text: search with metadata and semantic search is not discussed in this paper and further details can be found [1,8]. The main goal of this experimentation is to understand at what extent existing sim- ilarity techniques are able to support archaeologists in the their cataloging and classi- ﬁcation work and how these techniques are able to answer to the fragment similarity problem, discussing what is currently missing and what is necessary to be further de- veloped. One critical point when dealing with similarity assessment among archaeological fragments is that there is not a unique possible similarity criterion, but different aspects become relevant depending on what is the target of the similarity reasoning, and there- fore, what should be emphasized. For instance, the fragment’s thickness becomes an important search cue if the target is the reassembly with other parts, or details of the fragment’s decorations are relevant if the search is ﬁnalized to ﬁnd traces of an artist’s style in objects belonging to different collections. The visual search engine should be therefore designed with sufﬁcient ﬂexibility, giving the possibility to the users to tune and drive the search during their reasoning processes. The reminder of the paper is organized as follows. In Section we describe the case study in which the similarity questions of this paper take place. In Section we present the modeling that we adopted to address similarity evaluation, according to the requirements of the user group of the GRAVITATE project and to the characteristics of the fragments. In Section , we describe how the search engine was conceived as a combination of several the library of descriptors and how they act as independent ﬁlters on frag- ments. Finally, in Section , we draw some conclusions and sketch a plan for future development. 2 Similarity in the archaeological context Archeology is a challenging context for similarity assessment for various reasons. First of all, similarity is the guiding reasoning process for the analysis of artifacts and it involved all aspects which may contribute to the shape: geometry, color, textures, chemical properties of color pigments, and many more. Beside data related to the tangible essence of assets, artifacts come with all the contextual information, primarily in textual form, related to the archaeological description and documentation of the ﬁndings themselves. What complicates the computational model of similarity is the fact that, typically, assets are fractured, eroded, and their color faded. Moreover, pieces of fractured objects may be mixed-up. At the same time, there are a lot of visual cues that may suggest that fragments belong to the same object, and these cues are used jointly or in sequence to answer typical questions related either to re-assemblability (may these fragment match together?) or to re-association (could this fragment belong to that collection?) or simply to the stylistic analysis of the fragment itself (do these fragments share the same decoration?). In the initial GRAVITATE project phase, we worked with the archaeologists to understand what were the basic similarity criteria used in traditional studies, within the scope of the project case studies. The ﬁrst part of this section is dedicated to the criteria 2

text: search

text: with

text: metadata

text: and

text: semantic

text: search

text: is

text: not

text: discussed

text: in

text: this

text: paper

text: and

text: further

text: details

text: can

text: be

text: found

text: The

text: main

text: goal

text: of

text: this

text: experimentation

text: is

text: to

text: understand

text: at

text: what

text: extent

text: existing

text: sim-

text: ilarity

text: techniques

text: are

text: able

text: to

text: support

text: archaeologists

text: in

text: the

text: their

text: cataloging

text: and

text: classi-

text: ﬁcation

text: work

text: and

text: how

text: these

text: techniques

text: are

text: able

text: to

text: answer

text: to

text: the

text: fragment

text: similarity

text: problem,

text: discussing

text: what

text: is

text: currently

text: missing

text: and

text: what

text: is

text: necessary

text: to

text: be

text: further

text: de-

text: veloped.

text: One

text: critical

text: point

text: when

text: dealing

text: with

text: similarity

text: assessment

text: among

text: archaeological

text: fragments

text: is

text: that

text: there

text: is

text: not

text: a

text: unique

text: possible

text: similarity

text: criterion,

text: but

text: different

text: aspects

text: become

text: relevant

text: depending

text: on

text: what

text: is

text: the

text: target

text: of

text: the

text: similarity

text: reasoning,

text: and

text: there-

text: fore,

text: what

text: should

text: be

text: emphasized.

text: For

text: instance,

text: the

text: fragment’s

text: thickness

text: becomes

text: an

text: important

text: search

text: cue

text: if

text: the

text: target

text: is

text: the

text: reassembly

text: with

text: other

text: parts,

text: or

text: details

text: of

text: the

text: fragment’s

text: decorations

text: are

text: relevant

text: if

text: the

text: search

text: is

text: ﬁnalized

text: to

text: ﬁnd

text: traces

text: of

text: an

text: artist’s

text: style

text: in

text: objects

text: belonging

text: to

text: different

text: collections.

