I and U I. It is also true that W will perform all such associations. Thus, for

arbitrary i :

Wv , = (W\ + .. . + W, + ... + Wn h,

<sup>=</sup>(U \V \T + ... + <sup>u</sup> ·v .T <sup>+</sup>. .. <sup>+</sup> <sup>U</sup><sup>v</sup> T)y . I <sup>I</sup>

nn I

<sup>=</sup>(u \v th, + <sup>=</sup>U\ (vtv, ) + + (u ,ylh, + + u, (ylv, ) +

+ (u ny!h, + Un (v!v , )

The property of orthogonality was crucial here, because it forced the disappearance of all terms involving vectors other than Uj in the next to last step. The reader may find it useful to justify the steps in this

derivation.

When the set of input vectors is not orthogonal , the Hebb rule will not correctly associate output vectors with input vectors. However, a modification of the Hebb rule, known as the delta rule, or the Widrow-Hoff rule, can make such associations. The requirement for the delta rule to work is that the input vectors be linearly independent. The delta rule is discussed further in Chapter 11, and at length in Kohonen

(977) .

Earlier it was discussed how, at least for square matrices, knowledge of the eigenvectors of a matrix permits an important simplification to be made. The matrix multiplication of a vector can be replaced by scalar multiplication (cf. Equation 21) . I will now show that the Hebbian learning scheme fits nicely with the notion of eigenvectors. Suppose that we wish to associate vectors with scalar copies of themselves. This is what is done, for example, in an auto-associator like those discussed in J. A. Anderson et al . (977) ; see Chapters 2 and 17. In other

words , we want the vectors U ; to be of the form A; v i where <sup>v</sup>i are the input vectors. Let us further assume that the n scalars Ai are distinct.

Using the outer product learning rule, we have

where

If we now present the vector <sup>v</sup>I to the matrix W thus formed, we have

WV i = (W I + ... + WI + . . . + Wn hi

<sup>=</sup>(A IV IV { + . .. + Ajvjvl + ... + <sup>A</sup> <sup>n</sup> <sup>v</sup> <sup>n</sup> v!h,

<sup>=</sup>0 + .. . + Ai v, (V lv j ) + ... + <sup>0</sup>

This equation shows that <sup>v</sup>I is an eigenvector of W with eigenvalue Ai ' Let me summarize. When we calculate a weight matrix W using the

Hebbian learning rule and associate input vectors to scalar multiples of themselves, then those input vectors are the eigenvectors of W. It is

important to note that the matrix W need not even be calculated-as was stated in the section on eigenvectors, once we have the eigenvectors and eigenvalues of a matrix, we can throw away the matrix. All input-output computatiditPK4§'I"dcMft�rW using Equation 21. This

approach is in contrast to a scheme in which we first calculate a matrix W from the input vectors, and then calculate the eigenvectors from the matrix W. Here, the eigenvectors are avai lable in the statement of the

problem.

Why should one want to associate vectors with scalar copies of themselves? Essentially, the answer is that a system which learns in this way will exhibit the desirable property of completion. That is, when par-

tial versions of previously learned vectors are presented to the system, it will be able to produce the whole vector. Readers desiring more details on how this is done should consult Anderson et at . (1977) .

MATRIX INVERSES

Throughout this chapter, I have discussed the linear vector equation u = Wv . First, I discussed the situation in which v was a known vector

and W a known matrix. This corresponds to knowing the input to a system and its matrix, and wanting to know the output of the system. Next, I discussed the situation in which <sup>v</sup>and <sup>u</sup>were known vectors, and a matrix W was desired to associate the two vectors . This is the

learning problem discussed in the previous section. Finally, in this section, I discuss the case in which both u and W are known, but v is

unknown . There are many situations in which this problem arises,

including the change of basis discussed in the next section.

As we will see, the solution to this problem involves the concept of a matrix in verse. Let us first assume that we are dealing with square matrices. The inverse of a matrix W, if it exists, is another matrix

denoted W- l that obeys the following equations:

W-l <sup>W</sup>=I

WW- <sup>l</sup> <sup>=</sup>I

where I is the identity matrix.

Example:

[ 1 'hl W = -1 1 {2 <sup>3</sup>

W- l --

2

Copyrighted Materi 1

3

1 3

2 3

v

FIGURE 26.

We now want to show how to find the coordinates of a vector v ir. new basis Y h Y 2 , ... , Y n • These coordinates are simply the coef

cients Cj in the equation

(2

Let us form a matrix Y whose columns are the new basis vectors :

and let v· be the vector whose components are the Cj . Then Equati

23 is equivalent to the following equation:

v = Yv · (2

where v· is the unknown. The solution to the problem is now clear: . use the inverse matrix y-I to calculate the unknown vector as in t

previous section:

v· = y- 1v.

Example. Letting <sup>Y</sup> I = [\_! I and Y2 = [ I� I, we have Y = [\_! I,

-2 --1 3 3

and y- I = 2 2

3 3

w\* v· .. U ·

y - t y

v • U w

FIGURE 28.

Luckily, we already know how to make each of these

transformations- they are given by the equations:

v = yV · u =WV

u · = y- IU.

Putting these three equations together, we have

u· = y- Iu

= y- I Wv

= y- IWyv· .

Thus, W· must be equal to y-I WY. Matrices related by an equation of

the form W· = y-IWY are called similar.

One aspect of this discussion needs further elaboration. We have been treating matrices as linear operators on a vector space. However, as the results of this section make clear, a matrix is tied to a particular basis. That is, the numbers in the matrix are just as arbitrary as the numbers used for representing vectors. When the basis changes, the numbers change according to the equation W· = y-IWY. The under-

lying mapping, which remains the same when the matrix W is used in the original basis and the matrix W· is used in the new basis, is called a linear traniformation. The same linear transformation is represented by

different matrices in different bases .

It is interesting to recast the results on eigenvectors in terms of a

change of basis. For sOfflSP.w�mJVMJrerN§ consider changing basis to

the linear structure of the vectors shows that it is this linear structure that is relevant to the behavior of the model , and not the particular

basis chosen to describe the vectors.

NONLINEAR SYSTEMS

The use of nonlinearity occurs throughout this book and throughout the literature on parallel distributed processing systems (Anderson et aI., 1977; Grossberg, 1978; Hopfield, 1982; Kohonen, 1977) . In this section, I will indicate some of the reasons why nonlinearities are deemed necessary. 8 Although these reasons are based on the desire for behaviors outside the domain of linear models, it should be stated that linear systems have a great deal of power in themselves, and that many of the nonlinearities represent comparatively small changes to underlying models which are linear. Other models are more fundamentally nonlinear. Further discussions of nonlinear mathematics can be found

in Chapters 10 and 22.

One simple nonlinearity has already arisen in the discussion of a PDP system with one output unit. Such a system computes the inner product of its weight vector and the input vector. This is a linear system, given the linearity of the inner product. The geometrical properties of the inner product led us to picture the operation of this system as computing the closeness of input vectors to the weight vector in space.

Suppose we draw a line perpendicular to the weight vector at some point, as in Figure 29. Since all vectors on this line project to the same point on the weight vector, their inner products with the weight vector are equal . Furthermore, all vectors to the left of this line have a smaller inner product, and all vectors to the right have a larger inner product. Let us choose a fixed number as a threshold for the unit by requiring that if the inner product is greater than the threshold, the unit outputs ai , otherwise it outputs a O. Such a unit breaks the space into

two parts by producing a different response to vectors in the two parts. This use of a threshold is natural in using the unit to classify patterns as belonging to one group or another. The essential point is that the threshold permits the unit to make a decision. Other units in a larger

8 Since nonlinear systems in general are systems that are defined as "not linear," it is important to understand clearly what "lineat" means. A review of the section on linearity

\

FIGURE 29.

\ , , \ \

\ W

\ \

> \ , \ \

system that take their input from this unit could choose completely different behaviors based on the decision. Notice also that the unit is a categorizer: All input vectors that are on the same side of the space

lead to the same response.

To introduce a threshold into the mathematical description of the processing unit, it is necessary to distinguish between the activation of the unit and its output. A function relating the two quantities is shown in Figure 30. It produces a one or a zero based on the magnitude of the activation. It is also possible to have a probabilistic threshold. In this case, the farther the activation is above the threshold, the more

OUTPUT

o

ACTIVATION

likely the unit is to have an output of one, and the farther the activation is below the threshold, the more likely the unit is to have an output of zero. Units such as these are discussed in Chapters 6 and 7. The threshold unit is a good example of many of the nonlinearities that are to be found in PDP models. An underlying linear model is modified with a nonlinear function relating the output of a unit to its activation. Another related example of such a nonlinearity is termed subthreshold summation. It is often observed in biological systems that two stimuli presented separately to the system provoke no response, although when presented simultaneously, a response is obtained. Furthermore, once the system is responding, further stimuli are responded to in a linear fashion. Such a system can be modeled by endowing a linear PDP unit with the nonlinear output function in Figure 31. Note that only if the sum of the activations produced by vectors exceeds T

will a response be produced. Also, there is a linear range in which the system responds linearly. It is often the case in nonlinear systems that there is such a linear range, and the system can be treated as linear pro-

vided that the inputs are restricted to this linear range.

One reason why subthreshold summation is desirable is that it suppresses noise. The system will not respond to small random inputs

that are assumed to be noise.

All physical systems have a limited dynamic range. That is, the response of the system cannot exceed a certain maximum response. This fact can be modeled with the output function in Figure 32, which shows a linear range followed by a cutoff. The system will behave linearly until the output reaches M, at which point no further increase can occur. In Figure 33, a nonlinear function is shown which also has a

OUTPUT

T

ACTIVATION

FIGURE 32.

FIGURE 33.

M

OUTPUT

ACTIVATION

<sup>M</sup>-------- --.. .......

OUTPUT

ACTIVATION

maximum output M. This curve, called a sigmoid, is a sort of hybrid between Figure 31 and Figure 32. It combines noise suppression with <sup>a</sup>

limited dynamic range. Chapter 8 shows how such units are necessary for certain kinds of interesting behavior to arise in layered networks.

To summarize, I have described some of the ways in which linear systems are modified to produce nonlinear systems that exhibit certain desired behaviors. All of these systems have an important linear component and are sometimes referred to as semilinear. Furthermore, several of the systems have a linear range in which the nonlinearities can be ignored. The next chapter discusses more fundamentally non-

FURTHER READING

Halmos, P. R. (1974) . Finite-dimensional vector spaces. New York: Springer-Verlag. For the more mathematically minded. An excellent

account of linear algebra from an abstract point of view.

Kohonen, T. (1977) . Associative memory: A system theoretic approach.

Berlin: Springer-Verlag. This book has a short tutorial on linear algebra. The di scussion of associative memory depends heavily on the

mathematics of linear algebra.

Strang, G. ( 1 976) . Linear algebra and its applications. New York: Academic Press . A general textbook treating most of the essentials of linear algebra. It is especially good in its treatment of computational topics. A good place to find out about calculating matrix inverses and

eigenvalues.

vary is the same as the range of values over which the output of the unit (its activation) may vary. Another is that time may be ignored as <sup>a</sup> variable. The activation function of a unit will be taken to be a function that computes the output of the unit (at a fixed but unspecified time) as a function of its inputs (at a presumably slightly earlier but unspecified time). Thus, given a unit with n inputs whose activation values range over the set A, the activation function a for this unit is just a function from A n (the set of ordered n -tuples of elements of A)

to A, denoted a :An-A.

In order to avoid cluttering the presentation, detailed proofs of the results have been omitted; in their place are short sketches indicating the key steps. A more rigorous and abstract formulation of the basic concepts introduced here, along with detailed proofs of the results, may

be found in Williams (1983).

EXAMPLES OF ACTIVATION RULES

The following are some examples of activation functions from which

models have been constructed.

Example 1. A = (0,11 (the two-point set), a=fog, where g is linear into JR and f: JR-A is a thresholding function. (The operator <sup>0</sup> between two functions here denotes composition in a right-to-left manner.) A unit using this activation function is called a threshold logic unit or a linear threshold unit and is the basis of the simple perceptron

(Rosenblatt, 1962; Minsky & Papert, 1969).

Example 2. A = JR, a linear (Kohonen, 1977).

Example 3. A = I (the closed unit interval [0,1)), a=fog, where <sup>g</sup>is linear into JR and f is nondecreasiI-., into I. This is a commonly used variant of Example 1. Let liS call this a quasi-linear activation function. The function f is sometinu:!s called a squashing function for obvi-

ous reasons.

Example 4. A = I, <sup>a</sup>= fog. where f is nondecreasing into I and <sup>g</sup>

is a multilinear function into JR of the form

g(X\, ... ,xn) = X\X2+XJX4+ ... +Xn-lxn

(where n is assumed to be even). Such an activation function is suggested by Hinto<sup>n</sup>(1981 b). Note that this is similar to Example 3 Copyrighted Material

A

IX = 0

", ...

"""',' """','

.,

B c

FIGURE 3. a (x\,

X2) = max(O,xI+x2-l). A: Three-dimensional plot. The cube is

bounded by the planes where each coordinate is 0 or I. B: Contour plot. C: Some sections along xI' Note that the three-dimensional plot of this function consists of two planar surfaces. Clearly, each section along xI is a nondecreasing function; by symmetry the same is true of each section along x2' Thus this function is uniformly nondecreasi<sup>n</sup>g. Copyrighted Material

A

a

�X2

X,

a=l

�------------'-�

X 1

8 c

FIGURE 4. a (X!,

X2

) = xl+x2-xlx2

' A: Three-dimensional plot. The cube is bounded

by the planes where each coordinate is 0 or I. B: Contour plot. C: Some sections along xl' Note that each section along xl is a linear function with nonnegative slope; by symmetry the same is true of each section along x2' Thus this function is uniformly non-

decreasing.

A

a

�X2

X,

Cl

Cl = .5

Cl = 25

<sup>o</sup>n

o

Cl <sup>=</sup>

<sup>=</sup>.75

B

1

�

c

FIGURE 5. a(xJ,x2) = max(xJ,x2)' A: Three-dimensional plot. The cube is bounded by the planes where each coordinate is 0 or 1. B: Contour plot. C: Some sections along xI' Note that the three-dimensional plot of this function consists of two planar s<sup>u</sup>rfaces. Note also that each section along xI is a <sup>n</sup>ondecreasing function; by symmetry the same

A

a- a = 0

--,,-, a= 25�

I

a = .5 <sup>I</sup>

B

L-----------------��

Xt

c

FIGURE 7. a (xIoX2) - xl+x2-2xlx2' A: Three-dimensional plot. The cube is bounded by the planes where each coordinate is 0 or I. B: Contour plot. C: Some sections along XI ' Note the saddle shape of the three-dimensional plot of this function. Also note that the sections along XI are linear functions with slopes ranging from 1 to -I; by symmetry the same is true of the sections along Xl' Thus this function is monotonic-in-context but

not uniformly monotonic.

A

k::x

'

XI

o Ot = 0 o

Ot <sup>=</sup>----.....

B

I .751 I I I

X, �-----------------L�X,

c

FIGURE 8. ex (x1,X2) = Ih(xl+x2)' A: Three-dimensional plot. The cube is bounded by the planes where each coordinate is 0 or 1. B: Contour plot. C: Some sections along x I' Note that the three-dimensional plot of this func�on consists of a single planar surface. Each section along xI is a linear function with'slo\>e 'h, as is each section along x2, by

symmetry. Thus this function is uniformly nondecreasing.

A

(l

--��� :�

\,. "i C! = .5 5

C! .2 <sup>I</sup>5

(l = 0 <sup>I</sup>

8 c

FIGURE 9. cdx),x2) = (2x'I- t)2(2x'2- t)2. A: Three-dimensional plot. The cube is bounded by the planes where each coordinate is 0 or I. B: Contour plot. C: Some sections along xI' Note that the sections along XI are parabolas of varying widths. Evidently, this function is not monotonic-in-context since, for example, when x2- 0, a first

decreases and then increases as X I increases.

The mapping from Boolean expressions to real algebraic expressions

is defined by replacing:

1. True by 1. 2. False by O.

3. The disjunction operator by addition. 4. The conjunction operator by multiplication. S. The negation operator by subtraction from 1.

6. Xi by Xi, for each i.

For example, applying this function to the Boolean expression X,X2+ X, yields the real expression XI (I-x 2) + (I-XI). It should be emphasized that this is a function defined only on formal expressions; two expressions that are equivalent under Boolean algebra will not, in general, be mapped to the same real algebraic expression or even equivalent real algebraic expressions. In other words, it is not a map-

ping from Boolean functions to real functions.

A standard result from Boolean algebra is that any Boolean function may be expressed in a certain canonical form, called the disjunctive normal form. A simple prescription for this form is as follows: Form a disjunction of terms, each of which is the result of applying the verticesto-Boolean-expressions function described above to those vertices ofln for which the function takes on the value true. For example, the disjunctive normal form for the Boolean function /3(XbX2) = X,+X2 is

X,X2+ X,X2+ X,X2·

A closely related result for multilinear functions is the following:

Lemma. For any function assigning arbitrary real numbers to the vertices of P there is a unique multilinear function agreeing with

the given function on these vertices.

This function is formed in a manner generalizing the prescription given above for the disjunctive normal form: For each vertex of I

n,

form the corresponding Boolean conjunct; then apply the other function described above to turn each of these conjuncts into a real expression; finally, form the sum of these individual expressions with each one weighted by the value cf the given function at the corresponding vertex. It will be convenient to dub the result the vertex normal form for the given function. For example, the vertex normal form for a multi-

linear function a of two variables is

a (X "X2) = a (O,O)(l-x:)(I-x2)+a(O,O(I-x,)x2

+a (l,O)x,(1-x2)+a (l,l)x,x2. Copyrighted Material

This lemma has the following immediate consequence:

Theorem 1. Given any Boolean function, there is a unique multi-

linear activation function realizing it.

In contrast, not every Boolean function can be realized by a quasilinear activation function. Those Boolean functions that can be so realized are called linearly separable. It is easily shown that any linearly separable Boolean function is necessarily uniformly monotonic, but the converse is not true. A simple example of a function that is not linearly separable is the XOR function t31(X\.X2) = X1X2+X1X2. The easiest way to see that it is not linearly separable is to observe that it is not uniformly monotonic. An example of a function that is uniformly

monotonic but not linearly separable is

t32(X\.X2,X3,x4) = X1X2+ X3X4·

Our next result, also a consequence of the lemma, shows that the very general class of all activation functions may be represented up to vertex-equivalence by the narrower class of multilinear activation

functions.

Theorem 2. Every activation function is vertex-equivalent to a

unique multilinear activation function.

The next result suggests that monotonicity-in-context is enjoyed by a

fairly wide variety of activation functions.

Theorem 3. Every sigma-pi activation function is monotonic-in-

context.

This is an easy consequence of three facts: (a) that a multilinear function is linear in each variable when the other variables are held constant; (b) that a linear function is monotonic; and (c) that the com-

position of monotonic functions is monotonic.

The following result characterizes uniform monotonicity for multi-

linear activation functions.

Theorem 4. A multilinear activation function is uniformly monotonic if and only if its restriction to vertices is uniformly monotonic.

The key step in the proof of this result is the observation that a multilinear function may be built up inductively through linear interpolation, starting with the values at the vertices. This follows from the fact that a multilinear function is linear in each variable when the other variables are held constant. The remainder of the proof consists of verifying that each step of this inductive construction preserves uniform Copyrighted Material

monotonicity. This result may be extended to the sigma-pi case as well, under certain mild restrictions, using the fact that a strictly

increasing function has a monotonic inverse.

Corollary. Let a = fog be a sigma-pi activation function, where g is multilinear and f is a squashing function. If f is strictly increasing, then a is uniformly monotonic if and only if its restriction to ver-

tices is uniformly monotonic.

The results presented up to this point would seem to suggest that the class of multilinear activation functions provides us with sufficient power that we need not consider the more general class of sigma-pi activation functions. However, from the standpoint of uniform monotonicity, there may be some drawbacks in restricting ourselves to multilinear activation functions. One such potential drawback is that a uniformly nondecreasing multilinear activation function may have some negative weights. For example, the Boolean function {3 (X 1 ,X 2) = Xl + X 2 corresponds, by Theorem 1, to the multilinear activation function a (Xl,x2) = X\+X2-x\X2, which requires a negative weight even though it is uniformly nondecreasing. But what if a more general sigma-pi activation function were to be used? Is there a sigma-pi realization of this same Boolean function for which all weights are nonnegative? Of course there is in this case: The sigma-pi activation function a (Xl,x2) = min (xl<sup>+</sup> x2, n is one such realization; many others could be devised. (These two realizations of the OR function are displayed in Figures 4 and 6.) It seems reasonable to suspect that

the following is true:

Cor/iecture. Every uniformly nondecreasing activation function is vertex-equivalent to a sigma-pi activation function with nonnegative

weights.

Note that any sigma-pi activation function with nonnegative weights is certainly uniformly nondecreasing. The conjecture is that the converse is true (up to vertex equivalence). Under the assumption that the uniformly nondecreasing activation function is Boolean-like (as in the preceding example), the conjecture is indeed valid, as the following theorem shows. In fact, the conclusion may be made even stronger in

this case.

Theorem 5. Every uniformly nondecreasing Boolean-like activation function is vertex-equivalent to a sigma-pi activation function whose

weights are all 0 or 1.

The essential step in the proof of this result is showing that any uniformly nondecreasing �t.�a/may be expressed as a

disjunction of conjunctions containing no negated factors. Once such an expression is available, the desired sigma-pi activation function is obtained by converting this Boolean expression to a real expression and

then composing this with the function <sup>j</sup>(z) = min (z ,1).

This theorem may be generalized to cover arbitrary senses of uniform monotonicity by running any inputs for which the activation function is nonincreasing through the "inverter" j(x) = I-x. Thus the general class of all uniformly monotonic Boolean-like activation functions may be represented up to vertex-equivalence by a narrower class

of sigma-pi activation functions of a certain form.

It is instructive to contrast the sigma-pi activation functions which result from applying Theorems I and 5 to a particular uniformly monotonic activation function. Consider the Boolean function of six variables IHXJ,X2,x3,X4,XS,X6) - X1X2+ X�4+ XsX6' Theorem I realizes this using the vertex normal form, which, after simplification,

becomes

Q l(Xl,x2,x3,X4,xS,x6) = xlx2+xJX4+XsX6

- xlx�JX4 - xlx� sX6 - xJX4XsX6

+xlx�JX4XsX6'

In contrast, Theorem 5 implies a realization of this same function by

the gating activation function

Q2(Xl,x2,x3,x4,x5,x6) = min(xlx2+XJX4+XsX6, n.

CONCLUSION

As suggested in the introduction, the idea<sup>s</sup>and results presented here represent an exploratory set of concepts intended to help in understanding PDP networks. There is a clear need for a general language and set of concepts for describing and understanding PDP computation, both at the local, individual unit level, as explored here, and at the level of whole networks. (In fact, the greatest need is for a means of describing and understanding the relationship between computation at these two levels.) Whether the ideas contained in this chapter can extend naturally to become a useful framework for understanding the behavior of whole networks is difficult to foresee. One way that this gap between local and global computation might be bridged is by dealing with questions of learning in such networks. The goal of learning is generally to cause the ne���\$te�Xfa,erra5ticular global behavior, but

CHAPTER }}

An Analysis of the Delta Rule

and the Learning of Statistical Associations

G. o. STONE

The development of parallel distributed processing models involves two complementary enterprises: first, the development of complete models with desired operating characteristics� and second, the in-depth analysis of component mechanisms and basic principles. The primary objective in modeling is the development and testing of complete systems. In general these models are complex and their behavior cannot be fully deduced directly from their mathematical description. In such cas�s, simulation plays an important role in understanding the properties of a model. Although simulations are useful in determining the properties of a specific model, they do not, on their own, indicate how a model should be modified when a desired behavior is not achieved. An understanding of basic principles and a collection of potential mechanisms with known properties provide the best guides to the

development of complex models.

This chapter provides an analysis of one of the most popular components-namely, the error correction learning rule developed by Widrow and Hoff (1960). This learning rule which has been analyzed and employed by a number of authors (Amari, 1977a, 1977b; Kohonen, 1974, 1977; Sutton & Barto, 1981), has been called the Widrow-Hoff rule by Sutton and Barto (1981) and is generally referred to as the delta rule in this book. This rule is introduced in Chapter 2, discussed extensively and generalized in Chapter 8, and employed in models discussed in a number of chapters-most notably Chapters 17 and 18. In the present chapter I show how concepts from linear algebra Copyrighted Material

where, as usual, bold letters indicate vectors, uppercase indicates matrices and the superscript T indicates the transpose of a vector or matrix. This learning rule was described in some detail in Chapter 9 and that discussion will not be repeated here. It was shown there that if the input vectors are normalized in length so that ip . ip = 1 and are orthogonal, the product rule will, after the presentation of all of the

input/ output patterns, lead to the following weight matrix:

If the input vectors are orthogonal, there will be no interference from storing one vector on others already stored so that the presentation of

input ip will lead to the desired output tp' that is,

Wip = tp

for all patterns p from 1 to P. Unfortunately, we cannot always insure that the input vectors are orthogonal. Generally, the storage of one input/ output pair can interfere with the storage of another and cause crosstalk. For this case a more sophisticated learning rule is required. Fortunately, as we saw in Chapter 8, the delta rule is a rule that will work when the input patterns are not orthogonal. This rule will produce perfect associations so long as the input patterns are merely linearly independent (see Chapter 9) and will find a weight matrix which will produce a "least squares" solution for the weight matrix when an exact solution is not possible (i.e., the input patterns are not linearly

independent). In matrix notation the rule can be written as

W (n) = W (n - 1) + 1)8 (n)j T (n ) (1)

where W (n) is the state of the connection matrix after n presentations, i (n) is the input presented on the nth presentation, 1) is a scalar constant which determines the rate of learning, and 8 (n) is the difference

between the desired and actual output on trial n , such that

IS (n ) = t (n ) - W (n - l)i (n ) (2)

where t (n) is the desired output (or target) for presentation n and W (n - l)i (n ) = 0 (n) is the output actually produced on that presentation. W (0) is assumed to be the matrix with all zero entries. In other words, the weight matrix is updated by adding the outer product of the response error and the input. (See Chapter 9 for discussion of outer product.) Proofs concerning the convergence of this recursion to the optimum weight matrix (in the sense outlined above) are provided by

The Delta Rule in Pattern-Based Coordinates

To this point we have discussed the delta rule for what Smolensky (Chapter 22) has called the neural or unit level of representation. Before proceeding, it is useful to consider the form that the rule takes in the conceptual level of representation in which there is one vector component for each concept. In general, the input and output patterns correspond to an arbitrary set of vectors. Interestingly, it is possible to show that the delta rule applies only to the "structure" of the input and output vectors and not to other details of the representation. In a linear system, it is only the pattern of correlations among the patterns

that matter, not the contents of the specific patterns themselves.

We can demonstrate this by deriving the same learning rule following a change oj basis from the unit basis to the pattern basis. Since a detailed discussion of the process whereby bases can be changed is given in Chapter 9 and, in more detail, in Chapter 22, I will merely sketch tl1e concept here. Each pattern over a set of units corresponds to a vector. If there are N units, then the vector is of dimension N. In the unit basis, each element of the vector corresponds to the activation value of one of the units. Geometrically, we can think of each unit as specifying a value on a dimension and the entire vector as corresponding to the coordinates of a point in N-dimensional space. Thus, the dimensions of the space correspond directly to the units (this is why it is called the unit basis). Now, a change of basis amounts essentially to a change in coordinate system. This can be accomplished through rotation, as well as other linear transformations. Converting to the pattern basis merely involves transforming the coordinate system so that the patterns line up with the axes. Figure 1 illustrates a simple case of this process. In Figure lA we give the geometric representation of the patterns. Pattern 1, PI, involves two units, each with activation value + 1. Pattern 2, P2, has activation values < + 1 ,- 1 >. The patterns described

in the unit basis are

Figure 1 B shows the same two vectors, but now expressed with respect to a new coordinate system, the pattern coordinate system. In this case the axes correspond to the patterns not the units. The vectors

corresponding to patterns 1 and 2 now become

\* 1 = [?] and <sup>p</sup>

\* 2" [A]·

A Unit 2

-1--------�-------

-1

B -1

Pattern 1 <+1,+1>

Unit 1

Pattern 2 < +1,-1 >

+1

< 0,+1 >

-1

+1

FIGURE 1. An example of conversion from unit-based coordinates into pattern-based

coordinates.

In general, conversion to a new basis requires a matrix P which specifies the relationship between the new and old coordinate systems. For each vector, Pi' we write the new vector p\* i as P\* i = Pp i' If all of the vectors and matrices of the original system are converted into the new basis, we simply have a new way to describe the same system. For present purposes we have !wo tran"formation matrices, one that Cbpyngnred MaTerial

transforms the input patterns into a coordinate space based on the input patterns, which we denote P /, and one that transforms the target patterns into a coordinate space based on the target patterns, P r. In this case, we have i\*; = P/i; for the input vectors and t\*, = Prt, for the target vectors. Moreover, since the output vectors must be in the same space as the target vectors we have 0\*; = Pro,. We must also transform the weight matrix W to the new basis. Since the weight matrix maps the input space onto the output space, both transformations must be involved in transforming the weight matrix. We can see what this transformation must be by considering the job that the weight matrix must do. Suppose that in the old bases Wi = 0 for some input i and output o. In the new bases we should be able to write W\*i\* = 0

\*

Thus, W\*P/i = Pro and PT"IW\*P/i = 0 = Wi. From this we can readily see that P T" I W\*p / = Wand finally, we can write the appropriate

transformation matrix for W as

W\* = PrWP[I.

We can multiply both sides of Equation 1 by P r on the right and P [I

on the left. This leads to

PrWp[ l (n) = PrWp[l(n- 1) + PT1}8(n)ir(n)p/1

which, by substitution, can be written as

W\* (n) = W\* (n - 1) + 1/8\* (n ) [ P [I i \* (n ) r P /1,

where

8 \* (n ) = t\* (n ) - W\* (n - 1) i\* (n ).

(3)

Finally, by rearranging we have

(4)

where the matrix C, given by C = (p/I )Tp[l, is a matrix which holds the correlational information among the original input patterns. To see this, recall that we are changing the input patterns into their pattern basis and the target patterns into their pattern basis. Therefore, the <sup>v</sup>ector i\*j consists of a 1 in the jth cell and zeros ev<sup>e</sup>�<sup>y</sup>where else. Thus, since i· = P/l i\* j' we see that P /1 must be a matrix whose j th column is th� j <sup>t</sup>h original input vector. Therefore, C is a matrix with the inner product of the input vectors i I and i j occupying the ith row and j th column. This inner product is the vector correlation betwee<sup>n</sup>

the two patterns. Copyrighted Material

We have finally constructed a new description which, as we shall see, allows many insights into the operation of the delta rule which are normally obscured by the internal structure of the patterns themselves. Instead, we have isolated the critical interpattern structure in the

matrix C.

One advantage of this new description is that the output the system actually produces-even when it does not match any target exactly-can easily be interpreted as the weighted average of the various target patterns. The value in each cell of the output vector is the coefficient determining the amount of that target in the output. In this case the

sum squared error for input/output pattern p, given by

Ep = L (t\*j - O"jP'

measures the error directly in terms of the degree to which each target pattern is present in the output, rather than the degree to which each unit is present. It should be noted, of course, that this new patternbased error function is related to the old unit-based error by the same

change of basis matrices discussed above.

It might be observed further that under this description, the perfect associator-which results when the input and output patterns are linearly independent -will be the identity matrix, I, in which the main diagonal has a 1 in each entry and all other entries are O. It should be noted, however, that the preceding analysis of this new description has assumed the input and target output patterns were linearly independent. If they are not, no such pattern basis exists. However, there is an analogous, but somewhat more complex, development for the case in

which these vectors cannot form a legitimate basis.

I will now demonstrate some of the useful insights which can be gained through this analysis by comparing the unit and pattern basis descriptions for a sample learning problem. The upper portion of Figure 2 gives the representations of the four input/ output patterns to be learned in the unit basis. These patterns are all linearly independent and were generated under the constraint that each pattern has unit length and that the input patterns have the correlation structure given in the

matrix shown in the figure.

Figure 3 shows the states of Wand W\* after one, four, and eight sweeps through the four input/ output patterns. While inspection of the unit-based representations gives no direct information about the degree of learning and crosstalk between targets, this information is explicit in the pattern-based representation. For example, one can discern that the error for the pairs with highly correlated inputs (pairs 1 and 2) is greater at each stage than that for the pairs with slightly correlated input

.36 .20 49 .44 Xl -.0 <sup>1</sup>

> .28 .09 .55

.25 .48 -.48 .56 X3 .18 -.28 .0 4 -.22

1

.... Yl

3

.... Y3

.57

-.10 .05 -.66 .12 . 30 .23 -.08 X2 .08 -.45 .47 .50 -.36

.50

-.1 1

-.10 -.08 .40 .03 .27 .36 .27 X4 .0 <sup>1</sup> -.56 -.7 1 .6 1 -.54

.24

1.00 .75 0 <sup>0</sup>

2

-.07 -.39

....

.74 Y2 -.2 <sup>1</sup> 46 -. <sup>1</sup> 9

4

.47 .17

....

-.2 1 Y4 .0 <sup>1</sup> -.66 -.5 1

c== .75 1.00 0 <sup>0</sup>

o 0 1.00 .25 o 0 .25 1.00

FIGURE 2. Key-target pairs and the key correlation structure.

patterns (pairs 3 and 4). Moreover, there is no intrusion of targets associated with orthogonal inputs. In addition, the intrusion of targets from correlated pairs is least for the pair most recently learned, pairs 2 and 4. (The patterns were presented in order 1-2-3-4 on each sweep.) Finally, it is clear from inspection of the pattern-based weight matrix that after eight sweeps the patterns have been almost perfectly learned. The pattern-based formulation also allows a more detailed analysis of the general effect of a learning trial on the error. We can define the

.. potential error" to pattern j, a J \* as

Since the vectors i\* j and i\* [ consist of a 1 and the rest zeros, the entire expression i\* [e i\* j reduces to Ckj' the entry in the k th row and

jth column of matrix e. Thus, Equation 6 becomes simply

(7)

In other words, the decrease in error to the jth input/output pair due to a new learning trial is a constant times the error pattern on the new learning trial. The constant is given by the learning rate, Tt, times the correlation of the currently tested input and input from the learning trial. Thus, the degree to which learning affects performance on each test input is proportional to its correlation with the pattern just used in learning. Note that if Tt is small enough, the error to the presented pat-

tern always decreases. In this case Equation 7 can be rewritten

8k -en) = 8k -(n - D(1 - 'rjCkk)'

Recalling that Ckk is given by i[ik, the length of the kth input vector, we can see that the error will always decrease provided 11 - Tt i[ik I < 1. To summarize, this exercise has demonstrated that a mechanism can often be made more conceptually tractable by a judicious transformation. In this case, expressing the possible input and output representations in the appropriate pattern bases clarified the importance, indeed the sufficiency, of the input "structure" (i.e., the pattern of inner products among the input vectors) in determining the role of the input representations in learning. Furthermore, converting the weight matrix into a form from which the errors at any stage of learning can be read directly allowed us to "see" the learning more obviously. The result has been a clearer understanding of the operation of the delta rule for

learning.

STATISTICAL LEARNING

In this section we extend our analysis of the delta rule from the case in which there is a fixed target output pattern for each input pattern to the case in which sets of input patterns are associated with sets of output patterns. We can think of the sets as representing categories of input and outputs. Thus, rather than associate particular input patterns with particular output patterns, we analyze the case in which categories bf input patterns are associated with categories of output patterns. This, for example, might be the case if the system is learning that dogs bark. The representation for dog might differ on each learning trial with respect to size, shagginSi§p�htW8iNtatb;;afepresentation for the bark

might vary with regard to pitch, timbre, etc. In this case, the system is simultaneously learning the categories of dog and bark at the same time

it is learning the association between the two concepts.

In addition, when we have category associations, statistical relationships between the input and output patterns within a category can be picked up. For example, the system could learn that small dogs tend to have high-pitched barks whereas large dogs may tend to have low-

pitched barks.

In order to analyze the case of statistical learning, we now treat the input/ output pairs of patterns as random variables. In other words, each time pattern ij is selected as input, its entries can take different values. Similarly, the target output for pair j, Ij will have variable entries. The probability distributions of these random variables may take any form whatsoever, but they are assumed not to change over time. Moreover, we can consider the entire set of input/ output pairs to form a single probability distribution. We then assume that on each trial an input/ output pair is randomly sampled from this overall proba-

bility distribution.

We proceed with our analysis of statistical learning by computing the expected or average change in the weight matrix following a presentation. From Equations 1 and 2 we get the following form of the delta

rule:

W (n) = W (n - 1) + .,., [t (n ) - W (n - 1) i (n ) ] i T (n ).

Simplifying and taking the expected value of each side we have

Note, we may take

E[W (n - 1)i{n)i T (n)] = E[W (n - 1)]E[i{n)i T (n)]

since each trial is assumed to be statistically independent of all preceding trials, upon which W (n - 1) depends. Letting R J = E [Ii T] be the pattern of statistical correlations among the input patterns and R JO = E hi T] be the statistical correlations between the input and tar-

get patterns, we can rewrite Equation 7 as

E[W (n) ] = E[W(n- 1)](1 - .,.,RJ) + .,.,RJO.

If we solve the recursion by replacing W (n - I) with an expression in terms of W (n - 2) etc. down to W (0) and assuming that W (0) = 0, the matrix of all 0 entries, we can write the expected value of the weight

matrix after n trials as

E(W(n) ] = 7)RJOt(I - 7)R, )i.

(9)

i-O

Fortunately, in the limit, this matrix reduces to a simpler form. To see this, we must introduce the concept of the pseudo-inverse of a matrix. This is a matrix which, unlike the inverse, is certain to exist for all matrices, but which has a number of properties in common with an true inverse. (See Chapter 9 for a discussion of matrix inverses and the conditions under which they exist.) In particular, it is the true inverse, if the true inverse exists. The pseudo-inverse of a matrix B, designated

B+, is given by

B+ = 7)BTr, (I - 7)BBT)l i-I

provided 7) is sufficiently small. (See Rao & Mitra, 1971, and Kohonen, 1977, 1984, for a full discussion of the pseudo-inverse.) In order to convert Equation 9 into a form that includes the expression for the pseudo-inverse, we observe that since the square matrix R, = EUiT] has independent rows and columns, we can select a matrix P such that ppT = R, and P also has linearly independent rows and columns. Since the generalized inverse of P, P+, is also the true inverse of P, it satisfies (P T )-Ip T = I. Thus, taking the limit as

n-oo of Equation 9 and substituting P, we can write

00

limE(W(n)] = E(Woo] = RJO (pT )-1 (7)pT:E (I - 7)ppT F1. (10)

n-oo i=\

Now, by substituting in for the pseudo-inverse of P and simplifying we

get

Since the rows and columns ofR, are linearly independent, R,I = Rt.

So we finally get

(12)

Now we wish to show that, after training, the system will respond appropriately. Without further restrictions, we can demonstrate a minimal appropriateness of the response, namely, we can show that E [W J ] = E [t ]. In other words, we can show that the mean output of the system, after learning, is the mean target. Since the test trials

and learning trials are statistically independent we can write

that, given a particular input, the system will produce an output equal to the average of the targets paired with that input. In this sense, sys-

tematic covariation of input/ output pairs will be learned.

The Delta Rule and Multiple Linear Regression

Some readers may have already noticed the similarity of the learning task we have been analyzing to the problem encountered in multiple linear regression. In a linear regression problem the objective is to predict, to the degree possible, one variable, say y, from a set of variables x. In these problems we typically wish to find a set of coeffi-

cients, b, such that

(where Xo is taken to be 1) and the sum-squared error

is minimized. This is precisely the problem that the delta rule seeks to solve. In this case, each element of the target vector for input/ output

pair (p tp

) is analogous to a to-be-predicted observation Yj; our predic-

tion variables x j are analogous to our input vectors ip

; our regression

coefficients b correspond to a row of the weight matrix W; and the intercept of the regression line, bo, corresponds to the bias often assumed for our units (cf. Chapter 8). In our typical case the target vectors have many components, so we are simultaneously solving a multiple regression problem for each of the components of the target vectors. Now, the standard result from linear regression, for zero-mean random variables, is that our estimate for the vector b, b is given by

where X is the matrix whose columns represent the values of the predictors and whose rows represent the individual observations. (Again, we take the first column to be all Is.) Now, note from Equation 12

that the delta rule converges to

This equation is the strict analog of that from linear regression theory. 2 If we assume that each output unit has a bias corresponding to the intercept bo of the regression line, we can see that the delta rule is, in effect, an iterative method of computing the best, in the sense of least

squares, linear regression coefficients for our problems.

SUMMARY

To summarize, this chapter has shown that close examination of the delta rule reveals a number of interesting and useful properties. When fixed patterns are being learned, the rule's operation can be elucidated by converting from a unit-based description to a pattern-based description. In particular, the analysis showed that the correlations between the input patterns, and not the specific patterns used, determined their effect on the learning process. Thus, any alteration of the specific input patterns that does not alter the correlations will have no effect on learning by a linear delta rule. It was also shown that expressing the inputs and outputs in terms of the patterns being learned facilitated analysis of the learning process by allowing one to read directly from the output produced the degree to which each target was present in the

output generated by a given input pattern.

When the patterns being learned are variable, it was noted that the final weight matrix could be expressed simply in terms of the intercorrelations among the input patterns, RI, and the correlations between the input and output patterns, RIO' It was also shown that when several reasonable requirements for the distribution of the input/ output random variables are met, the delta rule will learn the pattern of covariation between the inputs and targets. Finally, we showed that the delta rule carries out the equivalent of a multiple linear regression from the input patterns to the targets. Those familiar with linear regression should conclude from this both the power of the rule and its weaknesses. In particular, wherever a linear regression is insufficient to provide a good account of the relationship between input and target patterns, the system will perform poorly. The solution to this problem is to have nonlinear units and intermediate layers of hidden units. Chapter 8 is a detailed discussion of the generalized delta rule and its

application to these situations.

2 Actually, th<sup>e</sup>re is a sli<sup>g</sup>ht difference in convention be<sup>t</sup>ween our development and that typical of linear regression. In our case, the stimulus vectors are the column vectors, whereas in linear regression the predictor variables are the rows of the matrix X. Thus this equation differs by a transposition from Equation 12. This has no consequences for

the points made here. Copyrighted Material

The preceding discussion does not, by any means, provide a complete analysis of the delta rule. Rather, it illustrates two important ideas. First, that a basic principle (in this case, the use of patternbased, rather than unit-based representations) can provide valuable insights into the operation of a useful mechanism; and second, that the analysis of component mechanisms which were designed for one use

can often reveal new applications.