text: The

text: visual

text: search

text: engine

text: should

text: be

text: therefore

text: designed

text: with

text: sufﬁcient

text: ﬂexibility,

text: giving

text: the

text: possibility

text: to

text: the

text: users

text: to

text: tune

text: and

text: drive

text: the

text: search

text: during

text: their

text: reasoning

text: processes.

text: The

text: reminder

text: of

text: the

text: paper

text: is

text: organized

text: as

text: follows.

text: In

text: Section

text: we

text: describe

text: the

text: case

text: study

text: in

text: which

text: the

text: similarity

text: questions

text: of

text: this

text: paper

text: take

text: place.

text: In

text: Section

text: we

text: present

text: the

text: modeling

text: that

text: we

text: adopted

text: to

text: address

text: similarity

text: evaluation,

text: according

text: to

text: the

text: requirements

text: of

text: the

text: user

text: group

text: of

text: the

text: GRAVITATE

text: project

text: and

text: to

text: the

text: characteristics

text: of

text: the

text: fragments.

text: In

text: Section

text: we

text: describe

text: how

text: the

text: search

text: engine

text: was

text: conceived

text: as

text: a

text: combination

text: of

text: several

text: the

text: library

text: of

text: descriptors

text: and

text: how

text: they

text: act

text: as

text: independent

text: ﬁlters

text: on

text: frag-

text: ments.

text: Finally,

text: in

text: Section

text: we

text: draw

text: some

text: conclusions

text: and

text: sketch

text: a

text: plan

text: for

text: future

text: development.

text: Similarity

text: in

text: the

text: archaeological

text: context

text: Archeology

text: is

text: a

text: challenging

text: context

text: for

text: similarity

text: assessment

text: for

text: various

text: reasons.

text: First

text: of

text: all,

text: similarity

text: is

text: the

text: guiding

text: reasoning

text: process

text: for

text: the

text: analysis

text: of

text: artifacts

text: and

text: it

text: involved

text: all

text: aspects

text: which

text: may

text: contribute

text: to

text: the

text: shape:

text: geometry,

text: color,

text: textures,

text: chemical

text: properties

text: of

text: color

text: pigments,

text: and

text: many

text: more.

text: Beside

text: data

text: related

text: to

text: the

text: tangible

text: essence

text: of

text: assets,

text: artifacts

text: come

text: with

text: all

text: the

text: contextual

text: information,

text: primarily

text: in

text: textual

text: form,

text: related

text: to

text: the

text: archaeological

text: description

text: and

text: documentation

text: of

text: the

text: ﬁndings

text: themselves.

text: What

text: complicates

text: the

text: computational

text: model

text: of

text: similarity

text: is

text: the

text: fact

text: that,

text: typically,

text: assets

text: are

text: fractured,

text: eroded,

text: and

text: their

text: color

text: faded.

text: Moreover,

text: pieces

text: of

text: fractured

text: objects

text: may

text: be

text: mixed-up.

text: At

text: the

text: same

text: time,

text: there

text: are

text: a

text: lot

text: of

text: visual

text: cues

text: that

text: may

text: suggest

text: that

text: fragments

text: belong

text: to

text: the

text: same

text: object,

text: and

text: these

text: cues

text: are

text: used

text: jointly

text: or

text: in

text: sequence

text: to

text: answer

text: typical

text: questions

text: related

text: either

text: to

text: re-assemblability

text: (may

text: these

text: fragment

text: match

text: together?)

text: or

text: to

text: re-association

text: (could

text: this

text: fragment

text: belong

text: to

text: that

text: collection?)

text: or

text: simply

text: to

text: the

text: stylistic

text: analysis

text: of

text: the

text: fragment

text: itself

text: (do

text: these

text: fragments

text: share

text: the

text: same

text: decoration?).

text: In

text: the

text: initial

text: GRAVITATE

text: project

text: phase,

text: we

text: worked

text: with

text: the

text: archaeologists

text: to

text: understand

text: what

text: were

text: the

text: basic

text: similarity

text: criteria

text: used

text: in

text: traditional

text: studies,

text: within

text: the

text: scope

text: of

text: the

text: project

text: case

text: studies.