CHAPTER 12

Resource Requlrements of

Standard and Programmable Nets

1. L. McCLELLAND

In several places in this book we have examined the capabilities of various models of parallel distributed processing. We have considered models that are guaranteed to do one thing or another-to learn, say, up to some criterion of optimality or to settle into global states with probabi lities proportional to the goodness of the states. In later chapters, we describe various models of psychological or neurophysiological processes and consider how well they account for the data. The models, then, are held up against various criteria of computational,

psychological, and sometimes physiological adequacy.

In this chapter I raise another question about PDP models . I consider the resources they require, in terms of units and connections, to carry out a particular amount of work. This issue is touched on in various other places in the book, particularly Chapter 3. There we showed that a distributed model can often perform even an arbitrary mapping with less hardware than a local model would require to do the same

task.

In this chapter I continue this line of thinking and extend it in various ways, drawing on the work of several other researchers, particularly Willshaw (1971, 1981). The analysis is far from exhaustive, but it focuses on several fairly central questions about the resource requirements of PDP networks. In the first part of the chapter, I consider the resource requirements of a simple pattern associator. I review the analysis offered by Willshaw (1981) and extend it in one or two small ways , and I consider how it might be possible to overcome some Copyrighted Material

FIGURE 1. A pattern associator consisting of a set of input units (across the bottom) and output units (a)ong the right side), with a connection from each input unit to each

output unit.

he assumed that the threshold of each output unit is set equal to the number of active input units. Given this assumption, only those output units with switched-on connections from all of the active input

units will reach threshold.

Now we can begin to examine the capacity of these networks. In particular, we can ask questions like the following. How many input units (n/) and output units (no) would be needed to allow retrieval of

the correct mate of each of r different input patterns?

The answer to such a question depends on the criterion of correct retrieval used. For present purposes, we can adopt the following criterion: All of the correct output units should be turned on, and, on the average, no more than one output unit should be turned on spuriously. Copyrighted Material

We want to keep this number less than 1. Adopting a slightly more

stringent criterion to simplify the calculations, we can set

or

I

I n

lo I mj � Pon

Rearranging, we get

For small positive x, logO-x ) = -x. If we restrict ourselves to cases where mj mol nj no < .1-that is, reasonably sparse patterns in the sense that m < nl.JTO-the approximation will hold for the right-hand side of

the equation, so that taking logs we get

We can solve this for r, the number of patterns, to obtain

(1)

Now, -log 1- [ <sup>n</sup>

� I �, I ranges upward from .69 for very sparse patterns

where mj = 10g2no' Using .69 as a lower bound, we are safe if we say:

or

This result tells us that the number of storage elements (that is, connections, nj na) that we need is proportional to the number of associations we wish to store times the number of connections (mj ma) activated in storing each association. This seems about right, intuitively. In fact, this is an upper bound rather greater than the true number of storage elements required for less sparse patterns, as can be seen by plugging values of mj greater than log2na into Equation 1.

It is interesting to compare Wilishaw nets to various kinds of local representation. One very simple local representation would associate a single, active input unit with one or more active output units. Obviously, such a network would have a capacity of only nj patterns. We can use the connections of a Willshaw net more effectively with a distributed input if the input and output patterns are reasonably sparse. For instance, in a square net with the same number n of input and output units and the same number m of active elements in each, if n = 1000 and m = 10, we find that we can store about 7,000 associations instead of the 1,000 we could store using local representation over

the input units.

Another scheme to compare to the Willshaw scheme would be one that encodes each pattern to be learned with a single hidden unit between the input and output layers. Obviously a net that behaved perfectly in performing r associations between mj active input elements and ma active output units could be handcrafted using r hidden units, each having m, input connections and ma output connections. Such a network can be economical once it is wired up exactly right: It only needs r (m;+mo) connections. However, there are two points to note. First, it is not obvious how to provide enough hardware in advance to handle an arbitrary r patterns of m, active input units and mo active output units. The number of such patterns possible is approximately (n, mil m, !)(no mol mo <sup>1</sup>) , and if we had to provide a unit in a,dvance for each of these our hardware cost would get out of hand very fast. Second, the economy of the scheme is not due to the use of local representation, but to the use of hidden units. In many cases even more economical representation can be achieved with coarse-coded hid-

den units (see Chapter 3 and Kanerva, 1984).

Randomly Connected Nets

Returning to the standard Wilishaw net, there are several minor difficulties with Willshaw's scheme. First, it assumes that each input unit sends one and only one connection to each output unit. In a neural network, we might aSSCDP¥r\_diMHI�rMllit sends out a randomly

distributed array of connections to the set of output units without any guarantee that each output unit actually receives a connection. Second, the analysis depends on a rather strict and sharp threshold for output unit activation. In a random net rather than a fully connected net, we could not actually guarantee that a given output unit would in fact receive mj inputs� and in realistic nets, we would expect there to be some inherent variability in the activations of the units. Thus, we would not be able to guarantee that all correct units would exceed the

sharp threshold, nor that all incorrect units would fall below it.

However, it turns out that we can reformulate the problem just slightly and get a handle on networks that have these properties. Assume that we have a square network of n input and n output units and that we wish to store associations between m active input units and m active output units. Suppose each input unit has f output connections which fall where they may among the n output units so that the output units have an average of f inputs each. Note again that the connections are randomly distributed without restriction so that there is

no guarantee that input unit i projects to output unit j.

To study the performance of this net, imagine storing some number , of patterns using the Willshaw learning scheme. During testing, we will examine the number of active inputs each output unit that should be turned on will receive and the number of active inputs each unit that should not be turned on will receive, and we will then calculate the signal-detection measure of sensitivity d' (Green & Swets, 1966) as an index of the ability of inputs reaching each output unit to distinguish between units that should be on and units that should not be on. Since d' is independent of the threshold, this measure allows us to bypass the

question of the threshold itself.

Let us first consider what happens in our random network as we train it with pairs of patterns using Willshaw's scheme. Pick an arbitrary connection in our net between an arbitrary input unit and an arbitrary output unit. Now, consider learning an arbitrary pattern. The probabi lity that a particular input unit will be on is m/ n. Similarly, the probability that a particular output unit will be on is m/ n. The probability that the units joined by the particular connection we are considering will be one of the ones turned on in learning a particular pattern, then, is m2/ n2 just as before. The rest of the earlier analysis still applies, and

we get

This is exactly the same value that we had before in the original Willshaw model, and it is independent of f, the number of connections Copyrighted Material

each unit makes. This factor will become important soon, but it does not affect the probability that a particular connection will be on after

learning r patterns.

Now consider what happens during the testing of a particular learned association. We activate the correct m input units and examine the mean number of quanta of activation that each output unit that should be on wi ll receive. The m active input units each have f outputs, so there are mf total "active" connections. A particular one of these connections reaches a particular output unit with probability lin, since each connection is assumed to fall at random among the n output units. Thus, the average number of active connections each output unit receives will simply be mf In. For output units that should be on, each of these connections will have been turned on during learning, so mfln is the average number of quanta that unit will receive. Assuming that n is reasonably large, the distribution of this quantity is

approximately Poisson, so its variance is also given by mfl n.

Units that should not be on also receive an arbitrary connection from an active input unit with probability lin, but each such connection is only on with probability Pon' Thus , the average number of quanta such units receive is (mf In )Pon' This quantity is also approximately Pois-

son, so its variance is also equal to its mean.

Our measure of sensitivity, d', is the difference between these means divided by the square root of the average of the variances. That is,

d' =mf I n (I - Pon )

J(mfln)(l + Pon)/2

Simplifying, this becomes

, ..Jm1Tii 1 - Pon <sup>d</sup>= mfln JO + Pon )/ <sup>2</sup>

We can get bounds on the true value of d' by noting that the denominator above cannot be greater than 1 or less than .Jiii . The largest value of the denominator sets a lower bound on d', so we find that

d' � .Jmf/n (t - Pon)'

Substituting for 1 - Pon, we obtain

d' � .J mf / <sup>n</sup>[ 1 - :: r. (3)

the units that should be on and false alarm to less than 1 % of the units

that should be off. 1

How big would a net have to be to meet these specifications? Assuming a fully connected net, and consulting Equation 4, we find that we need to set n equal to a value near about 106 to get r large

enough.

This number of units is not a serious problem since estimates of the number of units in the brain generally range upward from 1010 (see Chapter 20). However, individual units are not general ly assumed to have enough connections for this scheme to work as stated. If there are 1,000 to 10,000 connections per unit, as suggested in Chapter 20, we are off by two to three orders of magnitude in the number of con-

nections per unit.

Given this limitation on fan-out, we had better consult Figure 2. The figure indicates that the maximum capacity of a net with a fan-out of 1,000 and a d' of 5 is only about 150 patterns. With / - 10,000 we get up to a capacity of about 15,000 patterns, but we are still well short of the mark. It seems, then, that the fan-out of neurons drastically

limits the capacity of a distributed network.

A simple method for overcoming the fan-out limitation. But all is not completely lost. It turns out that it is a relatively simple matter to overcome the fan-out limitation. The trick is simply to use multiple layers of units. Let each input unit activate a set of what we might call dispersion units, and let each output unit receive input from a set of collection units. Let the / outgoing connections of each of the dispersion units be randomly distributed among the "dendrites" of the collection units. A miniature version of this scheme is illustrated in Figure 3. Note that it is assumed that each dispersion unit is driven by a sin gle input unit, and each collection unit projects to a Single output unit. Collection units are assumed to be perfectly linear so that the net input to each output unit is just the sum of the net inputs to the collector units that project to it. Assuming each input unit and each dispersion unit has a fan-out of /, the effective fan-out of the input and disper-

sion layers together becomes /2

. Simi larly, the set of collection units

feeding into each output unit col lect an average of /2 connections. To construct an associator of 1 million input units by 1 million output units assuming each unit has a fanout of 1,000, we will need 1 billion dispersion units and 1 billion collection units. The number of connections between the dispersion units and the collection units would be on

the order of 1012, or 1 trillion connections.

I It should be pointed out that any intrinsic noise in the units would reduce the actual

FIGURE 3. A diagram of a multilayer network consisting of an input layer, a dispersion layer, a collection layer, and an output layer. The network serves to square the effective

fan-out of each input unit, relative to the simple two-layer case.

The network would require about 20% of human cortex, based on the estimate that there are lOs neurons under each square millimeter of the brain and that there are about lOS square millimeters of cortical surface. This might be a little tight, but if the fan-out were 10,000, the network would fit handily. In that case, it would only require about 2

percent of the 1010 units.

There are, of course, a lot of reasons to doubt that these figures represent anything more than a first-order estimate of the capacity of real associative networks. There are several oversimplifications, including for example the assumption that the dispersion units are each driven by a single connection. We must also note that we have assumed a two-layer net along with an extremely simple learning rule. The intermediate layers postulated here merely serve to provide a way of overcoming the fan-out limits of individual units. However, as was pointed out in Chapters 7 and 8, a multilayer net can often learn to construct its own coding schemes that are much more efficient than the random coding schemes used here. Even simple two-layer nets can

profit if there are somedcf���}bed M1<sup>t</sup>

�?ra

fetwork and if they use a

sensible learning rule, as shown in Chapter 18. Thus random nets like the ones that have been analyzed in this section probably represent a lower limit on efficiency that we can use as a benchmark against which

to measure "smarter" PDP mechanisms.

Effects of Degradation and the Benefits of Redundancy

One virtue of distributed models is their ability to handle degradation, either of the input pattern or of the network itself. The d' analysis allows us to tell a very simple story about the effects of degradation. In this section I will just consider the effects of degradation by removal, either of a random fraction of the pattern or of a random fraction of the connections in the network; effects of added noise will be considered later on. In the case of removal, we can think of it either in terms of presenting an incomplete pattern or actually destroying some of the input units so that parts of the pattern are simply no longer represented. Consider the case of a network that has already been trained with some number of patterns so that Pan can be treated as a constant. Then we can write the equation relating d' to m, f, and n as

d' � k.Jmf!n.

Now, suppose that during testing we turn on only some proportion PI of the m units representing a pattern. The m in the above equation becomes mpI, so we see that the sensitivity of the network as indexed by d' falls off as the square root of the fraction of the probe that is presented. Similarly. suppose some of the connections leading out of each unit are destroyed. leaving a random intact proportion Pi of the mf active connections. Again. the sensitivity of the network will be proportional to the square root of the number of remaining connections. Thus, performance degrades gracefully under both kinds of

damage.

Another frequently noted virtue of distributed memories is the redundancy they tend naturally to provide. The ability of simple distributed memories to cope with degraded input patterns is really just a matter of their redundancy, as Willshaw (1981) pointed out. For. if a network is fully loaded, in the sense that it can hold no more associations and still meet some predetermined standard of accuracy with complete patterns, it will not be able to meet that same criterion with degradation. The only way to guard against this problem is to load the network lightly enough so that the criterion can still be met after subjecting the network or the inputs to the specified degree of degradation. Copyrighted Material

PROGRAMMABLE PATTERN ASSOCIATORS

In this section, I extend the sort of analysis we have performed on simple associator models to the resource requirements of connection information distribution (CID) networks of the type discussed in

Chapter 16.

The mechanism shown in Figure 4 is a distributed CID mechanism. The purpose of this network is to allow connection information stored in a central associative network to be used to set connections in several local or programmable networks in the course of processing so that more

Central Input Units

Central Output Units�

'-

Local Input Units

CA Units

o

FIGURE 4. A connection information distribution (CID) network consisting of two local, programmable networks; a central, standard network; and a set of connection activation (CA) units. Each local input unit projects to the corresponding central input <sup>u</sup>nit, and each CA unit projects to the corresponding connection in both local networks. Central output units turn on CA units releva<sup>n</sup> <sup>t</sup>to processing the patterns they program the local modules to process. �MlWe'ri8# are a few examples of each type.

than one input pattern can be processed at one time. The mechanism works as follows: One or more patterns to be processed are presented as inputs, with each pattern going to the input units in a different programmable network. The input pattern to each local net is also transmitted to the input units of the central associative network. When more than one pattern is presented at a time, the input to the central network is just the pattern that results from superimposing all of the input patterns. This pattern, via the connections in the central associative network, causes a pattern of activation over the central output units. The central output pattern, of course, is a composite representation of all of the input patterns. It is not itself the desired output of the system, but is the pattern that serves as the basis for programming (or turning on connections) in the local, programmable networks. The local networks are programmed via a set of units cal led the connection activation (CA) units. The CA units act essentially as switches that turn on connections in the programmable networks. In the version of the model we will start with, each CA unit projects to the one specific connection it corresponds to in each programmable network, so there are as many CA units as there are connections in a single programmable net. In the figure, the CA units are laid out so that the location of each one corresponds to the location of the connection it commands in

each of the programmable networks.

To program the local networks, then, central output units activate the CA units corresponding to the connections needed to process the patterns represented on the central output units. The CA units turn on the corresponding connections. This does not mean that the CA units actually cause activation to pass to the local output units. Rather, they simply enable connections in the programmable nets. Each active local input unit sends a quantum of activation to a given local output unit if

the connection between them is turned on.

The question we will be concerned with first is the number of CA units required to make the mechanism work properly. In a later section, we wilI consider the effect of processing multiple items simultane-

ously on the resource requirements of the central network.

Connection Activation Unit Requirements

Consider a CID mechanism containing programmable networks of n; by no units in which we wish to be able to associate each of s different output patterns with each of s different input patterns arising at the same time in different local networks. Input and output patterns consist of rn; by rno active units, respectively. Fol lowing the assumptions Copyrighted Material

for Willshaw nets, we assume binary units and connections, and we assume that output units are turned on only if they receive m quanta of

activation.

Now, let us consider how many CA units are needed to implement this mechanism. For now we bypass the bottom-up activation of CA units and assume instead that we know in advance which connections need to be turned on. If each local network must be as complex as a standard network capable of processing r different patterns, we are in serious trouble. In the previous analysis of Willshaw networks, we found that the number of connections we needed to process r associa-

tions of m by m active units was

It looks as though the number of connections required in each local network grows linearly with the number of known patterns times the content of each. If we had one CA unit for each programmable connection, a programmable version of our square I-million-pattern associator would require <sup>1012</sup>CA units, a figure which is one or two orders of magnitude larger than conventional estimates of the number of units in the brain. Just putting the matter in terms of the cost we must bear to use programmable connections, it appears that we need n 2 CA units just to specify the connections needed for a standard net that could do the same work with just the connections between n input and n output

units. 2

However, things are not nearly as bad as this argument suggests. The computation I just gave misses the very important fact that it is generally not necessary to pinpoint only those connections that are relevant to a particular association. We can do very well if we allow each CA unit to activate a whole cohort of connections, as long as (a) we activate all the connections that we need to process any particular pattern of interest, and (b) we do not activate so many that we give rise

to an inordinate number of spurious activations of output units.

The idea of using one CA unit for a whole cohort of programmable connections is a kind of coarse coding. In this case, we will see that we can reap a considerable benefit from coarse coding, compared to using one CA unit per connection. A simple illustration of the idea is shown in Figure 5. The figure illustrates CA units projecting to a single one

2 Many readers will observe that the CA un<sup>i</sup>ts are not st rictly necessary. However, the specificity of their connections to connections in local networks is an issue whether CA units are used as intermediaries or not. Thus, even if the CA un <sup>i</sup>ts were eliminated, it would not change the relevance of the following re<sup>s</sup>u<sup>l</sup>ts. In a later section, the CA units and central output units will be collapsed into one set of units; in that case, this analysis

will apply directly to the numI>edJfYllf}hlWl¥'lf.Mt�pe required.

o o o o

o

o

FIGURE 5. A programmable network with 8 input units and 8 output units and 64 programmable connections. Each of the 16 connection activation units is assumed to project to a random set of 4 pr<sup>o</sup>grammable connections. These connections are only drawn in for two of the CA units. The sets of connections are chosen without replacement so that each connection is programmed by one and only one CA unit. Whenever a CA unit is

on it turns on all of the connections it projects to.

of two programmable networks. Note that a given CA unit must activate the same connections in each programmable net when there is

more than one.

One Pattern at a Time

To see how much this scheme can buy us, I will start by considering the case in which we want to program some local nets to process a single pattern. We ask, how small a number nCQ of CA units can we ge� Copyrighted Material

by with, assuming that each one activates a distinct, randomly selected

set of ni nol nco connections?

First of all, the number of CA units that must be activated may have to be as large as mi mo, in case each of the different connections required to process the pattern is a member of a distinct cohort. Second, for comparability to our analysis of the standard network, we want the total fraction of connections turned on to allow no more than an average of 1 output unit to be spuriously activated. As before, this

constraint is represented by

[ 1 I �i Pon �;; •

As long as mi � log2n;, .5 will be less than the right-hand side of the expression, so we will be safe if we keep Pon less than or equal to .5. Since we may have to activate mimo CA units to activate all the right connections and since we do not want to activate more than half of the

connections in all, we conclude that

From this result we discover that the number of CA units required does not depend at all on the number of connections in each programmable network. Nor in fact does it depend on the number of different known patterns. The number of known patterns does of course influence the complexity of the central network, but it does not affect the number of CA units. The number of CA units depends on mimo, the number of connections that need to be turned on per pattern. Obviously, this places a premium on the sparseness of the patterns. Regard-

less of this , we are much better off than before.

Several Patterns at a Time

So far we have considered the case in which only one item is presented for processing at a time. However, the whole point of the connection information distribution scheme is that it permits simultaneous processing of several different patterns. There is, however, a cost associated with simultaneous processing, since for each pattern we need to turn on all the connections needed to process it. In this situation, we will need to increase the total number of CA units to increase the specificity of the set of connections each association requires if we are to keep the total fractiWSP9f;g1W�a�hat have been turned on

below .5. Formally, assume that we know which s patterns we want to process. Each one will need to turn on its own set -of mimo CA units out of the total number nca of CA units. The proportion of connec-

tions turned on will then be

<sup>I</sup>m,<sup>m</sup>o ]S p, on =1- 1- --nca

This formula is, of course, the same as the one we saw before for the number of connections activated in the standard net with s, the number of different patterns to be processed simultaneously, replacing r, the number of patterns stored in the memory, and with nca, the number of connection activation units, replacing nino, the total number of connections. Using Pon = .5 and taking the log of both sides we get

-.69 = slog 1 <sup>1</sup>\_ m�:o ].

Invoking the 10g (1- x) = -x approximation, we obtain

nca � 1.45sm 2.

This formula underestimates nca slightly for s < 3. With this caveat, the number of CA units required is roughly proportional to the number of patterns to be processed at one time, times the number of connec-

tions needed to process each pattern.

Overlapping the Programmable Networks

In Chapter 16, the CID scheme we have been considering thus far was generalized to the case where the programmable networks overlapped with each other. This allowed strings of letters starting in any of a large number of input locations to correctly activate units for the corresponding word at the appropriate location at the next higher level. Here I will consider a more general overlapping scheme using distributed representations in the overlapping local networks. A set of three overlapping local networks is illustrated in Figure 6. In this scheme, both the input and the output units can play different roles depending on the alignment of the input pattern with the input units. In consequence, some of the connections also play more than one role. These connections are assumed to be programmable by a number of different CA units, one for each of the connection's different roles. Obviously, this will tend to increase the probability that a connection will be turned Copyrighted Material

[J .J J Y .J Y .J .J

� y y y y y .J Y ()

LY y y y y .J .J Y lY � pr P' y � � �

y 15 � � � � .J � LY y ,;J .J Y 15 ;Y L.Y J ,;J y ,Y .J J J .J Y � Y lY y y y ;J y y y y 5 Y Y Y Y Y J Y Y J J Y

<sup>Y</sup>;J ;J ;J � y y y J Y Y Y {)

Y J J Y Y J Y Y Y Y Y J

,Y rY )' ,Y lY lY L.Y � ,;J ,;J � ,;J {)

y L.Y Y ;J LY [Y Y Y J Y LY Y

Y Y LY y lY lY y lY () Y Y -0 LY y lY lY y l5

y Y l5 J � ;J l5 � <sup>y</sup>15 Y y y ;J y y

0 0000000 00 0

FIGURE 6. Three overlapping programmable networks of 8 x 8 units each. The networks overlap every four units, so the input and output units can participate in two dif-

ferent, partially overlapping networks.

on, and therefore will require a further revision of our estimate of the

number of CA units required.

Unfortunately, an exact mathematical analysis is a bit tricky due to the fact that different junctions have different numbers of opportunities to be turned on. In addition, input patterns in adjacent locations will tend to cross-activate each other's output units. If the patterns to be processed are wel l separated, this will not be a problem. Restricting our attention to the well-separated case, we can get an upper bound on the cost in CA units of allowing overlapping modules by considering the case where all of the connections are assumed to play the maximum number of roles. This number is equivalent to the step size or grain,

example, for four-letter words, if the increments in starting places of successive overlapping networks were one letter wide, g would be 4. Assuming that the connections turned on for each slice of a pattern are independent of those turned on by each other slice, it is easy to show

that the formula for Pan becomes

Pan � 1 - 1 \_\_ , \_0 , [ m· <sup>m</sup><sup>I</sup>

Sg

nco

and the number of CA units required to keep Po" less than .5 is approx-

imated by

The cost goes up with the number of patterns to be processed simul-

taneously times the grain of the overlap.

Summary of CA Unit Requirements

In summary , the number of CA units required to program a programmable network depends on different variables than the number of connections required in a standard associator. We can unify the two analyses by noting that both depend on the number of patterns the net must be ready to process at any given time. For the standard associator, the number is r, the number of known patterns; for the programmable net, the number is sg , the number of patterns the net is programmed for times the grain of overlap allowed in the starting locations

of input patterns.

This analysis greatly increases the plausibility of the CID scheme. For we find that the "initial investment" in CA units needed to program a set of networks to process a single association is related to the content of the association or the number of connections required to allow each of the active input elements to send a quantum of activation to each of the active output elements. Incorporating a provision for overlapping networks, we find that the investment required for processing one association is related to the content of the association times the grain of the overlap. This cost is far more reasonable than it looked like it might be at first, and, most importantly, it does not depend on

the number of patterns known.

An additional important result is that the cost of programming a set of networks grows with the number of patterns we wish to program for at one time. This cost seems commensurate with the linear speedup we

would get by being abtc

o b�d<sup>h</sup>

��CJM

�lJ

��,1 patterns simultaneously.

The somewhat intangible benefit to be derived from mutual constraint among the patterns would come over and above the simple linear throughput effect. However, this benefit, as we shall see in the next section, is balanced by the extra cost associated with the possibility that there might be spurious patterns in the intersection of input elements

of the presented patterns.

The Cost of Simultaneous Access

So far, we have proceeded as though we already knew what patterns to prepare each local module for. However, the CID mechanism was intended to allow several inputs to access the central network simultaneously and thereby program the local networks in the course of processing. This simultaneous access costs something; in this section we consider how much. The discussion here is relevant to the general issue of the C:lsts of simultaneous access to a PDP network, as well as to the

specific question of the capacity requirements of CID.

For simplicity I will begin by considering local representations at the central output level. That is, I will assume that each central output unit represents a different pattern and that it is switched on only when all of

the central input units corresponding to its pattern are active.

Now, recall that a central input unit is switched on if the corresponding unit is active in any of the programmable nets. Thus, what the central output units actually see is the pattern of activation that results from the superimposition of the input patterns presented for simultaneous processing. The effect of this is that there is some possibility that ghosts of patterns not actually presented wil l show up in the result. This is just the kind of situation that is described in Chapter 16 when similar words such as SAND and LANE are presented to each of two programmable networks for simultaneous processing. When the activation patterns of the two words are superimposed, the central word units for LAND and SANE get turned on just as strongly as the central word units for SAND and LANE. Thus, the programmable networks end up being programmed to process any one of these four words, rather than

just any one of the two actually presented.

Is there anything that can be done to control the number of different patterns that show up when several patterns are superimposed? In fact, there/is. If we increase the number of input units in each programmable network or if we reduce the number of input units active in each pattern, we will reduce the possibility of spurious patterns showing up

in the superposition.

To get a quantitative grip on this matter, assume that the input patterns are random selections of m out of the n input units as we have been assuming throughout. The probability that a spurious pattern is present in the superposition of s patterns can now be easily calculated. First, we calculate the probability that a randomly selected unit will be

on; this is just

The probability that a particular spurious pattern is fully represented in the set of units activated by the s patterns is just this number to the power m, and the average number of such patterns out of r known patterns is just this probability times r - s. Thus, the average number of

spurious patterns present in the superposition is

Assuming r » s, we can simplify by replacing r - s with r. If we take acceptable performance to be an average of one or fewer spurious patterns present and therefore of spurious CP units active, we get

Rearranging and taking logs ,

log [ 1- (! )! 1= slog(1 - mi n).

Several things are apparent from this equation. First, the number of patterns that can be processed at one time increases with the number of input units. The effect is approximately linear as long as m/ n � .1. Second, though i t i s not quite as straightforward, s tends to increase with a decrease in m. For example, suppose n = 5,000 and r = 10,000. In this case, when m drops from 1,000 to 500, s increases from 21 to about 37; if m drops to 100, s goes up to about 120. Third, for a fixed m and n, especially for large m, we can make very large changes in r with only minimal impact on s. Thus, if we have, say,

n = 10,000 and m = 1,000 with r = 10<sup>6</sup>

, we get s = 43; if we reduce r

to lOS, we only get an increase of 2 in s, to 45.

If we allow overlapping local networks , and we assume that the patterns are random with independent subparts, we need only replace s in the preceeding equation with sg . While this is a fairly steep cost, it is still the case that reasonably moderate values of n (about 2. <sup>5</sup><sup>x</sup>lOS

)

would be sufficient to process 10 out of 106 known patterns of size

Simultaneous Access to Distributed Representations

The results just described, it must be remembered, depend on the use of local representations at the central output level . What happens if we consider simultaneously accessing distributed representations instead? Obviously this question remains relevant to general questions about simultaneous access, as well as to the situation that would arise using distributed central output units in CID. Furthermore, we should note that the central output units in Figure 4 simply mediate a mapping from one distributed representation-on the central input units-to another-on the CA units. The present analysis describes what would happen if we simply collapsed these two sets of units into one, activat-

ing the connections directly from the central output units.

We consider a case exactly like the one we were just considering, except that now the output representation is not a single unit per pattern, but mo active units on out of no central output units. We consider two somewhat separate questions. First, if we superimpose

several input patterns, what effect does this have on d '

at the central

output level , relative to the case where only a single pattern is shown ? Second, what is the probability that ghosts of whole patterns not

presented will show up in the output of the central network ?

To begin our analysis of the first question, recall from Equation 2 the

expression for d' in random nets with full fan-out (n = j):

d'= .rm; <sup>1</sup> - Pon

I .J ( 1 + Pon )/ 2 .

We first ask, what is the effect on d' of turning on spurious input units with probabi lity p , in addition to the m units representing a particular pattern to be processed? The number M; of input units that wi ll then

be on is

M; == mj + (nj - mj )p.

Consider first, output units that should not be on. These will receive M; active inputs, and each of these connections will be on with probability Pon . The output units that should be on will receive mj inputs on the input lines whose connections were turned on in learning the presented pattern plus (nj - mj )p inputs to connections that will have been turned on in learning other patterns with probabi lity Pon ' The numerator for our revised expression wi ll then simply reduce to its old value, with the (n; - mj )p term canceling out. However, there will be

an increase in variance, and hence a decrease in d '

. The denominator

means, which are, also as before, equal to the means. The expression

for d' therefore becomes

d' = m

( 1 - Pan ) J (m; + m;pon + 2(n; - m;)p 1/ 2

We get a simpler expression if we approximate by replacing Pan in the denominator with its maximum value of 1; this gives us a slight overestimate of the variance and therefore a slight underestimate of d':

> d' � ( <sup>1</sup>- Pan ) ? <sup>m</sup>.j m; +( n; - m; )p .

The variance goes up with the mean number of spuriously activated units, and d' goes down with the effect of this on the square root of the

variance.

To determine the effect of presenting several patterns on d', we note that from the point of view of the units that belong to one of the patterns, all the units activated by the other patterns are spurious. The

number of such units is

1 - ( 1 - m;/ ny - I

Inserting this for p in the previous equation gives

d' = m

( 1 - Pan ) .Jm; + (n; - m; )[I - ( 1- m;/ ny- I]

Using this equation we can examine the effects of increasing s on the value of d'. Not too surprisingly, d' does go down as s goes up, but

the effect is relatively benign. For example, with n = 106

, r = 106

,

m = 1,000, and s = 1, d' is about 11.6. It drops to half that value at <sup>s</sup>= 4, and drops much more gradually thereafter. With n = 2 x 106 units and the same values of r and m, we can get an acceptable value

of d' (� 6) with s as high as 16.

The final issue we will consider is the possibility that new spurious output patterns have been introduced in the superposition of the s output patterns simultaneously activated in processing the s mput patterns. For simplicity, we will just consider the probability of a "ghost," given that all and only the correct ma units are active for each of the s patterns. The analysis is entirely the same as the one we gave before for the probabi lity of ghosts showing up in the input patterns. We get an

average of one ghost when

<sup>1</sup>� r [ 1 - (1 - m/ n ) S 1m •

As before, the number of simultaneous patterns we can tolerate increases with n and decreases with m and is relatively insensitive to

the value of r.

In general , it appears that the probabi lity of ghosts occurring can be kept small with sufficiently large dedication of resources, but these trade off approximately linearly with s. With fixed n, we must simply

make the patterns sparser or tolerate some loss of sensitivity.

Discussion

This analysis of the resource requirements of networks like the eIn model has discovered a number of basic results. In essence, the picture is really a very simple one. The resource requirements of eIn depend on the number of programmable networks one wants to program for at once. The number of connections needed in each local network depends on the number of patterns to be programmed for and is independent of r. the number of known patterns. In the central network, the number of units required to keep spurious activations under control grows with s, as does the number of units required to keep ghosts from emerging in the input and output patterns. It is worth noting, also, that the probabi lity of ghosts increases as we increase m. The fact that the resource requirements of the local networks are independent of the number of patterns known is obviously important. Relative to the central network, it means that the local networks are very cheap. The number of distinct inputs that are needed to program them is quite reasonable, and, as I will explain, we can even get by with far fewer units in the local networks than we need at the central level . On the other hand, the results concerning the costs of simultaneous access to the central network are much less encouraging for the eIn scheme. Using local or distributed representations in the central module, the unit requirements grow with the product of s and r-a very expensive proposition since the number of central connections will

then grow as sr2.

However, there are several important further observations. One is that, at fixed numbers of units and patterns known, the degradation of sensitivity as a function of s is rather gradual . And, given a lightly loaded network, one can take s up to reasonable values without catastrophe. Simultaneous access by multiple patterns is very much like degradation: a network can handle it without a noticeable decrement of function if it is lightly loaded. A second observation concerns the limits of coarse coding. Fot[�l1teYw!5le9jaF of m essentially amounts

to a question of how coarse the code is: Large m corresponds to very coarse coding, and small m corresponds to very fine coding. As we saw in Chapter 3, the ability to represent several patterns at a time goes down as the coding gets coarser. For simultaneous processing we need sparse patterns, with each unit serving as a rather sharply tuned con-

junctive detector.

The final observation is that large costs are associated with simultaneous access to the central network. This fact has lead me to the view that it is probably most reasonable to imagine that we must probably restrict simultaneous access, except perhaps in the case of small, compact and well-differentiated subpatterns like letters. I incorporated this idea of restricted access in the programmable blackboard model of reading by assuming that we program successive parts of the programmable blackboard sequentially, using only the contents of the spotlight of attention to access the central network; but that the local networks so programmed continue to process and hold patterns of activation and to allow those patterns to interact with one and other after the spotlight of attention has moved on. In this way we get the best of both worlds: sequential access to central knowledge, combined with interactive parallel processing of several stimuli in the programmable nets. Another point is that it may be a good idea to dissociate the inputs to the local networks and the inputs to the central networks . Throughout this chapter and Chapter 16, I have assumed that the units in each local network would be isomorphic to units in the central network. However, there is no reason for them to be. The central network needs much higher "resolution " (n proportional to r) than the local networks (n proportional to s ) . Thus, the units in the programmable modules need only provide a few primitive clues to which of the s patterns are to be represented in their outputs , while the units in the central network

would require a much higher-resolution representation.

CONCLUSION

This chapter has indicated how Willshaw's fruitful analysis of simple pattern associator models can be extended in several directions. These extensions have lead to several interesting observations , particularly into the effects of limited connectivity (Mitchison, personal communication, 1984) and into the capacity requirements of programmable networks. A large number of issues remain to be explored. I hope that this discussion and elaboration of Willshaw's analysis will aid in this

ACKNOWLEDGMENTS

This work was supported by Contract N-0001 4-82-C-0374 , NR 667- 483 with the Personnel and Training Research Programs of the Office of Naval Research, by a grant from the System Development Foundation to the Institute for Cognitive Science at UCSD, and by an NIMH Research Scientist Career Development Award (MH-00385) . This chapter was developed in response to a number of questions raised by Geoff Hinton and Scott Fahlman about the resource requirements of programmable nets. I thank Dave Rumelhart for several useful discussions and for encouraging me to pursue the issues descri bed herein. The material descri bed in the section entitled "Randomly Connected Nets" was developed in col laboration with Dave, and the application of the d' analysis to the problem of simultaneous access to a distributed

memory network was Dave's suggestion.

CHAPTER 13

P3: A Parallel Network Simulating System

D. ZIPSER and D. RABIN

Research on parallel distributed processing is to a large extent dependent upon the use of computer simulation, and a good deal of the researcher's time is spent writing programs for this purpose. Virtually all the PDP systems described in this book require special-purpose computer programs to emulate the networks under study, In writing programs of this type, it is usually found that the basic algorithms of the PDP network are easy to program but that these rather simple "core" programs are of little value unless they are embedded in a system that lets the researcher observe and interact with their functions. These user interface programs are generally tedious and very time consuming to write. What is more, when they are directed toward one particular system they can be quite inflexible, making it difficult to easily modify the PDP network being studied. Also, because of the time involved, particularly for interactive graphics programs, the researcher often makes do with very limited facilities for analyzing the performance of the network. In this chapter we will describe a general-purpose parallel system simulator called P3. It was developed with PDP research explicitly in mind and its major goal is to facilitate simulation by providing both the tools for network description and a powerful user interface that can be used with any network described using the tools. There are many problems to be faced and tradeoffs to be made in designing such a system but in the process of doing this we feel that not only has a useful system been developed, but also that we have learned a great

In P3 networks, each computing element, called a unit, contains a computer program that reads inputs from connections and sets outputs on other connections, possibly also modifying some local state parame-

ters. The major components of P3 are:

• The plan language, which describes the collection of units in a model and specifies the connections between them. This

description is called a "plan."

• The method language, an extension to LISP, which implements the internal computational behaviors of the units in a model.

• The constructor, which transforms the plan and associated methods into a computer program and, when run, simulates the

network.

• The simulation environment, which provides an interactive display-oriented facility for observing and testing P3 models.

Input to units described in a P3 plan can come only from other units in the plan. That is, there is no "outside world" in a P3 plan language description of a network. This means that at the level of description of the P3 plan language, the P3 network is closed. Access to the outside world must occur inside a unit through its method. Methods may access the world outside the P3 system through any available computer peripheral. The only thing that methods are not allowed to do is to reconfigure with the P3 system itself or communicate with other methods through "underground connections" not mentioned in the P3

plan.

In any simulation, the relationship between real and modeled time is of key importance. A real unit, such as a neuron, would read inputs continuously and update its outputs asynchronously, but this cannot be simulated exactly on a digital computer. Many simulations use a simple synchronous approximation to real time. However, sometimes this produces unwanted artifacts and a closer approximation of asynchrony is required. Often, in fact, what the investigator really wants to do is to experiment with the effect of different kinds of time simulation on the network under study. Since there is no way for the system designer to know in advance all the possible ways that the investigator will want to handle time, some strategy has to be used that allows great flexibility. The approach taken by P3 is that this flexibility can come through the methods that can use conditional updating. The P3 system itself is completely synchronous and updates all units on each cycle. Since updating a unit involves invoking its method, the question of whether or not the outputs of ��I"5R!mDe on any P3 cycle can be

decided by the method. For example, to model asynchronous updating, each unit can have an additional input that controls whether or not it is updated on a cycle. Then the decision as to which units are to be updated can be given to a control unit that is connected by a separate line to the update inputs of all the other units. The method program inside this control unit decides which units in the network will be updated on each cycle. Note that this approach is very flexible since small changes in the method program of the control unit can imple-

ment a large range of possible update time schemes.

A typical P3 plan might contain a large number of simple neuron-like units forming the core of the network together with a few special purpose units to generate input to the core network and control its function. The master control unit, used above to implement asynchronous updating, is an example of this kind of special-purpose unit. They can also be used to sequence simulated experiments and to interpret output of other units. How all this can be done will become clearer as we describe the use of P3 in detail. They key point here is that the P3 "style" is to include within the P3 plan all aspects of the simulation including input to and control of the core network. This approach simplifies the problem of constantly having to interfere special-purpose

routines to a general-purpose modeling environment.

It often happens that n�tworks are modular, that is, made up of distinct subnetworks. P3 facilitates the use of modularity by allowing subnetworks to be treated as single processing units. This feature is of particular use when several P3 units are used to simulate a single object such as a "realistic" neuron. The modular feature also facilitates "topdown" and "structured" definition of the plan even when the underly-

ing networks are not particularly modular.

The P3 plan language has an additional feature that is not directly concerned with describing the functional aspects of a parallel network. Every unit in a P3 plan has a location in a three-dimensional Euclidean reference frame call P3 space. This means that every P3 plan not only describes a network, but it also describes a geometrical structure. Since the functioning of a P3 network does not depend on its geometrical structure, it might seem odd to go to all the trouble of describing the geometry. There are two main reasons for locating P3 units in space. The first reason is to facilitate visualizing a P3 network while observing its function during the simulation of a model. The units can be placed so that they appear at the same relative positions on the computer display during simulation as they have in the investigator's conceptual image of the model. The second reason to give each unit a position in space is to make it possible to specify connections between units implicitly on the basis of their spatial locations rather than explicitly. This latter feature is of particular importance when modeling systems in Copyrighted Material

which the connectivity is described in terms of geometrical relations. This is often the case when dealing with realistic neuronal modeling,

especially of primary sensory processing s tructu res.

The P3 Plan Language

The job of the P3 p lan language is to describe the units and connections that constitute the network being simulated. To do thi s , the language uses a small but rich set of s ta tements that make it possible to succinctly describe large groups of complex, connected units. The three fundamental constituents of the plan language are the UNIT TYPE, UNI T, and CONNECT s ta temen ts . The UNIT TYPE s ta teme nt name s and describes a kind of unit. The UNIT statement instantiates and names actual units. This statement can instantiate either a single unit or a whole array of units of the same type. The CO NNECT st ate ment makes connections. Since the statement can be used inside of loops , a single connect statement can make an arbitrarily large number

of connection s using the available array features.

A unit in P3 can have any number of inputs and outputs together with any number o f parameters. Before the start of a si mula tion , values must be given to all parameters and to a ll outputs. Each value is always a single computer word in length. The interpretation of thi s word depends on how the methods use it. As the simulation proceed s, these initial values are continuously updated. Taken together, the values of the parameters and the outputs constitute the state of the sy stem at any time during simulation. The major difference between parameter values and output values is that outputs are a <sup>v</sup>ai lable to other units in a network through connections, while the value of parameters can only be read by the unit to which they belong. P3 units can have two classes of parameter s: unit parameters and terminal parameters. The unit parameters apply to the whole unit, for example, the threshold in a linear threshold unit. The terminal parameters are associated with individual inputs or outputs and correspond, for example, to

weights.

An impor tant function of the P3 plan language is to de scribe the connections between units. Since units can have multiple inputs and outputs there has to be some way to name them so that the CO N NECT statements will know which connections to make. These names are also used within the method programs to read inputs and set outputs.

The basic form of the CONNECT stateme nt is

(CONNEC T < un it -name > OU TPU T < output-name> TO < unit-re�hlflM1Ie1Jainput-name > )

For units with only a few inputs or outputs each input or output can be given a separate name. When a unit has a large number of inputs or outputs it is more convenient to group them together in input or output arrays. The individual items in these arrays are referenced by giving the array name and a set of subscript values. These arrays can be used

in iterative statements in plans and methods.

An output value can serve as input to any number of units, i.e., the fan-out is arbitrarily large. Each individual input can receive only one value. This is easy to enforce as long as it is known that just one connection is to be made to each input. This works well in many cases but it often happens that it is very hard or impossible for the programmer to know exactly how many connections will be made. This is the case, for example, when connection decisions are being made implicitly by some computational procedure such as "connection by location" or random connection. To overcome this, P3 secretly treats each individual input as an array and automatically adjusts its size to fit the num ber of inputs. This process is transparent to the programmer which means that multiple connections can be made freely to a single input. There is a special iteration statement in the method lang uage to access these multiple inputs. Each individual input that actually gets generated is called a TERMINAL and there are procedures for associating parame-

ters with terminals and initializing their values.

The method programs that im plement the internal functionings of units are written in the form of ordinary computer programs in an appropriate language. In the current implementation, which runs on the Symbolics 3600, the language is LISP. In order to allow the methods to use the values of inputs and parameters in their computations, a set of special access statements is incorporated into this system and is available to LIS P programs. These statements make it possible for methods to read and set inputs, out puts, and parameters more or

less as if they are ordinary variables.