text: The

text: ﬁrst

text: part

text: of

text: this

text: section

text: is

text: dedicated

text: to

text: the

text: criteria


text: Color Color Color Color+Thickness Figure 8: The query is the leftmost model of each row. First three rows: examples of results by color histograms. Last row: the same query of the 3rd row, reﬁned also by thickness. fragment like the leftmost in the ﬁrst row of Figure 9, the roughness ﬁlter discards fragments that are a jagged decoration on the external skin, even after the relaxation of the similarity criteria. When, the use of multiple ﬁlters seems to be a bit restrictive (Figure 9(Middle row)), the relaxation option allows to add more similarity options (based on the selected criteria). Other examples of color matchings are shown in the Figures 10 and 11. In both cases, the fragments come from the Naukratis collections and are grouped by the cura- tors according to their provenance. In the example in Figure 10, using only color and relaxation on the automatic color threshold, we are able to select all the fragments in the dataset that belong to the same group. One of the peculiar characteristics of this group is that all its elements are made by a stone. In the example in Figure 11, the query results obtained with the color ﬁlter contain two false positives: the complete vase in the leftmost position in the second row and the fragment in the rightmost position in the second row. Anyway. the CH experts judged that these false positive elements are stylistically compatible with the elements of the group. The bottom row highlights the fragments that are also compatible by thickness. Addressing the overall shape matching problem is really challenging. When deal- ing with broken fragments of statues there is a large intra-class variability, often em- 11 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Color

label: paragraph

text: Color

text: Color

text: Color+Thickness

text: Figure

text: The

label: caption

text: query

text: is

text: the

text: leftmost

text: model

text: of

text: each

text: row.

text: First

text: three

text: rows:

text: examples

text: of

text: results

text: by

text: color

text: histograms.

text: Last

text: row:

text: the

text: same

text: query

text: of

text: the

text: row,

text: reﬁned

text: also

text: by

text: thickness.

text: fragment

text: like

text: the

text: leftmost

text: in

text: the

text: ﬁrst

text: row

text: of

text: Figure

text: the

text: roughness

text: ﬁlter

text: discards

text: fragments

text: that

text: are

text: a

text: jagged

text: decoration

text: on

text: the

text: external

text: skin,

text: even

text: after

text: the

text: relaxation

text: of

text: the

text: similarity

text: criteria.

text: When,

text: the

text: use

text: of

text: multiple

text: ﬁlters

text: seems

text: to

text: be

text: a

text: bit

text: restrictive

text: (Figure

text: row)),

text: the

text: relaxation

text: option

text: allows

text: to

text: add

text: more

text: similarity

text: options

text: (based

text: on

text: the

text: selected

text: criteria).

text: Other

text: examples

text: of

text: color

text: matchings

text: are

text: shown

text: in

text: the

text: Figures

text: and

text: In

text: both

text: cases,

text: the

text: fragments

text: come

text: from

text: the

text: Naukratis

text: collections

text: and

text: are

text: grouped

text: by

text: the

text: cura-

text: tors

text: according

text: to

text: their

text: provenance.

text: In

text: the

text: example

text: in

text: Figure

text: using

text: only

text: color

text: and

text: relaxation

text: on

text: the

text: automatic

text: color

text: threshold,

text: we

text: are

text: able

text: to

text: select

text: all

text: the

text: fragments

text: in

text: the

text: dataset

text: that

text: belong

text: to

text: the

text: same

text: group.

text: One

text: of

text: the

text: peculiar

text: characteristics

text: of

text: this

text: group

text: is

text: that

text: all

text: its

text: elements

text: are

text: made

text: by

text: a

text: stone.

text: In

text: the

text: example

text: in

text: Figure

text: the

text: query

text: results

text: obtained

text: with

text: the

text: color

text: ﬁlter

text: contain

text: two

text: false

text: positives:

text: the

text: complete

text: vase

text: in

text: the

text: leftmost

text: position

text: in

text: the

text: second

text: row

text: and

text: the

text: fragment

text: in

text: the

text: rightmost

text: position

text: in

text: the

text: second

text: row.

text: Anyway.

text: the

text: CH

text: experts

text: judged

text: that

text: these

text: false

text: positive

text: elements

text: are

text: stylistically

text: compatible

text: with

text: the

text: elements

text: of

text: the

text: group.

text: The

text: bottom

text: row

text: highlights

text: the

text: fragments

text: that

text: are

text: also

text: compatible

text: by

text: thickness.

text: Addressing

text: the

text: overall

text: shape

text: matching

text: problem

text: is

text: really

text: challenging.

text: When

text: deal-

text: ing

text: with

text: broken

text: fragments

text: of

text: statues

text: there

text: is

text: a

text: large

text: intra-class

text: variability,

text: often

text: em-

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: - early sixth century BC [15]. Most of these statues are fractured, shards are faded and eroded: in the project, we worked on 241 digital models of Salamis statues fragments. Another interesting case study is represented by the Naukratis collection at the British Museum. The peculiarity of this dataset is that curators provided over the years a series of rich comments, identifying fragments that belong to the same object and documenting the reasons for these conclusions. Those comments also reveal contra- dictions between curators and inconsistencies in the data. Based on the curator’s con- clusions, fragments that are likely to belong to the same object are grouped together, but they do necessarily join physically. The collection in our study is made of 72 digital scan of the fragments. Figures 1 and 2 show some examples of the fragments of our collection. Figure 1: Examples of fragments of Salamis terracotta ﬁgurines. 3 Conceptual modelling of the similarity engine in GRAV- ITATE In the ﬁeld of 3D model search and retrieval, a plethora of techniques have been developed to search for 3D objects according to similarity [3, 6, 34] or for part-based search and retrieval in 3D object databases [23, 28] or speciﬁc contexts, like product design [7,14]. When reasoning about similarity in a collection of fragmented artifacts, however, there are several aspects of interest and novelty with respect to the traditional 3D ob- ject retrieval operations. For instance, the nature of CH data calls for methods dealing with multimodal information in combination (e.g., geometry and texture), which is necessary to effectively group artifacts or their parts into meaningful clusters. This is partially in contrast with the current scenario: many state-of-the-art methodologies for comparing, retrieving, or classifying objects in repositories are based on a single 4 ##LTFigure##

text: early

text: sixth

text: century

text: BC

text: Most

text: of

text: these

text: statues

text: are

text: fractured,

text: shards

text: are

text: faded

text: and

text: eroded:

text: in

text: the

text: project,

text: we

text: worked

text: on

text: digital

text: models

text: of

text: Salamis

text: statues

text: fragments.

text: Another

text: interesting

text: case

text: study

text: is

text: represented

text: by

text: the

text: Naukratis

text: collection

text: at

text: the

text: British

text: Museum.

text: The

text: peculiarity

text: of

text: this

text: dataset

text: is

text: that

text: curators

text: provided

text: over

text: the

text: years

text: a

text: series

text: of

text: rich

text: comments,

text: identifying

text: fragments

text: that

text: belong

text: to

text: the

text: same

text: object

text: and

text: documenting

text: the

text: reasons

text: for

text: these

text: conclusions.

text: Those

text: comments

text: also

text: reveal

text: contra-

text: dictions

text: between

text: curators

text: and

text: inconsistencies

text: in

text: the

text: data.

text: Based

text: on

text: the

text: curator’s

text: con-

text: clusions,

text: fragments

text: that

text: are

text: likely

text: to

text: belong

text: to

text: the

text: same

text: object

text: are

text: grouped

text: together,

text: but

text: they

text: do

text: necessarily

text: join

text: physically.

text: The

text: collection

text: in

text: our

text: study

text: is

text: made

text: of

text: digital

text: scan

text: of

text: the

text: fragments.

text: Figures

text: and

text: show

text: some

text: examples

text: of

text: the

text: fragments

text: of

text: our

text: collection.

text: Figure

text: Examples

text: of

text: fragments

text: of

text: Salamis

text: terracotta

text: ﬁgurines.

text: Conceptual

text: modelling

text: of

text: the

text: similarity

text: engine

text: in

text: GRAV-

text: ITATE

text: In

text: the

text: ﬁeld

text: of

text: model

text: search

text: and

text: retrieval,

text: a

text: plethora

text: of

text: techniques

text: have

text: been

text: developed

text: to

text: search

text: for

text: objects

text: according

text: to

text: similarity

text: or

text: for

text: part-based

text: search

text: and

text: retrieval

text: in

text: object

text: databases

text: or

text: speciﬁc

text: contexts,

text: like

text: product

text: design

text: When

text: reasoning

text: about

text: similarity

text: in

text: a

text: collection

text: of

text: fragmented

text: artifacts,

text: however,

text: there

text: are

text: several

text: aspects

text: of

text: interest

text: and

text: novelty

text: with

text: respect

text: to

text: the

text: traditional

text: ob-

text: ject

text: retrieval

text: operations.

text: For

text: instance,

text: the

text: nature

text: of

text: CH

text: data

text: calls

text: for

text: methods

text: dealing

text: with

text: multimodal

text: information

text: in

text: combination

text: (e.g.,

text: geometry

text: and

text: texture),

text: which

text: is

text: necessary

text: to

text: effectively

text: group

text: artifacts

text: or

text: their

text: parts

text: into

text: meaningful

text: clusters.