In order to illustrate how P3 works, we will describe a model of a simple competitive learning network of the type described in Chapter S. The basic network contains two types of units: a pattern generator to supply stimuli and a cluster of com petitive learners connected to it, which spontaneously discover some features of the patterns. Since learning is spontaneous and does not require a teacher, the functioning of the network is simple and straightforward. Thr. pattern generators sequentially produce output patterns that serve as input stimuli to the cluster units. Each pattern is an activation vector specifying which of the in puts are active and which are not. Each cluster unit prod uces an output indicating its res ponse to the current stimulus which is transmitted to all other members of the cluster to create a "winner take all" network. The cluster unit which wins is the only one that learns and it Copyrighted Material

uses the weight redis tribution procedure described in the competitive

learning chapter, that is,

aWl} <sup>=</sup>CI <sup>0</sup>if uni <sup>t</sup>j loses on stimulus k

g...!!... - gWi} if unit j wins on stimulus k

nk

where Clk is e qual to 1 if in stimulus pattern Sk, element; in the lower layer is active and zero otherwise , and nk is the number of acti <sup>v</sup>e ele-

ments in patte rn Sk (thus nk = l:Clk)'

i

The first step in creating a P3 plan is to supply the UNI T TYPE statements. The UNIT TYPE statement for the pattern generator is

given below:

(unit type dipole

parameters flag i 1 j 1 i2 j2 outputs (d array i j)

method < update routine code in lisp> )

In this, and all our other examples, words in italics are part of the P3 plan language while the nonitalicized words are supplied by the user. The UNIT TYPE statement gi ves the type a name that will be used throughout the plan. The name for the pattern generator type is "dipole." There are five parameters that are used internally for pattern generation . The technicalities of the use of these parameters is irrelevant here. The UNIT TYPE statement describes the output of the unit. This ou t put is a two-dimensional array of lines called "d." This array of outputs is the retina on which stimulus patterns are generated which serves as an input to the competitive learning cluster units. The "i" and "j" that follow the word array are dummy variables that tell P3 how many dimensions the array has. The actual size of the array is variable and is initialized when we instantiate units of the type dipole. Note that the unit type dipole had no inputs since it is itsel f the source

of patterns.

The second basic unit type is the competitive learning unit, which in our plan we call "competitor." The unit type statement for it is given

below:

(unit type competitor parameters p g flag

inputs (C array i j terminal parameters W)

(i-A) outputs (o-A)

Note that the input array"C" of this unit corresponds exactly in form to the output array "d" of the dipole unit described previously. This correspondence will make it possible to make one-to-one connections between the output of dipole type units and the input of competitor type units. Also notice that a terminal parameter "W" has been associ-

ated with the input array"C." The competitor unit needs an additional

input called "i-A" which will receive information from the outputs of all

the other members of the cluster.

We have described the two unit types we will need. We can now go ahead and instantiate units of these types. The statement that creates a

pattern generator unit of type dipole is shown below:

(unit stimulus of type dipole at (@ 000)

outputs (d array (i 0 5) (j 0 5»»

The unit statement names the unit it is creating. This is the name of a real unit that is actually going to exist in our model and it is the name that will be referred to when this unit is connected to other units. For P3 to build such a unit, it has to be told the type. There can be any number of units of the same type and they can all have different names. Since every real unit in P3 has a location in P3 space, we must specify it in the unit statement that instantiates the unit. The at clause is used for this. The at is followed by a location specifier that simply evaluates to the x-, y-, and z-coordinates of the unit in P3 space. For simplicity we locate the pattern generator at the origin of P3 space which will initially be located at the center of the display window when we simulate the model. Since we are building a real unit, we have to give a size to its array of output lines. This is done in the outputs clause of the UNIT statement. Each subscript specifier consists of a subscript name and initial value, which in the current implementation must be 0, and final value, which in this example is 5 for both the "i " and the "j" subscripts. This statement will generate a 6 x 6 array of output lines on

connector" d."

Now that we have a source of patterns, we need to create a cluster of units that will receive these patterns. The statement that instantiates

these units is given below:

(unit cluster array (k 0 - cluster-size 1) of type competitor at (@ ( ... k (+ cluster-size 4»)( + cluster-size 10) 0)

initialize (g = 0.05)

inputs (C array (i 0 (- stimulus-size 1)(j 0 (- stimulus-size 1))

In this case, we are not instantiating a single unit but an array of units. In the competitive learning model, the learning cluster always consists Copyrighted Material

of two or more units, so we want a way to vary the number of units in a cluster. In the first line of the unit statement we give the name cluster to the array and then we indicate the size of the array with a subscript specifier. The name of this subscript is "k"; its initial value is O. Its final value is one less than the global constant "cluster-size." Th<sup>e</sup> value of cluster-size, which will occur at various points in the plan, is set by a statement at the beginning of the P3 plan that determines the value of global constants. This feature means that we can change the parameters such as cluster-size globally throughout the plan by only fiddling with a single value. The upper bound of the stimulus input line array has also been set with the use of a global constant "stimulus-size" rather than with an integer as was done previously. Also notice that the variable "k" is used in an at clause to place each unit of the array at

a different place in P3 space.

Our next task is to connect the units together in the appropriat<sup>e</sup> fashion. We have two classes of connections: those that go from the stimulus generator to the learning cluster and those that interconnect the units within the learning cluster. Each of these classes of connections has many individual connections within it, but these individual connections can be specified algorithmically in such a way that only a few CONNECT statements are needed to generate the entire network. What is more, the algorithmic specification of these connections makes it possible to change the size of the cluster or the size of the stimulus array without altering the CONNECT statements at all. The code required to connect the stimulus to the learning cluster is given below:

{for (k 0 (+ 1 k})

exit when (> k cluster-size) do

{for (i 0 (+ 1 j)}

exit when (> i stimulus-size) do

{for (j 0 (+ 1 j»

exit when (> j stimulus-size) do (connect unit stimulus output d i j to unit (cluster k) input C i j

terminal initialize (W - (si:random-in-range

0.0 (! I 2.0 (expt (+ stimulus-size 1) 2»»»»

There are three nested loops. The first ranges over each member of the cluster, and the next two range over each dimension of the stimulus array. Inside these three nested loops is a single CO NNECT statement. The CO N NEC T statement has the job of initializing the value of any terminal parameters. In our model we have a very important terminal parameter, "W," the weight between a stimulus line and a cluster unit, which we �byR�il1Matl!n;lt random value which sums

to one for the whole input array. This is accomplished by setting the

initial value of "W" with a LISP function that evaluates to the required quantity. In general, in a P3 plan wherever a number is required, a function (in our case a LISP function) that evaluates to a number can replace the number itself. The sum of the random numbers generated by our simple LISP function is not exactly one, but only averages one. This is satisfactory for the competitive learning algorithm because it is self-normalizing and will force the sum to one in the course of learning. The connections that link the members of a cluster are a bit more complex. Each member of the cluster must receive input from all other members except itself. The code for doing this in a completely

general way for clusters of any size is given below:

{for (k 0 (+ 1 k»

exit when (> k cluster-size) do

{for G 0 (+ 1 j»

exit when (= j k) do

(connect unit cluster k output o-A to unit cI uster j input i-A»

{for G (+ k 1) (+ 1 j»

exit when (> j cluster-size) do (connect unit cluster k output o-A to unit cluster j input i-A»)

The idea here is that we first connect each unit to those units whose subscripts are lower than it and then to each unit whose subscript is higher than it. This requires two separate loops, each with its own CONNECT statement, both nested within an outer loop that ranges over all units in the cluster. Note that this is a case of making multiple connections to a single input line. We don't have to know how many connections there are because within the method there is code that will examine all connections on this line to decide if the unit has won. This feature is very useful and can be applied whenever a method needs to

know the value of an input but not its originating unit.

We have now specified all the features of a plan that describes the

basic competitive learning network. Of course, this plan can only be

used to construct a running model if we have available the appropriate method programs. Since these are ordinary computer programs written in LISP, we won't analyze them in detail. The code for the methods used here is given in the appendix of this chapter, which shows a complete plan for a simulation of competitive learning. It is worthwhile, however, to see how the method language accesses the inputs and outputs of the units about which we have been saying so much in the

The only difference between the arguments to a P3 method and the arguments to a normal LISP function is that the P3 arguments are accessed by special access functions. For example, to get the value of a

parameter, the following form is used: (read-un it-parameter flag)

This form returns the current value of flag. To read an input from an

array of input lines the following form can be used:

(read-input (C i j»

In this case the value of "i" and "j" must be bound at the point in the program where an expression using them occurs. There are corresponding forms for reading terminal parameters, setting outputs,

and setting parameter values.

The P3 Simulation System

The P3 simulation system is the environment in which models in P3 are simulated. It is highly interactive and makes extensive use of the window system and the "mouse" pointer of the Symbolics 3600. The first step in simulating a model is to compile the methods and construct the plan. The constructor is a program similar in purpose to a compiler. However, the input is a P3 description of a network, rather than a computer language description of a program. The output of the constructor is a data structure containing all the relevant information about the network that can be used by the P3 simulation system to run a simulation of the model. As with any form of computer programming, a model must be debugged before it can be simulated. There are really two

levels of debugging for network models. First, the user wants to know

that the network that has been created is connected up in the way

intended. Once this has been established, the actual functioning of the

network can be debugged. P3 provides tools for both of these phases

of the debugging process.

To check the correctness of connections, P3 provides a display that shows each unit in the model at its location in P3 space. The user interacts with this display with a mouse pointing device. Clicking on a particular unit provides a menu that enables the user to trace out any of the connections emanating from that unit. This facil ity for tracing out connections, one at a time, has proved much more useful than simply presenting a user with the wiring diagram of the model. Once the user is convinced that the constructed model corresponds to the envisioned network, the job of anal:ebMRt'¥PRJ�99n9Ji the model can begin.

Analyzing the running of a complex simulation is a demanding task. It is in this analysis that we have found that all the features of the P3 system come together and begin to justify their existence. Because every object in the model has a location in P3 space that corresponds to the user's mental image of the network, the simulation system can display values representing the state of the system at locations on the screen that have meaning to the user. This means that during the course of a simulation, meaningful patterns of P3 variables can be displayed. This approach is widely used in analyzing the function of parallel systems. What P3 has done is to standardize it and relieve the user of the need to implement the details of this display strategy for

each new model.

In the current implementation of P3, each object in the model is represented at its designated location by a small rectangular icon. By the use of a mouse pointer driven menu system, the user can assign the icon representing a unit the variable whose value is to be displayed. Thus, for example, the icons representing the input terminals of a cluster unit in our example can be assigned either the value of the input to that terminal or the value of the weight on that terminal. These assignments can be made or changed at any time during a simulation run. They can be set to be updated continually as a simulation proceeds, or they can be examined in detail when the simulation is temporarily interrupted. The current P3 implementation displays the relevant state values at two possible levels of precision. The approximate value of the state value is indicated by the degree of darkening of the icon. There are five levels of intensity. Their range is under user control and can be changed at any time. This enables the user to adjust the range so that the difference between the lightest and the darkest icons will optimize the information content of the display. There is also a high precision display that permits the exact value of any P3 variable to be

examined.

Figure 1 shows how the screen of the Symbolics 3600 looks after 588 P3 cycles of simulation of a competitive learning model with a 6 x 6 stimulus array and a dipole stimulus. There are six windows displayed and each shows a different aspect of the simulation. Window A shows the three units in the model at their respective positions in P3 space. The upper narrow rectangle is the pattern generator. It is not displaying any value. The lower two rectangles represent the two cluster units. They are displaying the approximate value of their outputs by the size of the contained black rectangle. Clearly the unit on the left has a lower output value than the one on the right. Window B shows the output of the pattern generator unit, which was called "stimulus" in the plan. The lines form a square array because that is the way they were specified in the plan. The two dark rectangles show the current dipole Copyrighted Material

In addition to the special functions of P3, the user also has available all the powerful program development tools of the Symbolics 3600. For example, suppose that the user believes that an observed bug is due to an error in the code of a method. It is possible to interrupt the simulation, go directly to the editor buffer that contains the method code, alter it, recompile the alteration, and then return to the simulation at exactly the point at which it was interrupted. This facility has

proved invaluable in debugging.

As we work with the P3 simulation system, we constantly find new features that are useful in the analytical process. We view the implementation of each of these new analytical techniques as analogous to adding a new instrument to a laboratory. Thus, we call the features of P3 that enable the user to analyze a functioning model "instruments." Each of these instruments can be called up at any time. Every instrument has a window that displays the results of the instrument's analysis. For example, one instrument is the "strip chart recorder" used in Figure 1. The strip chart recorder has a probe that can be connected to any particular state variable of any unit. Since multiple instances of any instrument can be created, any number of strip charts can be running at the same time. In addition to instruments that display the values of variables, we also envision a class of instruments that record these variables. Clearly, it is very important for a serious modeler to be able to record the results of a simulation. The instrument concept will enable the modeler to record just those variables required. This is a very important feature since simply recording the entire state of the model as it develops in time would produce an

overwhelming flow of data.

Performance

So far we have said nothing about the speed at which simulations run. This is a problem of tremendous importance for PDP models. Big models inherently run slowly on serial computers. Generally, parallel programming systems like P3 stress ease of model definition and simulation. How much penalty must we pay in model performance? There is always some performance penalty for a general-purpose system. For any given piece of computer hardware, it is generally possible to write a specially tailored program that will run some particular model faster than any general system will run it. However, this special tailoring itself takes considerable time and makes it much harder to change the details of the model structure. Thus, we envision that programs like P3 will be useful in the early stages of model development when the size Copynghted Material

of the models are modest and there is frequent need for changes in structure. When the structure and parameters of a model have been decided upon and it is necessary to scale the model up and have it run extremely rapidly. it may in some cases be advantageous to write a spe-

cial program to implement the model.

The general-purpose systems, however, have several things going for them with respect to model performance. First of all, since the data structure has the same form for models, it is possible to put a lot of effort into optimizing running speed for the particular hardware on which the system is implemented. This optimization only has to be done once rather than for each model. A second way in which generalpurpose systems can improve performance is through the use of special-purpose hardware. The models generated by the P3 system are inherently parallel models and map well to some parallel computer architectures. The one way to get blinding speed from parallel models is to implement real parallelism in parallel computers. In some cases, array processors can also be highly beneficial. Since all the P3 models are of the same sort. a constructor can be made that will provide the appropriate data structures to run any P3 model on these kinds of hardware. This will make the hardware transparently available to the user of systems like P3. This, we believe, is a significant plus, since it is notoriously difficult to program any particular application for array

processors or truly parallel hardware.

In conclusion, the P3 system illustrates some of the general issues that arise in any attempt to simulate PDP models, and provides a number of useful tools that can greatly facilitate model development. General-purpose systems like P3 have promise for speeding and facilitating the programming of parallel models and the ultimate ability to

run these models very fast using specialized hardware.

APPENDIX A

'" '" '"

'" P3 Plan for Competitive Learning

'" (NOTE the use of the "plan constant" and "include" statements.)

'" '"

••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••

'"

'" Unit types

••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••

'"

;;; ........ Dipole pattern generator ...... ..

(unit type dipole

parameters flag il j1 j1 i2 j2

outputs (d array i j)

include dipole-generator) ... (see code file on p. 506)

;;; .......... Learning unit ....... .

(unit type competitor parameters p Q flag

inputs (C array i j terminal parameters W)

(i-A) outputs (o-A)

include comp-learn) ... (code on p. 504)

••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••

'"

'" Unit instances

••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••

'"

(plan constant stimulus-size = 6) (plan constant cluster-size = 2)

;;; ........ Dipole pattern generator········

(unit stimulus of type dipole

at (@ 000)

;;; ........ Learning units ...... ..

(unit cluster array (k 0 (- cluster-size 1) of type competitor at (@ (. I (+ cluster-size 4» (+ cluster-size 10) 0)

initialize (q = 0.05)

inputs (C array (i 0 (- stimulus-size 1) GO stimulus-size»)

.....................................................................

Connections

.....................................................................

;;;<sup>u</sup>

...... Stimulus to both clusters ....... .

(for (k 0 (+ 1 k»

exit when ( <sup>=</sup>k cluster-size) do

(for (j 0 (+ 1 j»

exit when ( <sup>=</sup>i stimulus-size ) do

(for G 0 (+ 1 j)

exit when ( <sup>=</sup>j stimulus-size) do

(connect unit stimulus output d i j to unit cluster k input C i j

terminal initialize

(W = (si:random-in-range

0.0 (! / 2.0 (expt (+ stimulus-size 1) 2»»»»

;;; ....... Interconnect the clusters to implement competition ....... .

(for (k 0 (+ 1 k»

exit when ( <sup>=</sup>k cluster-size 1) do

(for G 0 (+ 1 j»

exit when ( <sup>=</sup>j k) do

(connect unit cluster k output o-A to unit cluster j input i-A»

for G (+ k 1) (+ 1 j»

exit when ( <sup>=</sup>j cluster-size 1) do (connect unit cluster k output o-A to unit cluster j input i-A»)

APPENDIX B

'" ,,, '"

'" Competitive Learning: Methods

'"

•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• ",

to, Method for unit in cluster of competitive learners

•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• to,

method

(let «imax (input-dimension-n C 1) Gmax (input-dimension-n C 2»

(win t) (N 0»

;; ........ Is this a learning iteration? ••••••••

(cond

;; ........ No ....... .

;; Accumulate the weighted sum of the pattern inputs into ;; unit parameter p, and set the competition output p-A to

;; that value

«> (read-unit-parameter flag) 0)

(loop initially (set-unit-parameter p 0)

for i from 0 below imax do

(loop for j from 0 below jmax do (set-unit-parameter p

(+ (read-unit-parameter p)

(. read-terminal-parameter (C i j) W)

(read-input (C i j»»» finally (set-output o-A (read-unit-parameter p»)

;; .......... Yes ••••••••

;; Figure out whether this unit wins on this cycle. Winning ;; requires that this unit's parameter p be greater than those ;; for the other units of this type. Those values are available

;; on the terminals of input i-A.

;; NOTE: On iteration 0, everything is 0, so no unit thinks it

;; wins, and hence all avoid learning.

(t

;; ........ Find out whether we won ....... .

;; Win was initialized to t in the let at the top level of this method.

(for-terminals k of input i-A

(if « .. (read-unit-parameter p) (read-input (i-A terminal k»)

(setq win nil))

(when win

;; ........ Accumulate sum of all inputs into N ....... .

;; This will become a normalizing constant. (loop for i from 0 below imax do (loop for j from 0 below jmax do

(setq N (+ N (read-input (C i j»»»

;; .. •• .... Compute new weights ....... . ;; But only if the total input was greater than O.

(if (> NO)

(loop with q-factor = (read-unit-parameter g)

for i from 0 below imax do (loop for j from 0 below jmax do

;; ........ Compute one new weight ........

(let· (old-weight

(read-terminal-parameter

(C i j) W» (new-weight

> (+ old-weight (. g-factor

;; Update the terminal parameter to the new weight

(set -termi nal-parameter (C i j) W

new-weight»») )

;; ........ Flip the iteration-parity flag ....... .

(set-unit-parameter flag 1»»

••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••

, "

", Dipole pattern generator method

••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••

method

;; ........ Do we need a new pattern on this iteration? ....... .

(cond

;; ........ Yes. Erase old dipole and make new one ........ .

« < (read-unit-parameter flag) l)

Oet « imax (- (output-dimension-n d 1) 2» (jmax (- (output-dimension-n d 1) 2»)

(set-output (d (read-unit-parameter i 1) (read-unit-parameter i 1) 0) (set-output (d (read-unit-parameter i2) (read-unit-parameter i2» 0)

(set-unit-parameter i 1 (+ (random imax) 1» (set-unit-parameter jl (+ (random jmax 1)

(cond «> (random 2) 0.5 (cond «> (random 2 0.5)

(set-unit-parameter i2 (+ (read-unit-parameter il) 1))

(t

(set-unit-parameter i2 (- (read-unit-parameter i 1 1) )

(set-unit-parameter i2 (read-unit-parameter j 1»)

(t

(cond «> (random 2) 0.5)

(set-unit-parameter i2 (+ (read-unit-parameter j 1) 1))

(t

(set-un it-parameter i2 (- (read-un it-parameter j 1) 1»»

(set-un it-parameter i2 (read-unit-parameter i 1»» (set-output (d (read-unit-parameter il) (read-unit-parameter j 1) 1) (set-output (d (read-unit-parameter i2) (read-unit-parameter j2» 1)

(set-unit-parameter flag 1»)

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Index

Page numbers in roman type refer to Volume 1; page numbers in

italic type refer to Volume 2.

Abeles, M., 36 7, 369, 385, 553 unit, 425 Abelson, R. P., 18, 19, 574 logic of

Abrams, T. W., 552, 561 conclusions on, 442-443 Abramson, A. S., 94, 560 description of, 363 Abstraction models of memory, examples of, 428-429 199-206 main concepts of, 426-428 comparison of experimental and as major aspect of PDP model s,

simulation results, 203-205 51-52

other findings on, 205-206 results of, 429, 439-442 summary of training and test rules , examples of, 425-426 stimuli, 202-203 three-dimensional plots, 429,

Whittlesea's experiments on, 201 - 430-438 (Figs.) 202 views of, 423-425 Ackley, D. H., 264, 306, 313, 317 , Activation rule. See Activation

335,507, 510, 393, 562 functions

ACP model, 32, 124, 127 Activation vector of harmony Activation functions theoretical model , 21 5-216 examples of Active representation in PDP

gating activation function, 426 models. See PDP models, active

semilinear, 52, 324-328 representation in

quasi-linear, 425 Addition problem, simple binary, quasi-multilinear, 426 341 -346. See also Deltarule,

Algebra, linear. See Linear algebra Algebraic properties of inner products, 382-383. See also

Vectors

Allman, J.. 34 7, 374. 375. 553, 554 Alternative activity patterns, 82-83 AM. See Monocular occlusion Amari, S. A., 42, 45, 444, 507, 489.