text: This

text: is

text: partially

text: in

text: contrast

text: with

text: the

text: current

text: scenario:

text: many

text: state-of-the-art

text: methodologies

text: for

text: comparing,

text: retrieving,

text: or

text: classifying

text: objects

text: in

text: repositories

text: are

text: based

text: on

text: a

text: single

text: ##LTFigure##


text: Figure 6: A fragment and the corresponding, concatenated color histogram with respect to the L, a and b channels. Figure 7: Left: The non-normalized D signatures of two fragments; Right: the persis- 2 tence diagram with respect the average, geodesic distance of a fragment of feet. of the artifacts (statues are often at different scales), the presence of cracks and missing parts, the deﬁnition of a geometric signature is an hard challenge. The quite limited number of elements per class also prevents the use of learning tech- niques. At the end, we opted for a combined shape description that mixes rough ﬁlters with scale invariant and non-rigid descriptions. Namely, we adopted a combination of multiple global shape descriptors, like compactness and the min- imal bounding box aspect ratio [10], the spherical harmonics indices [16], a non- normalized variation of the shape distributions [24] and the persistences spaces computed according to the average geodesic distance and the distance from the main axis of an object [5]. Figure 7 represents two histograms corresponding to the non-normalized distribution on the shape of the D chord descriptor and 2 the persistence diagram with respect to the average of the shape geodesic dis- tances [13]. All the distances adopted in this setting are metrics in the descriptors space. 3.2 Modelling fragments The Salamis 3D models are acquired by laser scans, while the Naukratis models are obtained with photogrammetric scans. All these 3D models are part of the GRAV- ITATE use cases [1]. Each 3D model is represented as a 3D mesh, equipped with colorimetric information on the vertices. 8 ##LTFigure## ##LTFigure##

text: Figure

text: A

text: fragment

text: and

text: the

text: corresponding,

text: concatenated

text: color

text: histogram

text: with

text: respect

text: to

text: the

text: L,

text: a

text: and

text: b

text: channels.

text: Figure

text: Left:

text: The

text: non-normalized

text: D

text: signatures

text: of

text: two

text: fragments;

text: Right:

text: the

text: persis-

text: tence

text: diagram

text: with

text: respect

text: the

text: average,

text: geodesic

text: distance

text: of

text: a

text: fragment

text: of

text: feet.

text: of

label: list

text: the

text: artifacts

text: (statues

text: are

text: often

text: at

text: different

text: scales),

text: the

text: presence

text: of

text: cracks

text: and

text: missing

text: parts,

text: the

text: deﬁnition

text: of

text: a

text: geometric

text: signature

text: is

text: an

text: hard

text: challenge.

text: The

text: quite

text: limited

text: number

text: of

text: elements

text: per

text: class

text: also

text: prevents

text: the

text: use

text: of

text: learning

text: tech-

text: niques.

text: At

text: the

text: end,

text: we

text: opted

text: for

text: a

text: combined

text: shape

text: description

text: that

text: mixes

text: rough

text: ﬁlters

text: with

text: scale

text: invariant

text: and

text: non-rigid

text: descriptions.

text: Namely,

text: we

text: adopted

text: a

text: combination

text: of

text: multiple

text: global

text: shape

text: descriptors,

text: like

text: compactness

text: and

text: the

text: min-

text: imal

text: bounding

text: box

text: aspect

text: ratio

text: the

text: spherical

text: harmonics

text: indices

text: a

text: non-

text: normalized

text: variation

text: of

text: the

text: shape

text: distributions

text: and

text: the

text: persistences

text: spaces

text: computed

text: according

text: to

text: the

text: average

text: geodesic

text: distance

text: and

text: the

text: distance

text: from

text: the

text: main

text: axis

text: of

text: an

text: object

text: Figure

text: represents

text: two

text: histograms

text: corresponding

text: to

text: the

text: non-normalized

text: distribution

text: on

text: the

text: shape

text: of

text: the

text: D

text: chord

text: descriptor

text: and

text: the

text: persistence

text: diagram

text: with

text: respect

text: to

text: the

text: average

text: of

text: the

text: shape

text: geodesic

text: dis-

text: tances

text: All

text: the

text: distances

text: adopted

text: in

text: this

text: setting

text: are

text: metrics

text: in

text: the

text: descriptors

text: space.