495, 553 Ambiguity

> lexical. 302-304 structural, 304-305

and shading of feature patterns,

305-306

Ambiguous characters, 8, 155-157. See also Reading, programmable

blackboard model of Amnesia. See also Memory, a distributed model of

anterograde, 504, 505-506. 512-513 bitemporal, basic aspects of, 505-

506

conclusions on, 52 7 description of, 503-505

neurochemistry of synaptic change. hypothetical account of, 50 7-509 paradox of, resolution to, 506-50 7 quantitative model, formulation of,

509

relation to other accounts of, 510-

5/l

residual learning in, 518-524 simulations, 511-518

anterograde amnesia, 512-513 lost memories, recovery of, 515-

516

retrograde amnesia, 513-515 retrograde and anterograde amnesia, correlation of, 51 3

studies on, 516-518 summary of, 518

spared learning in. 207-208, 518,

524-526

Anandan, P., 322, 508 , 550, 554 AND function, 429. See also Boolean function

Anderson. C. W., 539, 554

Anderson, J. A., 33 , 42, 56,62,66, 68 ,80, 82, 101 ,406, 409, 410, 418, 507,510, 51 1, 8, 173, 183, 200,226,277, 311, 380, 389, 394,399,431,551, 553, 554,

561,564

Anderson, J. R., 32, 124, 127, 25 1, 46 1 ,507, 173,106,209, 524,

549, 554, 558 Anlezark, G. M., 477, 554

Annealing, simulated. See Simulated

annealing Aragon, C. R., 235, 508 Arbib, M. A., 362, 577

Arbitrary mapping, implementing.

96- 104

grapheme strings. 97·99, 102 sememe units, 97-98, 99 , 10 1·102 three-layer network, 97-98 . See also Boltzmann machines Arguments, missing, filling in, 301- 302. See also Case role assignment, model of

Ary . M., 477, 564 Asanuma, H., 35 7, 554 Asynchronous update vs. synchronous update, 61 ATN. See Augmented transition

network parsers

Attention

networks for focusing, 114·1 18 shifting focus of, 147·151 Augmented transition network parsers (A TN), 11 9

Auto·associative paradigm. See also Memory, a distributed model of

definition of, 55 , 161 modification of, 211-212

Back'propagation simulator , description of, 328 ·330 Bagley, W. C., 58, 98, 1 06 , 554

Bahl, L. R., 293, 508 Baker, J. F., 34 7, 554 Baker, R. G., 24, 515

Boolean function (continued) linearly separable functions, 440 mapping from expressions, 439 neurons treated as devices, 424-425

XOR function , 429

Bottom-up approach to analysis of information processing systems , {3 units, simulations of (App.), 469- 470. See also {3-coefficient model

Bunt, A. H., 35 7, 566

Bybee, 1. L., 221, 247. 249, 250, 251.254, 255,256,556

123 CA. See Connection activation unit

Bottom-up presentation of harmony Caan, W., 368, 571

theory, 199, 213-26 1. See also Cajal, S. Ramon y, 336, 350, 358, Harmony theory 359,361, 363,36 7, 556 Bottom-up processing system, 51 Caminiti, R., 377-378, 560 matrix representation of, 57-58 Canonical feature detectors, 114

Boycott, B. B. , 336, 558 Caplan, C. J. , 478, 570 Brachman, R. 1., 313, 555 Carew, T. J., 552, 561 Brain. See Cerebral cortex, anatomy Carlson, M., 275, 573 and physiology of; Cerebral Carman , 1. B., 350, 556 cortex, questions about Carnes , K. M. , 477, 554

computation in Carpenter, P. A., 153, 16 1, 549,564,

Brain damage. See also Amnesia; 577

Damage, effects of on a network Carrol l, Lewis, 97 graceful degradation and, 29, 134 Cascade model, 42

neuropsychological investigation of Case role assignment, model of patients with, 134-135 architecture of, 27 7-289

simulated effects of, 304-313 case role representation, 286-288 Brain state in a box model (BSB), semantic microfeatures, 278-283 66-68 sentence processing and learning,

Branch. M., 435, 571 detai ls of, 288-289

Brayton, R. K., 381. 382, 575 sentence-structure units, 283-286

Bresnan , J .• 274, 559, 564 descriptions of, 3-4 Brightman, M. W., 336, 572 discussion of, 313-325 Broadbent, D. E., 1 2 1 , 508, 97, 532, context-sensitive coding,

555 iteration. and center embedding,

Brock, L. G., 36 7, 555 323-325

Brodmann, K., 345, 346, 555 fixed-length, 320-322 Brodmann's areas of the neocortex, limitation myths, 322-323

345-346. See also Cerebral meaning, representation of. 315-

cortex, anatomy and physiology 316

of model. basic features of, 314-315 Brooks. C. McC.. 336 , 553 parsers, conventional interfacing

Brooks. L. R .• 171.556 with,317-318 Brown, P .• 295 recursion, 318-320 Brown , R .• 219, 241, 556 goals of, 276-277 Bruce, C. J., 368. 556. 558 studies on , 277

Brugge. J. F .• 356, 563 multiple constraints on, 272-275 BSB model . See Brain state in a box studies on. 273. 275, 277

COHORT model (continued) cerebral cortex in, 387-389

resolution of, 101-106 Computational temperature (n, 211 Cole, K. S., 364, 556 Computational viewpoint on PDP

Cole, R. A., 58, 61, 98, 99, 100, models, 397-398

107,111, 556, 557 Conel, J. LeRoy, 358, 557 Collins, A. M., 85, 508 Confabulation , as a stable pattern, 81 Colonnier, M., 348, 361, 557, 571 Conjuncti ve encoding, 90-91, 96. Colon Osorio, F. c., 424, 510 See also Coarse coding; Coltheart, M., 102, 508 Distributed representations Competence theorem, 226-229 CONNECT statements for P3 Competitive inhibition, 21. See also system, 495-496. See also P3

Phoneme identification , factors system

influencing Connection activation system, 130-

effect of at phoneme level, 89-90 134

Competitive learning Connection activation unit (CA) . architecture of competitive learning See also Connection information system, 162- 166 distribution mechanism basic components of, 151-15 2 definition of, 132-134

characteristics of, 166- 167 multiple pattern processing, 477-

conclusions on , 190 478

definition of, 147 requirements of, 474-476

dipole experiments, results of, single pattern processing, 476-477 170-177 summary of requirements, 480-48 1 correlated teaching inputs, 182- Connection information distribution 183 (CID) mechanism , 473-486, letter similarity effects, 182 129-134, 137, 139-140 number of elements, effect of per benefits of, 164-166

serial position , 181-182 central knowledge store in, 130-134 position-specific letter detectors, computer simulation of word 180-18 1 recognition using, 134-136 word detection units, 18 1 extensions of, 167-168

word perception model, 177-180 resource requirements of, 473 -486,

formal analysis of, 168-170 166-167

geometric interpretation of, 164- Connectionist models, 72 166 Connolly, M., 353, 355, 566 history of, 152- 159 Connors, B. W., 364, 369, 557 horizontal and vertical lines, Constructor, as major component of experiments with, 184-190 P3 system, 489. See also P3

mechanism of, 159 system

neural-like processing systems, Content-addressable memory, 79-80 160- 16 1 Continuity and uniqueness in P3 simulation system constraints in depth perception,

methods (App. B), 504-506 19-20

plans for (App. A), 502-503 Cool ing schedule, 244-245. See also

Completion task. example of. 206- Electricity problem-solving; 208 Simulated annealing Computational level. 121 Coombs. 1. S .• 367. 555

Cooper, L. N., 42, 180, 508, 480, 487, 489, 499, 555, 557, 573

Corkin, S., 519, 524, 556 Cortical areas of the neocortex, Brodmann's cortical parcellation

scheme, 345 input systems, 346

irregularities within, 353·356 thalamus, as gateway to cerebral

cortex, 349·352

topographic representation, 352·

353

output systems, 356·357 Cortical plasticity, processes that modulate, 476·478 Cotton, S., 99, 105, 557

Cottrell, G., 165, 277, 3lJ, 314, 557

Coulter, 1. D., 357, 570 Cowan,1. 0., 381, 389,578 Cowan, W. M., 350, 351, 556, 572

Cowey, A., 363, 576 Crain, S., 274, 557 Crane, A. M., 350, 572

Crick, F. H. c., 296, 297, 509, 351,

384, 387, 557

Critical or sensitive period of plasticity, 473·484

cortical plasticity, processes that

modulate, 476·477 effect of environment on, 478 hypothesis for, 483·484

ocular dominance statistics, 474·

476

other examples of, 477·478 Crosstalk, 139·142. See also Reading, programmable blackboard model of Crow, T. 1., 476, 477, 554, 557 Crowder, R. G., 64, 94, 209, 557 Cynader, M., 475, 476, 478, 492,

498, 557, 558

in Boltzmann machines, 304·313

Daniel, P. M., 354, 558 Daniels, 1. D., 477, 554 Darien·Smith, J., 480, 574 Davis, E., 432, 558

Daw, N. W., 497, 500, 558, 577 Deacedo, B. S., 519, 524, 556 Decay process in increments to

weights, 181. See also Memory, a

distributed model of Decision·making and freezing in

harmony theory

coherent interpretation, robustness

of, 240

idealized task, 238·239

phase transitions, computational significance of, 236·237 symmetry breaking, 237·238 thermodynamic limit, 239

Deep dyslexia, 102

Degradation, effects of and benefits of redundancy, 472. See also Standard pattern associators Deliberate conscious control (DCC> , 543-545. See also PDP approach,

reflections on

Delta rule, 43, 53, 62, 63, 363, 417·

418

Delta rule, analysis of. See also Delta rule, generalized and multiple linear regression,

457·458

in pattern-based coordinates, 447·

453

statistical learning and, 453·457

summary of, 458·459 in vector notation, 445·447 Delta rule, generalized, 149 application of, 327·328 conclusions on, 361·362 general cases, 352·361 recurrent nets, 354·361 and gradient descent, 322·324

Dahlstrom, A., 350, 553 problem of, 318·323 Damage, effects of on a network, 29, XOR problem, 319·321

134, 472. See also Amnesia; for semilinear activation functions Boltzmann machines; Brain in feedforward networks, 324·328

Delta rule, generalized (continued) simulation results, 328-330, 352 encoding problem, 335·339 negation problem, 346-347

parity, 334-335

simple binary addition problem,

341-346 symmetry, 340-341 T-C problem, 348-352 XOR problem, 330-334 use of to determine size and direction of changes in connections, 179-181 generalized, 209

DeMarzo, P. M. , 241, 242, 509 Demps ter, A. P., 293, 509

Dendrites , studies on , 381-382. See also Cerebral cortex, questions

about computation in

Denes , P., 87, 558

de Riboupierre, F., 385, 553 de Ribou pierre, Y. , 385, 553 Derthick, M., 298, 299, 509, 313 Desimone, R., 368, 374, 556, 558 Di pole experiments for competitive learning mechanisms, 170-174 formal analysis of, 174-177 homogeneous stimulus patterns ,

191·193

Distal landmarks for biologically plausible models of place

arbitrary pairing, implementing,

96-104

binding problem, 88-90 coarse coding, 91-96 con junctive encoding, 90-91 Distributed view-field model for goal location , 449-460. See also Place recognition and goal location, biologically plausible models of

descri ption of, 451-453 network used in, 450-451 P3 simulation of, 453-454 Symbolics 3600 LISP machine, use of for, 454-456

problems of, 451 properties of, 453 testing of, 453 Dobson, V., 97, 100 Dostrovsky, J., 435, 571 Dowling, J. E. , 336, 558, 578 Drager, U. c., 490, 558 Dretske, F., 195, 509 Duckrow, R. B., 477, 561 Duda, R. 0., 380, 558 Dukes, K., 61, 110, 570 Durham, D. , 500, 558

Dynamic functional system, <sup>41</sup> Dynamical systems, perspective of PDP models as, 397-398

recognition and goal location , 445 Eccles, 1. C., 336, 367, 370, 553,

Distributed representations. See also 555, 558

Memory, a distributed model of; Edelman, G. M. , 387, 558 Schemata, concept of Edmonds , D., 153, features of Ehrlich , S., 163, 573

generalization effects, 82·85 Eichenbaum, H., 385, 519, 524, 556,

memory, 79·81 565

new concepts, 86·87 Eigenvalues . See Eigenvectors and

status of, 78·79 eigenvalues

structure in representations and Eigenvectors and eigenvalues, 399· processes, 104·108 403. See also Matrices and linear

constituent structure, systems

representing, 105·106 Eimas , P. D. , 94, 473, 558, 560

sequential symbol processing, Eisen , M. , 456, 509 106·108 Elbaum, c., 499, 573

summary on , 108·109 Electricity problem·solving, 240·250 technical details of, 87·104 Electroencephalography (EEG), 335 Copyrighted Material

Freezing and decision-making in harmony theory. See Decisionmaking and freezing in harmony

theory

Freund, T. F., 363, 576 Friedlander, M. J. , 351 , 560

Fry, D. B., 94, 560 Fuchs, A. F., 357, 566

Fujimura, 0., 62, 473, 560, 569 Fujisaki, H., 77, 94, 560

Fukushima, K. , 42, 152, 162, 300,

313,350, 509

Future directions for PDP, 54 7-552 Fuxe, K., 350, 364, 553, 560 Fuzzy map theory, 432-433. See also Place recognition and goal location, biologically plausible

models-of

Gallistel, C. R., 141, 509 Gamma factor in bitemporal amnesia, 50 7 Ganong, W. F., 60, 78, 560 Garey, L. J. , 348, 497, 500, 555,

560,5 77 Garnes, S., 61, 555 Garrett , M. F., 274, 559 Garrud, P., 434, 466, 569 Gary-Bobo, E., 476, 478, 556

Gati, I. , 499, 560

Gating activation function, 426. See also Activation functions Gee, J., 60, 63, 107, 115, 561 Gelatt, C. D., Jr ., 48, 235, 287, 511 Geman, D., 148,264,271,278,289,

509

Geman, S., 148,264,268,271,278,

289, 509

Generalization, 30, 85. See also Memory, a distributed model of Glaser, E. M., 366, 577 Glass, L., 494, 571 Glorioso, R. M., 424, 510 Gluck, M. A., 383, 552, 561 Glushko, R. J., 43, 510, 171, 561 Goal location. See Place recognition and goal location, biologically

plausible models of Goldman-Rakic, P. S., 350, 572 Goodness-of-fit function, 14- 16, 31 , 32-33, 35-36. See also Schemata,

concept of

Graceful degradation, 29, 134. See also Damage, effects of on a

network Gradient descent

and delta rule, 322-324 use of to solve hard optimizing problems on conventional computers, 287-288

Grapheme strings, 97-99, 102. See also Arbitrary mapping,

implementing Gray, E. G., 339, 361, 561 Green, D. M., 466, 510, 225, 561

Green, K., 96, 569 Greenberg, Z., 254 Greenway, A. P., 477, 554 Grinvald, A., 385, 561

Grosjean, F., 60, 63, 99, 100, 105, 10 7, 114, 11 5, 55 7, 561 Gross, C. G., 368, 556, 558 Grossberg, S., 42, 53, 70, 116, 146, 152, 162, 164, 167,418,510,90,

389, 494, 535, 556, 561 Grossberg's models, 70-71 Guillery, R. W., 351, 561 Gutnick, M. J., 364, 369, 55 7

Georgopoulos, A. P., 377-378, 560 Haggard , M., 85, 576 Gernsbacher, M. A., 79,559 Hale, B. L., 159, 563 Gerstein, G. L. , 385, 560 Halle, M., 24, 508, 168, 56 1 Gibbs sampler, research with, 264 Halmos, P. R., 422, 510 Gibson, E. J., 198 Hamiltonian function (fl), 211

Gilbert, C. D., 356, 358, 366, 385, Hamori, J., 351, 577

480, 560, 561, 577, 578 Hard learning, example of, 299-303. Ginsburg, H. P., 241, 509 See also Boltzmann machines Copyrighted Material

Horizontally hierarchical networks,

21 7

Houde, J. , 353, 355, 566 Houser, C. R., 351, 562

Hubel, D. H., 340, 353, 356, 367, 373, 474, 475, 483, 497, 500,

562, 563, 566, 578

Huberman , B. A., 313, 511, 389, 562 Hughlings-Jackson, J., 41, 141-142 Human information processing. See also PDP approach, reflections on

essential properties of, 537 new understanding of, 545-546 Human memory. See Memory, a distributed model of; Memory, retrieving information from Hummel , R. A., 285, 286, 511, 514

Hunt, S. P., 364, 559

Huppert, F. A., 516, 517, 563 6-hydroxydopamine (6-0HDA),

477-496

Identity/ role combinations, 1 06. See also Case role assignment, model of; Distributed representations

Imbert, M., 476, 478, 556 Imig, T. J. , 356, 563

Implementational (physiological) level of distributed memory, 121 Inner products . See also Vectors algebraic properties of, 382-383 in two-dimensional space, 380-382

Input units, definition of, 48 Interactive activation model of word perception, 43, 71 -72, 177-180, 216-2 1 7,125-129, 137-1 38 Interactive model of reading (Rumelhart) , 43, 122

Interactive models, representation of in matrix form, 57, 59-61. See also Interactive activation model

of word perception layered systems , 60 one-step system, 60 single-level , 60

associators� Reading,

programmable blackboard model of� Standard pattern associators Interference of memories, 208-209. See also Memory, a distributed

model of

Internal representations. See Delta rule. generalized� Representation,

learning of IS-A hierarchy, 105 Isenberg, D .• 7. 511

Isomorphism hypothesis. See also PDP models, neural and conceptual interpretation of of conceptual and neural levels.

395-396

failure of in nonlinear models,

422-424

levels and localized damage, 413-

416

levels, learning of connections and,

416-418

in linear systems, 411 -413 Iverson, L. L., 340, 563

Jackson , J. H., 41, 141,511 Jacoby, L. L. , 171, 172, 192, 193,

556,563

Jakimik, J., 61 , 98, 99, 100, 107,

111, 557 Jelinek, F., 293, 508 Jenkins, 1. J., 473, 569 Jenkins, W. M. , 385, 563 Johnson, D. S., 235, 508 Johnson, E., 324, 563 Johnston, D., 477, 562

Johnston, J. C., 159, 160, 563, 574 Jones, E. G., 351 , 352, 35 7, 363,

563,564

Jones, R. S., 56, 406, 409, 410, 418, 507,173, 226, 311 , 551 , 554

Jouvet, M. , 350, 555 Julesz, B., 18, 511

Just, M. A., 153, 161,549, 564,577

Interference of patterns, 139-142. 387, 569, 570, 578 See also Programmable pattern Kahneman, D., 536, 564 Copyrighted Material

Kaiserman·Abramof, I. R., 336, 361,

365,571

Kalaska, J. F., 378, 560

Kandel, E. R., 333, 364, 507, 552,

561 , 564, 576 Kanerva, P., 76, 465, 51 1 Kant, E., 17, 19,564

Kaplan, R. M., 119, 516,274,559,

564

Kasamatsu, T., 476, 477, 492, 497,

564

Kawamoto, A. H., 101, 511,277,

311,564

Kawashima , T., 77, 94, 560 Keele, S. W. , 171, 183, 200, 203,

572

Kelly,1. P. , 480, 560 Kewley·Port, D. , 85, 564

Kienker, P. K., 302 , 350, 511,515 Kinematics of the dynamical system, 398·399, 400·403. See also PDP models, neural and conceptual

interpretation of

Kirkpatrick, S., 148, 235, 287, 511

Kisvardy, Z. F .• 363, 576 Klatt, D. H., 59, 60,62. 564

Kliegl, R., 273. 567 Knapp, A., 173, 200, 564 Knowledge atoms definition of. 202

and electricity problem·solving, 242 Koch, C., 365, 381, 383, 564. 565 Kohonen. T .. 42. 45. 62. 63. 152, 406.409, 418, 422. 424, 425, 444, 446,455,461, 5Il. 222,

226, 380, 565 Konishi, M., 473, 565 Kratz, K. E., 490, 565 Krnjevfc, K .• 339. 565

Kubie. J. L., 435. 441. 445, 447.

565.570

Kucera. H., 74. 240, 565

Kuczaj, S. A., 219, 220. 221. 253,

257. 258, 565 Kuffler, S. W .• 367. 565 Kuipers, B., 433, 565

Kuperstein, M .• 385, 565 Kupperman, B.; 478, 572 Kurtzman. H. S., 275, 565 Kuypers. H. G. 1. M., 352, 571

Laird, N. M., 293, 509

Lakoff, G., 97

LaManna , 1. c., 477, 561 Lamperti, 1., 277, 511 Lane, H. L., 91 , 565 Lang, K., 127, 510 Lang, M., 511

Language acquisition device (LAD), 217. See also Verbs , learning past

tenses of Larkin, 1. H., 24'1 , 512 Larsson, K., 350, 553 Lashley , K. S., 41,512

Law of mass action, 509·510. See

also Amnesia

Layered systems in interactive

models , 60 Lazarus, J. H. , 94, 572 Le. See Locus coeruleus Leaky learning model. 179

Learnability theorem, 235·236, 260 Learning. See also Boltzmann

machines; Delta rule, generali zed; Memory, a distributed model of; Neural plasticity and learning

in Boltzmann machines easy and hard, 290-299 hard, example of, 299-303 learning algorithm, derivation of,

31 5 - 318

establishing features through, 258·

260

in symbolic and subsymbolic paradigms, 261·262 Learning machine (Minsky and Edmonds), 153-154 Learning paradigms, 54-57

Learning rule, general form of, 52·53

Leavitt, R. Y., 351, 563 Le Cun, Y .• 322, 512 Lehiste, 1.,61. 565

Kullback, S., 294, 511 Lenneberg, E. H .• 473, 565 Kuno, M .• 336, 565 Copyrighted MMMfa/U. <sup>R</sup>.• 63, 559

Letter-perception model. See Interactive activation model of

word perception

LeVay, S .• 350. 353. 355,361.374. 497.500.563. 565. 566 Levels of analysis. 121·129. See also Isomorphism hypothesis Marr's levels. 122-124

other notions of levels. 124-127 reductionism and emergent properties. 127-129 Levin. 1. A.. 43. 85, 512. 90. 566 Levin's proteus model. 43 Levine. R. D., 227, 512 Lewis. B. , 341, 566 Lewis, C. H. , 251. 512 LGN cells. 480

Liberman. A. M .• 61. 62. 73. 92, 94, 95. 473. 560. 566. 569, 573. 576

Lichten. W .• 63. 569 Licklider, 1. C. R., 106, 566 Lieberman. F .• 489. 557 Limited fan-out. See Fan-out. limited; Standard pattern

associators

Limited increment hypothesis. 520.

See also Amnesia Lin, C. S .• 351. 560 Lindsay. P. H .• 8. 206. 512

Linear algebra

matrices and linear systems

basis for vector space, change of.

413

description of, 385-386 eigenvectors and eigenvalues ,

399·403 linearity. 393 matrices. 386-390 matrix inverses , 410-413 matrix multipl ication and multilayer systems. 395-399 one-layer PDP system. 390-393 PDP systems, 2 examples. 406-

410

vectors

basic operations of. 367-370 concepts , use of in analysis of simple PDP model, 383-385 description of, 365·366 inner products, 375 linear combinations and independence, 370-373 vector spaces, 374-375 Linear models, simple, 61-63. See

also Linear algebra auto-associator version of, 63 pattern associator, 63 simple linear associator, 62

weaknesses of, 63

Linear threshold units, 63-66, 425, 224. See also Activation

functions perceptron, 65 XOR function, 64-65

LISP programming language, 124 use of in P3 system, 492, 496-497 Llinas, R., 364, 370, 558, 566 Local representations, 77, 85, 96 Locus coeruleus (LC), 476-4 77, 493

Loeb, G. E., 383, 566

Loftus , E. F., 85, 508, 208, 566 Logical computation. notion of. See

Activation functions Logogen model, 43, 192-193 Loomis. L. H .• 402, 566

Lorente de No, R., 344. 358, 566

Lovins. 1. B., 62, 560 Luce, R. D., 75. 93. 195, 566 Luce choice rule. appl ication of, 90·

91, 195

Lund, J. S., 35 7, 358, 361, 366, 374,

566,5 73

Lund, R. D., 336, 35 7, 566 Luria, A. R., 41,79, US, 512 Luria's dynamic functional system,

41

Lynch , J. c., 366, 56 7

transposes and the outer product, Ma , S. -K., 388, 56 7 403-405 Macchi, G., 350, 567 nonlinear systems, 418·420 Macrodescription of harmony

Copyrighted Mat9lrgranism. 246-258

problem sol ving for, 246-247 one-layer PDP system. 390-3Q� productions and expertise, 251-253 transposes and outer prod uct, 403-

schemata. 253 405

approximation , 256-258 Matrix inverses. 410-413. See also for rooms, 255-256 Matrices and linear systems two-choice model , 253-255 Matrix mapping, algebraic properties MacWhinney, 8.. 273, 274, 554.56 <sup>7</sup>of, 389-390. See also Matrices

Macy, A., 481, 567 and linear systems

Magistretti. P. 1., 364, 569 Maunsell.1. H. R., 345,352,353,

Mann, V. A., 95, 118. 56 7 356. 500. 568, 577

Mapping from expressions. 439. See Maximum-likelihood model , 292-294 also Boolean function McCarthy, J., 157-158, 512 Marcus. M. P., 317, 324, 567 McClelland, 1. L.. 20, 22, 24, 27, 28. Marin, O. S. M., 134,514 42,43,71 , 79, 85, 120, 121, 133, Marin-Padilla, M., 363, 567 177, 199,202,216,321,512. Mark, R. F., 490, 555 514,8,58.59, 63, 69. 71. 75, 77. Marr, D., 18, 19.20.42, 113, 116, 80,90, 119, 123, 126, 127. 130. 117.121-122,123,196,283,512, 131,135,138,140, 141, 142, 63, 119.378,567 170,172,195, 217, 370,380, Marr's levels of analysis. 122-124. 381,383, 394.532.558,559,

See also Levels of analysis 568, 574

Marrocco, R. T., 480, 567 McClurkin, 1. W .• 480, 567 Marshall, 1. c., 102,508 McConkie, G. W., 161, 163, 568,

Marshall. W. H .• 352, 579 573

Marslen-Wilson, W. D., 43. 512. 63. McCulloch, W. S., 152,424.512 77, 79,80, 97, 98, 99, 275, 56 7 McDermott, D., 432.568 Martin , K. A. c., 363, 56 7, 576 McGeoch, L. A .• 235, 508 Massaro, D. W .• 77, 81, 86. 94, 567. McGill. J. , 145, 575

568,5 71 McGuinness, E. , 374, 375. 553 Massey, T., 378. 560 McNaughton , B. L. , 448, 568 Matrices and linear systems. See also MD. See Monocular deprivation Linear algebra Medin, D. L., 171, 181,200.205.

basis for vector space, change of, 536, 568

413-418 Meditch , J. S., 456,512

descriptions of, 385-386 Memory, a distributed model of. See eigenvectors and eigenvalues, 399- also Distributed representations; 403 Memory, retrieving information

examples of PDP systems, 406-410 from

linearity, 393-395 conclusions on, 214-215

matrices, 386 detailed assumptions of, 176-182 addition of. 387-389 experimental results of simulations matrix mapping. algebraic repetition and familiarity effects,

properties of, 389-390 192-199

scalars, multiplication by, 386 representation of general and matrix inverses. 41 0-413 specific information, 199-206

Memory, a distributed model of Miller, W., 254

extensions of model (continued) Milner, B., 504, 506, 510, 514 , 569

interference and fan effects, 208- Minciacchi, D., 350, 56 7

209 Minsky , M., 65, 76, 96, 111-1l3, memory blends, 208 153, 154, 157-158, 160, 233, 29 1, regularities of behavior, 319,321, 334, 348, 354, 36 1, emergence of, 207 424, 513, 18, 19, 535,569 semantic memory , emergence of Mitchell, D. E., 475, <sup>558</sup>

from episodic traces , 206 Mitchinson, G., 296, 297, 468, 486,

general properties of, 173- 175 509

modular structure, 174-1 75 Mitra , S. K. , 455, 513 paltern of activation, 175 Miyuwaki , K., 473, 569

hidden units, augmenting model Models, simi larities and differences

with, 209-214 between, 4-5

model 's behavior, key aspects of, Modifications of patterns of

182- 192 interconnectivity in PDP models,

relation to basic concept in, 176 52-53. See also Learning theories of, 2-3, 170-1 71 Modular structure of memory, 79, Memory, retrieving information 174- 175. See also Memory , a from. See also Distribu ted distributed model of representations, fealures of; Molinari, M. , 35, 567

Memory, a distributed model of Monocular deprivation (MD) , 474,

content addressability, 25-29 475

de fault assignment, 29-30 00 shift under, <sup>48</sup> <sup>7</sup>

graceful degradation, 29 rearing, 489

spontaneous generalization , 30-31 Monocular occlusion (AM) , 474 Mercer, R. L., 293, 508 Monotonicity concepts for acti vation Merzenich , M. M., 385, 386, 38 7, functions, 427-428. See also 563, 569 Activation functions Mesulam, M. M., 350, 569 Monotonicity-in-context , 426-428 Metropol is, N., 230, 277, 513 Montero, V. M., 351, 569 Meyer, D. R. , 352, 579 Moran, 1., 374 , <sup>558</sup> Michael , C. R. , 36 7, 368, 569 Morrell, F. , 379, 569

Micro- and macrolevels of harmony Morris, R. G. M., 434, 466, 569 theory, 203-204, 210 Morrison, J. H., 364, 569 Microdescription , problem-solving Morton , J., 43, 513, 97, 106, 192,

of, 246. See also Electricity 570

problem-solving Moser, M. G., 313, 570

Microfeature, 80-81 Motor control

Microfeatures, semantic. See example of multi ple constraints in,

Semantic microfeatures 4-6

Microstructure of cognition, 12-13 PDP models for, examples of, 13-

Miezin, l, 374, 375, 553 18

Miller, D. T. , 536, 564 finger movements in ski lled

Miller, G. A., 63, 106, 323, 566, 569 typing , 14-16

Miller, J. L., 62, 96, <sup>569</sup>reaching for objects, 16-18 Miller, J. P., 381, 382, 569, 575 Mountcastle, V. B., 356, 366, 373,

Moussouris , J., 289, 513 Mower, G. D., 478, 570

Mozer, M. C, 68, 127, 507, 513 , 125, 139, 140, 142, 5 70 Muller, R. U. , 435, 441, 445, 4<sup>4</sup> 7 ,

<sup>565</sup>, <sup>5</sup> 7<sup>0</sup>

Multiple linear regression and the delta rule, 457·458. See also Delta rule, analysis of Multiple simultaneous constraints, 4·9, 58, 124, 272·2 75 mutual constraints, 7 operation of in syntax and

semantics , 6-7 duplication of connection

information , need for to exploit ,

124

role assignment, use of in , 272-2 <sup>75</sup> simultaneous mutual constraints in

word recognition , 7-9

speech perception, role of in, 124 Munro, P. W., 42, 180, 508, 480, 487, 489, 495, 496, 555, 570

Murray, E. A. , 35 7, 570 Mutt, V., 364, 560

groups of neurons behavior in cerebral cortex, 365-366 neurons wi thout many spines, 361 neurons wi th spines, 360-361 si ngle·neuron behavior in cerebral

cortex, 364- 365 special cell types , 362-364 Network damage. See Damage ,

effects of on a network; Graceful

degradation

Neural and conceptual interpreta tion of PDP models. See PDP models ,

neural and conceptual interpretation of Neural-like processing systems , common learning paradigms in,

1 <sup>6</sup>0·16 1 Neural modeling, 130

Neural nets, biological limits on storage capacity of, 469-470. See

also Fan-out, limi ted Neural or unit level of

representation. See Pattern-based

coordinates , delta rule in Neural plasticity in cerebral cortex, 385·38 7. See also Cerebral

Nadel, L. , 435, 445, 510, 52 7, 571, cortex , questions about 576 computation in Nakatani, L., 6<sup>1</sup> , llO, 570 Neural plasticity and learning Nathan, P. W. , 514, 515 , 574 critical period , effect of Nativism vs. empiricism, 139·142 environment on, <sup>478</sup> Nauta, W. J. H. , 334, 342, 570 critical period of, 473·4 78 NE. See Norepinephrine ocular domi nance , measuring Necker cube, ll, 13, 16, 28. See also changes in, 474·4 76

Schemata, concept of plasticity· modulating processes ,

Neely, R. B., 63, 122, 573 476·4 78

Negation problem, 346-348. See also discussion of, 471·473, 494·499 Delta rule, generalized existing modification rules, state

Neisser, U., 81, 11 3. 11 6, 212 , 513 , dependence and, 494·496 515 global modulators as dynamic

Nelson, R. J. , 352, 354, 385, 477, variables, 496·497 569, 570 predictions for, 497·499 Nelson, S. B., 477, 554 experimental data on, comparison

Neocortical neurons, nature of with, 489·494

cell types in, 358·360 ocularity state and its effect on,

Neural plasticity and learning

(continued)

state dependencies of plasticity,

484-489

PDP models, learning rate as factor

in. 472

single-unit learning. focus on, 471

summary of, 501 terminology. 473

visual cortex, ideas described with respect to ocular dominance in,

499-501

Neural processing in cerebral cortex . 378-381. See also Cerebral cortex , questions about computation in

Neural realism, lack of in PDP models, 136-13 8

Neurons. See also Cerebral cortex, anatomy and physiology of; Neocortical neurons. nature of behavior in the cerebral cortex

isomorphism. failure of in, 411-414 quasi-linear systems with, 418-422

use of. 41 8-42 1

limited dynamic range and, 420-

42 1

subthreshold summation , 420 Nonpyramidal cel ls. 358-360. See also Neocortical neu rons, nature

of

Norepinephrine (NE) , 476-477, 478,

480, 493

as potential modulator of neural plasticity, 476-477. 491-494, 497

Normal rearing (NR) . 474 Norman, D. A .. 8, 9, 14, 15, 79, 11 6. 133, 206, 512, 513 , 51 4, 18, 19, 53 7, 539, 540. 543, 545, 555,

570, 574

Norris, D. , 63, 100, 57/ NR. See Normal rearing Nusbaum , H. C. 110. 57/

groups of neurons, 365-366 Ochs, M. T. , 385, 563 single neuron , 364-365 Ocular dominance (OD)

classical description of, 335-33 7 class, 484

connections between, in brain, histograms, 475, 479. 489, 490 132-133 shift, under MD. 48 7 communication among, 132 statistics , 474-4 76

continuous output of, 133- 134 Ocular dominance index. 481-483,

graceful degradation of. 134 484, 490

neocortical , 358-365 vs. responsivity plot, 491

number of, 131 Ocularity plane, power of as tool in speed of, 130-131 theoretical analyses of ocular systems, 131- 132 dominance plasticity, 483-484 treated as devices, 424-425 (see Ocularity state and its effect on also Boolean function) plasticity, 478-480. See also without many spines, 361 Neural plasticity and learning

with spines , 360-361 ideal neuron. 479-480 Newell , A., 108, 195, 513 ocularity plane, 480-484 Newsome, W. T. , 347, 356, 554, 568 plasticity, state dependence of,

Newtonian mechanics, 125 484-489

Nicholson , C. , 364, 566 OD. See Ocular dominance Ninteman , F. W., 493, 572 Oden, G. C. , 77, 275, 568, 571 Nonlinear systems. See also PDP ODH. See Ocular dominance,

models, neural and conceptual histogram

interpretation of 001. See Ocular dominance index dist ributed, natural competition in, 6-0HDA. See 6-hydroxydopamine 414-428 . d Oh�awa l., 481, 491, 560, 567 Copynghte Matena

Oja, E., 489, 557

O'Keefe, J., 434, 435, 445, 448, 466,

568, 569, 571 O'Kusky, 1., 348, 571 Olson, C. R. , 475, 571 Olson, L. , 350, 553

Olton, D. S., 435, 522, 554, 571 One-layer PDP systems , use of matrices in analyzing, 390-393 . See also Matrices and linear

systems

One-layer perceptron , 111 -113 limitations of, 11 2- 11 3 One-step system in interactive

models, 60

Optimization problems, use of linear programming for, 284-286 Organization of Behavior, 53, 152

Ortony, A., 19, 32, 574

Output units definition of, 48

as major aspect of PDP models, 46,

48 -49

system), 453-454

competitive learning methods (App.

B), 504-506

description of, 364, 488-489 major components of, 489-490 performance of, 500-501

plan for competitive learning, (App.

A), 502- 503

plan language of, 490, 49 1 -497 simplified diagrams of, 454-455,

464

simulation system, 497- 500 typical plan for, 490 Parity networks, 159, 160 Parity problem, 334-335 See also Delta rule, generali zed Parker, D. B., 322, 36 1, 513 PARSIFAL (Marcus ), 31 7 Parsing in PDP models, 31 7-323 Part/ whole hierarchy, 105

Past-tense acquisition , three stages of, 21 9-221. See also Verbs , learning past tenses of

Patte, P., 38 7, 556

Pattern of activation , in distributed

P3 system. See Parallel process model of memory, 175 programmer memory traces as change in

PABLO. See Reading, programmable weights, 176

blackboard model of mental state as, 176

Palay, S. L., 338, 571 prior, retrieval as reinstatement of,

Palmer, R. G. , 296, 511, 513 176

Palmer, S. E. , 11 7, 513 and response strength, 1 94- 195 Pandemonium, 42 Pattern association paradigm ,

Pandya, D. N., 352, 571, 573 defin ition of, 55

Papert, S., 65, 76, 96, 11 1-1 13, 158, Pattern associators. See also

160, 233, 29 1, 319, 32 1, 334, Programmable pattern associators; 348, 354, 361 , 424, 513 , 535, 569 Standard pattern associators; Paradigms of learn ing Verbs , learn ing past tenses of associative, 54-55 basic properties of, 33-37, 226-228

regularity discovery, 55 definition of, 161

Paradiso, M. A., 477, 554 learning regular and exceptional

Parallel distributed processing. See patterns in, 226

PDP. models, attractive properties of, 38

Parallel Models of Associative Memory, restrictions of, 228-233

33, 533 Pattern-based coordinates, delta rule

Pattern completion device, use of Boltzmann machine as, 289-290. See also Boltzmann machines Pattern of connectivity as major aspect of PDP models, 46, 49-5 1

Patterson , K. , 102, 508

PDP approach and cognitive science

conclusions on, 1 45- 146 natural applications of, 144

objections to

analysis, wrong level of, 121-127 cognitive approach, lack of, 120-

121

problems of, 543-545 new directions in, 545-546 strengths of, 535-539

description , levels of, 538-539 human information processing, essential properties of, 53 7

schema, 536-53 7 useful result of, 537-538 weaknesses of, 539-543

evaluative structure , need for,

541 -542

multiple systems, required for,

542-543

cognitive theories, constraints of, type-token problem , 539-540

129- 136 variables, 540-541

conscious knowledge and explicit PDP models, active representation

reasoning, 143- 145 in, 31 -40, 175-1 76 human vs. rat intelligence, 143 local vs . distributed, 32

lack of neural realism in, 136- 1 38 pattern associator models attractive

Marr's levels, 122-124 properties of, 38 nativism vs. empiricism, 139- 142 pattern associ ators, 33-37 other notions of levels, 124-127 structure of ensemble of patterns,

reductionism and emergent extracting, 39-40

properties, 127-129 PDP models and the brain. See also weaknesses of, 111- 120 Cerebral cortex , anatomy and status of various models, 144- 145 physiology of; Cerebral cortex , PDP approach , introduction to questions about computation in active representation in, 31 -40 relationship between, 328-330 as alternati ve to serial models, 12- models of neural mechanisms,

13 330-331

appeal of, 3-4, 10- 12 neurophysiology relevant to, 32 7-

examples of 328

memory retrieval , 25-31 PDP models, future directions for,

motor control , 13- 17 54 7-552

perception, 1 8-24 PDP models, general framework for history of, 41 -44 bottom-up processing, 57-59 local vs . distributed representation, conclusions on, 74-76

32-33 hierarchical organizations, 57 pattern associator models interactive models , 59-60 attractive properties of, 38 learning paradigms, 54-57 ensemble of patterns, extracting major aspects of, 46-54 structure from, 39-40 activation rule, 51-52 learning multiple patterns, 37-38 learning rule, 52-54 learning rules of, 36- 37 propagation rule, 51 workings of, 33-36 sigma-pi units, 73-74 PDP approach, reflections on synch ronous vs . asynchronous

versions, specific

brain state in a box model (BSB) ,

66·68

Feldman and Ballard ' s units , 72 Grossberg's units, 70·7 1 interactive activation model, 71 linear threshold units, 63·66 simple linear models, 61·63 thermodynamic models, 68·70 PDP models, learning rate as factor in, 472·4 73. See also Neural plasticity and learning PDP models , neural and conceptual

interpretation of conclusions on , 429·431 consideration of, 390·391 distributed nonlinear models, natural competition in, 424·429 as dynamical systems, 39 7·398 interpretations, 391 ·396

isomorphism hypothesis, fail ure of in nonlinear models, 422·424

kinematics , 400·403

kinematics and dynamics, 398·399

learning connections and

isomorphism of levels, 41 6·418 linear systems, isomorphism of

levels in, 41 1·413

localized damage and, 413·416 nonlinearity, quasi·linear systems

with, 418·422

pattern coordinates, 406·41 1 vector space, structure of, 403·406

Pearlmutter, B., 298 Pearson , 1. c., 385, 575 Peck, C. K. , 475, 571

Peptides in the cerebral cortex, 340

Perez, R. , 494, 571

Perception, examples of for PDP

models

familiar patterns and, 20· 23 novel patterns, completion of, 23·

25

stereoscopic vision, 18-20 Perceptrons (Rosenblatt) , 154· 15 8 convergence procedure, 41 ·42 , 65,

learning rule, 53, 154·15 7 parallel reorganizing elements,

158·159

Perceptua l processing, centrality of, 197·198. See also Harmony

theory

Perkel, D. H., 381, 571 Perkel, D. J., 381 , 571 Perrett, D. I., 368, 571 Perry, 1., 195, 508 Peterhans, E., 376, 578

Peters , A., 336, 338, 361, 363, 365,

559, 571, 572 Petersen , S. E. , 34 7, 554 Petitjean, F., 350, 555

Pettigrew, J. D., 476, 477, 487, 490,

492, 555, 564

Phase transitions, computational significance of, 236·237 Phill is, J. W. , 339, 565

Phoneme identification , factors

influencing

categorical perception, 84, 88·95 detectors, retuning of by context,

95· 97

lexical effects on , 77· 78 factors influencing, 78-80 simulations, summary of, 97 trading relations, 84·88

use of phonotactic rules in, 81 ·81 Phonemic restoration effect, 20

Piaget, J., 17, 19, 572 Piercy, M., 516, 51 7, 563 Pillsbury, W. B., 20, 513 Pinker, S., 21 7, 572 Pinto-Hamuy, T. , 352, 579 Pisoni, D. B., 60, 77, 92, 94, 100,

572, 574

Pitts, W, 15 2, 424, 512

Place-field model. See also Place recognition and goal location , biologically plausible models of

description of, 436-441 goal location , 449 properties of, 439·440

scope of, 448

shape and size, 445-446

Place recognition and goal location, biologically plausible models of � -coefficient constants (App.) ,

46 7-468

�-coefficient model , 460-466 � units (App.) , 469-4 70 conclusions on , 466 distal landmarks , 445

distributed view-field model, 449-

460

goal location, 449

location parameters, 441 -443 model , description of, 436-441 properties of, 439-440 place-field location , 446-448 place-field model, scope of, 448 place-field shape and size, 445-446

place recognition, 434-436 simulated experiments, 443-445 Plan language, as major component of P3 system, 489, 490, 49 1-497 creating P3 plan for, 493-497

features of, 490 functions of, 491-493 Plasticity, neural . See Neural

plasticity Podgorny, P., 501 , 575

Poggio, T., 18, 19, 20, 42, 117,512 , 513, 129, 365, 381, 383, 564,

565, 572 Poincare, H., 213, 513

Pollack, 1. B. , 277, 278, 31 1, 314,

578

reading. See Reading,

programmable blackboard model

of

Programmable pattern associators. See also Connection information distribution mechanism; Reading, programmable blackboard model

of

CA units, requirements of, 474-476 CID networks, requi rements of,

473-474

conclusions on, 486 discussion of, 485-486 distributed representations , simultaneous access to, 483-485 multiple pattern processing, 477-

478

overlapping, 478-480

simultaneous access , cost of, 481-

482

single pattern processing, 476-477 summary of CA requirements,

480-48 1

Proskauer, C. c. , 336, 361, 363, 571,

572 Prototypes

> coexistence of and repeated exemplars, 189- 192

learning from exemplars, 182-184 multiple, nonorthogonal, 184- 188 Pyramidal cells, 358-360. See also Neocortical neurons, nature of

Porrino, L. J., 350, 572 Quasi-linear activation function , 52, Poschel, B. P. H. , 493, 572 425. See also Activation

Posner, M. I., 171, 183, 200, 203, functions

572 Quasi-linear systems with Powell , T. P. S., 344, 348, 350, 351 , nonlinearity, 418-422. See also 352, 556, 560, 563, 564, 572, 5 73 Nonlinear systems; PDP models,

Prince, D. A., 364, 369, 557 neural and conceptual Principles of Neurodynamics, 41-42, interpretation of

154 Quasi-local interpretations, 394-395. Probabi lity theory, 209-2 10 See also PDP models, neural and Processing units, set of for PDP conceptual interpretation of

model , 46-48 Quasi-multilinear activation function ,

vs. one-unit-one-concepl 426. See also Activation

Rader, R. K., 497, 5 77 conspiracies of, /57-/59 Rail, W., 336, 381 , 382, 569, 572, Realizability theorem, 230-235 575 Receptor, cartoon of, 508-509. See

Ralston, H. 1., 336, 572 also Amnesia

Ramachandran , V. S., 478, 572 Recurrent networks, 354-360. See Ranck, J. B. , Jr., 435, 441 , 445, 44 7, also Delta rule, generalized 565, 570 performance of, 359-360 Random-dot stereograms, 18 sequence completion , learning of,

Rao, C. R. , 45 5, 513 358-359

Rauschecker, J. P. , 476, 478, 576 shift register problem and, 357-358 Rawlins, J. N. P., 434, 466, 569 Recursion, 11 9-120, 318-320 Rayner, K., 153 , 161 , 163, 164, 275, Reddy, D. R., 63, 107, 122, 573 560, 572, 573 Reductionism, PDP models as

Reading, programmable blackboard exercise in, 127- 129

model of (PABLO) Redundancy, benefits of, 472 . See ambiguous characters, 155-157 also Standard pattern associators

amount of feedback, effect of Reese, T. S., 336, 572

display length on, 159-161 Regularity detector paradigm, 1 61

cm computer simulation , results Reich , P. A. , 323, 573 of, 136- 141 Rei lly, D. L., 499, 573 conclusions on, 168-169 Reitboek, H. 1. P., 385, 573 connection information Relaxation searches, use of parallel distribution , networks to perform , 283-290 benefits of, 164- 166 learning, difficult and easy, 290-292 computer simulation of word maximum likelihood models, 292-

recognition using, 134-136 294

cost of, 166-167 optimization problems and, 284·

extensions of, 167-/68 286

mechanism, details of. /29-136 pattern completion, 289·290 (see also Programmable pattern . probabilistic decisions, use of to associators) escape from local minima, 287 -

description of, 2, 125 288

fixations, sequences of, /61-164 simulated annealing, appl ication of fixations, single and multiple, 153- to Hopfield nets , 288·289 155 Relaxation system, 135- 136 interactive activation model , Relearning, speed of in Boltzmann

bottom-up activations of word machines, 308-310

units in cm version of, /3 7-138 Repetition and famil iarity effects,

interference and crosstalk, 139-1 41 /92- /99

PABLO simulation model alternative interpretation of, 193-

coarse coding, 146-14 7 /94

details of, /5/-153 pattern action and response feedback, 147 strength, 194-195

focus of attention , shifting, /47- time-accuracy curves, effects of /5/ experimental variables on , /95-

.overlappirig slots in, 143-14 7 /99

Repp, B. H., 95, 118, 56 7, 573 Representation , distributed. See Distributed representation Representation of environment as major aspect of PDP models, 53-

54

Representation vector of harmony

theory, 213-214 Representational features of

harmony theoretical model , 213 -

214

Representation , learning of. See Boltzmann machines;

Competitive learning; Delta rule,

general ized

Representation , learning of new in

harmony theory

procedure and abstract features,

258-260

in symbolic and subsymbolic

paradigms , 26 1

Reproduction trial s , 520. See also

Amnesia

Residual learning skills in amnesia,

518-524.

Retinocentric feature detectors , 11 4 Retrograde amnesia, 505-506, 513-

515

Reverse learning , occurrence of,

See also Reading, programmable

blackboard model of Rolls, E. T., 368, 571 Rosch , E., 171, 573 Rose, J. E., 345, 573 Rosen, I., 357, 554

Rosenblatt, F. , 41 , Ill , ]52, 153 - 154, 15 5- 157, 158, 29 1, 424, 514,

226, 289, 535, 573

Rosenbluth, A. W., 230, 277, 513 Rosenbluth, M. N., 230, 277, 513

Rosenfeld, A., 285, 514 Rosenthal , M., 477, 561 Ross , B. H., 206, 554 RS. See Reverse suture Rubin, D. S., 293, 509 Rudnicky, A., 58, 55 7

Rule of propagation as major aspect of PDP models, 46, 51 Rumelhart, D. E. , 6, 9, 14, 15 , 20, 22, 24, 43, 71, 11 6, 1 20 , 121, 130, 133, 177, 199, 202, 216, 300, 32 1, 512, 51 4, 7, 18, 19, 31 , 59, 71 , 75, 77, 90, 170, 195, 21 7, 316, 380, 394, 532, 539, 540,

545, 568, 573, 574 Russell, W. R., 514, 515, 574

Ryder, 1. M., 7, 511

296-298 Sachs, M. B. , 383, 574 Reverse suture (RS) , 474, 48 7-488, Saffran, E. M., 134, 514 489 Said, S., 364, 560 Reverse suture paradigm, time Sakaski , K., 370, 558

course of, 489-490. See also Salasoo, A., 60, 100, 197, 198, 559,

Visual cortex, developmental 574

models of Sal vert , D., 350, 555 Ribak, C. E., 363, 572 Samuel , A. G. , 95, 160, 574 Riley, M. S. , 24 1, 264, 514, 515 Sanderson , K. 1., 480, 574 Rinzel, J., 381 , 382, 569, 575 Sayers , F. c., 324, 563

Ritz, S. A., 56, 406, 409, 410, 41 8, Scalars, multiplication by, 367, 386. 507, 1 73, 226, 311 ,399, 551, 554 See also Matrices and linear

Roberts, E. , 351, 562 systems ; Vectors Rock, I., 11 7, 5 1 4 Schacter, D., 519, 574

Rocke! , A. 1., 344, 573 Schaffer, M. M., 17/, 200, 536, 568 Rockland, K. S., 352, 366, 374, 573 Schank, R. c. , 6, 9, 201, 202 , 514,

Role-assignment model . See Case 18, 19, 3]4, 574

role assignment, model of Scheibel , A. B., 351 , 574 Role-specific letter un its, I�/J#fghted n§�P5WcfJ, M. E. , 351, 574

Sentence processing in PDP 418, 507, 173, 226, 311, 399,

networks. See Case role 55 J, 554

assignment, model of Simon, H. A., 24 1, 508 Sentence-structure (SS) Simple binary addition problem, representation , 283-286. See also 34 1 ·346. See also Delta rule,

Case role ass ignment, model of generalized

Sequential symbol processing, 106- Simulated annealing, 287, 288-289, 108 313. See also Cool ing schedule;

Sequential thought processes in PDP Harmony theory

models. See also Recurrent Simulation system environment, as networks; Sequential symbol major component of P3 system,

processing; Seriality, emergent 489 , 497-500

account of mental processing, Singer, W. , 476, 478, 575, 576 development of, 38-39 Single·level interactive model , 60 conclusions on, 53-57 Single-shot algorithms, 380. See also consciousness, contents of, 39 Cerebral cortex, questions about

control , problem of, 39-40 computation in

conversations , 42-44 Single-unit learning, focus on , 472. external representations and formal See also Neural plasticity and

reasoning, 44-48 learning important aspects of, 47 Singley, 209

role of language in, 47 Skirboll, L. R. , 350, 577 mental models and, 40-42 Slater, P. c., 513, 515, 576 mental simulations and practice, 42 Siobin, D. 1., 221, 24 7, 249, 250, summary of, 48 251 ,254,255, 256, 556 thinking, goal direction in, 48 Siowiaczek, L. M., 120, 571 tic-tac-toe example of, 48-53, 54 Smal l, S. L. , 277, 3/4, 55 7, 576

Seriality, emergent, 247 , 249. See Smith , D. c., 490, 565 also Electricity problem-solving Smith , E. , 24, 515 Settlage, P. H., 352, 579 Smith , P. T. , 24, 515

Shall ice, T., /45, 543, 570, 575 Smolensky, P. , 12 5, 24 1, 259, 264 , Shankweiler, D. , 92, 94, 566 277, 289, 447 , 514, 515 Shannon , C. E., 195, 267 , 515 Sokolov, 1. L., 274, 56 7 Sharma, v. K. , 477, 561 Somogyi , P. , 362, 363, 56 7, 576 Shaw, G. L., 385, 575 Spared learning in amnesia, 518-519, Shepard, R. N. , 198, 515, 501, 536, 521-524. See also Amnesia 575 Spear, P. D., 490, 565 Shepherd, G. M., 336, 381, 382, Speech , fundamental aspects for 572, 575 development of TRACE model Sherk , H., 350, 374, 566 architecture for TRACE model,

Sherman , S. M., 351, 560 importance of, 63-64

Shiffrin, R. M., 197, 198, 559, 574 context effects , left and right, 60

Shlaer, R., 494, 571 cues , sensitivity of, 62 Sickels, E. R. , 324, 563 lack of boundaries , 60, 61 Sigma-pi units, 72, 73·74, 426 speech signal , noise and generalized delta rule for, 353 indeterminacy in, 62-63

Spencer, W. A., 364, 576 Spoehr, K. , 24, 515

Spoken word recognition , study of,

97- 98

Spontaneous learning (Rosenblatt) ,

15 5- 156

Squashing function , 425 , 485-489. See also Activation functions; Ocularity state and its effect on

plasticity

Squire, L. R., 505 , 510, 513, 515, 516, 51 7, 525, 52 7, 556, 576 Standard pattern associators. See also Programmable pattern associators

conclusions on , 486 degradation, effects of, 472 limited fan-out, effects of, 468-472 programmable nets , resource requirements, 465-468 redundancy, benefits of, 472 resource requirements of, 460,

46 1 -465

computations of, 463-465 Stanford, L. R., 351, 560

State of activation as major aspect of

PDP model , 46, 48 State space (S) , 398·399. See also Kinematics of the dynamical

system

coordinate system for, 400 of general nonlinear activation

model, 402 pattern view of, 406

unit coordinates for, 400-401 Statistical learning, analyzing case of, 453·457. See also Delta rule,

analysis of Steedman, M., 274, 55 7 Sternberger, 1. P., 248, 576 Sternberg, S. , 13 3, 51 5, 402, 566 Stevens, K. N. , 61, 168, 561, 5 76 Stimulus equivalence, problem of,

11 3·1 14

Stochastic generative model, 293·

294, 31 3 Stochastic units, 81

Structure in representations and processes , 104· 108 constituent structure, 105·106 sequential symbol processing, 106-

108

Studdert-Kennedy, M., 92, 94, 566,

576

Subsymbol ic and symbolic

paradigms, learning in, 261 ·262 Subth reshold summation , 420. See also Nonlinear systems, use of

Summerfield, Q., 85, 576 Sur, M. , 352, 354, 385, 56 9 , 570 Surround effects in visual cortex, 374-3 77. See also Cerebral cortex, questions about computation in

Sutton , R. S., 43, 53, 57, 444, 508, 515 , 383, 539, 554, 576 Swanson , L. W. , 350, 576 Swets, 1. A., 466, 510, 225, 56 1 Swinney , D. A., 109 , 577 Symbol ic and subsymbolic

paradigms, learning in, 26 1 ·262 Symmetry problem , 340·34 1. See also Delta rule, generalized Synapses in the cerebral cortex , basi c types , 338·339

Synaptic change, neurochemistry of, 50 7·509. See also Amnesia

Synchronous update vs . asynchronous update, 61 Syntactic processing in PDP models,

31 7·323

Szentagothai, 1., 351, 361, 577

T-C problem, 348·352. See also Delta rule, generalized Tabula rasa, 139, 141 Takeuchi, A., 489, 495, 553 Talbot , W. H., 366, 56 7 Tank, D., 389, 562 Tash, 1., 92, 572 Teitelbaum , P. , 142, 515 Tel ler, A. H., 230, 277, 513

Strang, G., 41 1, 422, 515 Tel ler, E., 230, 277, 513 Strange, W. , 473, <sup>569</sup>Copyrighted �aum , J. M. , <sup>63</sup> , <sup>554</sup>

Terminal parameters, 49 1 . See also 57-59

P3 system Top-down theoretical strategy Terrace, H. S., 183, 515 pursued in harmony theory, 196,

Theorems, harmony, (App.) 199-213

Boltzmann machines, second-order Torre, V., 11 7 , 51 3, 129, 365, 381,

observables and, 273 - 275 565, 572 harmony function H, cogn itive Touret , M., 350, 555

systems and, 267-268 Touretzky, D., 322, 548, 577 terminology, 268-269 TRACE model of speech perception overview of definitions, 264-267 compared to Fanty's parser, 321-

proofs of, 275-28 1 322

retrieving information from H, conclusions on , 120-121 269-272 deficiencies of, 119-120 storing information in H, 2 72-273 descri ption of, 2, 58-59, 64-68, Thermal equilibrium of network, 123- 124, 143, 153- 154

290- 29 1, 313 phoneme identification , factors

Thermodynamic limit, 239 influencing

Thermodynamic models, 68-70 categorical perception , 84, 88- 95 Boltzmann machines , 68, 69 lexical effects of on , 77-81 harmony theory, 68 phoneme detectors, retuning of

Thibadeau, R., 161, 549, 577 by context, 95- 97

Thomas , G. 8. , Jr., 276, 515 simulations, summary of, 97 Thompson , H., 6 0 , 6 3 , 8 4 , 107, 577 trading relations, 84-88

Thompson , R. F. , 383, 552, 561 use of phonotactic rules in, 81 -83 Th ree-dimensional plots , 429, 430- phoneme units, context-sensitive

438. See also Activation tuning of, 68-69

functions programmable version of, 167-168 Three-layer network, 97-98. See also speech, fundamental aspects for,

Arbitrary mapping, implementing 59-63

Th ree-level interactive model, 59 spoken word recognition, study of, Three-stage learning curve, 240-245. COHORT model , 98- 101, 102,

See also Verbs , learning past 103

tenses of word segmentation , lexical basis

Th reshold logic unit, 425. See also of, 106-115

Acti vation functions word segmentation simulations,

Tic-tac-toe, as example of sequential summary of, 115-11 7

thought processes in PDP models , successes, summary of, 117-/19 48-53, 54 TRACE I model , 69, 70- 71 , 75, 76 Tigges , J., 348, 577 TRACE II model , 69, 70, 71 - 75,

Tigges, M. , 348, 577 76, 102, 110 Timing Travis, A. M., 352, 579 problem of, 378-380 Tri bus , M., 227, 512 of sensory stimulation , 383-384 T'so, D. Y., 385, 577 temporal binding, 383-385 Turner, M. R., 385, 560 Timney, B. N., 475, 558 Tversky , A., 499, 560

TombOl , T., 35/, 36 /, 363, 577 Two-dimensional space, inner Top-down approach , 12 3 products in, 380-382. See also

Top-down processing syst��ghted MateYfifl.ors

Verbs, regular, 245-246 , 24 7, 254- 257, 2j8-260. See also Verbs, learning past tenses of Verbrugge, R., 473, 569 Vertical and horizontal lines, experiments with, 184-190 Vertical ly hierarchical networks, 217

Videen, T. 0. , 497, 577

View-field model. distributed. See Distributed view- field model

Vincent, S. R., 350, 577

Visual cortex , developmental models

of. 489-494

alternating monocular occlusion ,

490, 491

norepinephrine, role of, 492-494 responsi vity, relation of to connectivity, 490, 492 reverse suture paradigm, time course of, 489-490 Visual system, 282-283. See also Boltzmann machines

Vital-Durand. F .• 497. 500. 555, 577

Voight, H. F. , 383, 574 Volman, S. F., 36 7, 574 von der Heydt. R .• 376. 578 von der Malsberg, C .• 42. 1 47. 15 2. 162, 164, 51 6. 384. 489. 494. 578 Von Neumann architecture. 534-535. See also PDP approach ,

reflections on

Von Neumann computer metaphor.

195

Vygotsky. L. S. , 43, 47, 578

Walker, A. E., 349, 578 Walker, E. C. T., 7, 511 Walker, J. A .• 435. 554 Wall. J T., 385, 569

Waltz, D. L., 277. 278, 3lI, 314.

578

Warren, R. M .• 20, 516 Webster. H. de F .• 338, 571 Weight-decay. use of in Boltzmann

machine. 298·299. Weller. R. E. , 345, 578

Welsh, A., 43 , 512 , 63, 77, 79, 80,

Werbl in , F. S., 336, 578 Wessels, J, 99, 104, 577 Westrum, L. E. , 336, 578

Whitteridge, D. , 354, 363, 558, 56 7,

576

Whittlesea, B. W. A., 171, Ill, 200, 201, 202, 203, 204, 556, 578 Wickelfeature representation

(Wickelgren) blurring, 238-239 details of, 234-239

Wickelgren , W. A., 62, 181 , 233, 510, 5lI, 527, 578 Wickelphones, 233-234, 236-238

Widrow, G., 3 2 1 , 516

Widrow- Hoff rule, 53, 291 -292 . See

also Delta rule

Wiener, 534

Wiesel, T. N., 340, 353, 356, 358, 366, 36 7, 373, 374, 385, 474, 475,483, 497, 500, 561, 562,

563, 566, 577, 578 Williams, R. J., 425, 516

Willshaw, D. J. , 42, 97 , 100, 460, 46 1, 465-466, 468, 472, 475, 516

Wi lls haw nets, 46 1 -465

comparison of to various kinds of local representation , 465 difficulties with, 465-468 Wi lson , H. R., 381, 389, 578 Winston, P. H., 32, 51 6, 524, 578

Wise, R. A., 493, 578 Wise, S. P., 35 7, 564 Wolverton , G. S., 161, 568 Wood, C. c., 102, 516, 551, 578 Woods , W., 11 9, 516. 322. 578 Woolsey, C. N., 352, 579

Woolsey , T. A., 355, 500. 558. 579 Word identification simulations in TRACE, summary of. lI5- 1 17 Word-order and semantic cues to role assignment. 293-300. See also Case role assignment. model

of

Word perception model (McClelland & Rumelhart). See Interactive activation model of word

Word segmentation , lexical basis of for TRACE, 106-1 15 identification , word, 112- 115 nonwords , end of, 11 1-112 short sentence example, J 15 word inputs, single and multiple,

107-1 11 Wu, T. Y., 350, 577 Wyatt, H. J., 500, 558

XOR problem, 319-32 1, 330-334. See also Delta rule, generalized

Yin, T. C. T. , 366, 56 7 Young, E. D., 383, 574

Zadeh , L. A., 423, 516 Zeki, S. M., 368, 374, 577, 579 Zipser, D., 97, 300, 51 4, 432, 43 7, 438, 440, 442, 443, 444, 44 7, 579

Zola, D., 161, 568

Zucker, S. W., 285, 286, 511, 514