text: Modelling

text: fragments

text: The

text: Salamis

text: models

text: are

text: acquired

text: by

text: laser

text: scans,

text: while

text: the

text: Naukratis

text: models

text: are

text: obtained

text: with

text: photogrammetric

text: scans.

text: All

text: these

text: models

text: are

text: part

text: of

text: the

text: GRAV-

text: ITATE

text: use

text: cases

text: Each

text: model

text: is

text: represented

text: as

text: a

text: mesh,

text: equipped

text: with

text: colorimetric

text: information

text: on

text: the

text: vertices.

text: ##LTFigure##

text: ##LTFigure##


text: Published as a conference paper at ICLR 2018 Item pool: [5, 5, 1] A utilities: [8, 7, 1] B utilities: [8, 3, 2] Turn Agent Proposal 1 A [3, 4, 4] 2 B [4, 2, 0] 3 A [3, 4, 0] 4 B [4, 1, 0] Table 1: Actual transcript of two self-interested agents negotiating using the proposal channel. In this setting, no linguistic message is decoded at all. Agent sociality Proposal Linguistic Both None ± ± ± ± Fraction of joint reward 0.87 0.11 0.75 0.22 0.87 0.12 0.77 0.22 Self-interested 25th & 75th percentiles [0.81, 0.95] [0.61, 0.94] [0.81, 0.95] [0.62, 0.97] ± ± ± ± Turns taken 3.55 1.12 5.36 1.20 3.43 1.10 3.00 0.13 ± ± ± ± Fraction of joint reward 0.93 0.10 0.99 0.02 0.92 0.11 0.95 0.11 Prosocial 25th & 75th percentiles [0.89, 1.0] [1.0, 1.0] [0.88, 1.0] [0.93, 1.0] ± ± ± ± Turns taken 3.10 0.99 3.71 0.58 2.98 0.97 2.27 0.69 Table 2: Joint reward success and average number of turns taken for paired agents negotiating with random game termination, varying the agent reward scheme and communication channel. The re- sults are averaged across 20 seeds, with 128 games per seed. We also report the standard deviation ± as the number and the interquartile range. which is in line with the theoretical analysis of cheap talk in Crawford & Sobel (1982), who ﬁnd that, once agent interests diverge by a ﬁnite amount, no communication is to be expected. Our ﬁndings also corroborate the hypothesis found in the language evolution literature that cooperation is a prerequisite for the emergence of language (Nowak & Krakauer, 1999). We conjecture that this ﬁnding is partially due to the fact that the agents operate in a non-iterated environment (i.e., they have no explicit memory of previous interactions). In the classic prisoner’s dilemma game, rational agents playing a single-shot version of the game learn to always defect, whereas rational agents playing iterated prisoner’s dilemma can learn more mutually beneﬁcial co- operative strategies under the right conditions (Sandholm & Crites, 1996; Wunder et al., 2010). More sophisticated players will often have an incentive to take a short-term loss in reward to teach or punish another player for long-term gain in the repeated setting (Fudenberg & Levine, 1998). We suspect that, by iterating the game, there is an incentive for the cheap talk to be more veriﬁable, as there is a more credible threat of punishment if the accepted proposal and the linguistic utterance do not correspond. 4 EXPERIMENT 2 - CAN PROSOCIAL AGENTS LEARN TO COORDINATE? 4.1 EXPERIMENT DESCRIPTION In the previous experiment, we showed that self-interested agents can divide up items fairly such that each agent gets equal reward. However, this does not correspond to an optimal joint allocation where items go to the agent with the higher utility for that item (equivalently, maximising the sum of both agents’ rewards). The prosocial reward scheme aligns exactly with this optimal outcome, as learning to negotiate equates to ﬁnding the optimal joint item allocation. In this setting, there is an in-built bias towards communication, as solving the game optimally requires pooling hidden utilities. Table 2 reports the turns taken as well as the joint reward optimality, i.e., the sum of both agents’ rewards, divided by the maximum possible reward given the item context, where the latter corresponds to each agent getting all the items for which they have a higher utility. 4.2 RESULTS 6 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Published

text: as

text: a

text: conference

text: paper

text: at

text: ICLR

text: Item

label: table

text: pool:

text: A

text: utilities:

text: B

text: utilities:

text: Turn

text: Agent

text: Proposal

text: A

text: B

text: A

text: B

text: Table

text: Actual

text: transcript

text: of

text: two

text: self-interested

text: agents

text: negotiating

text: using

text: the

text: proposal

text: channel.

text: In

text: this

text: setting,

text: no

text: linguistic

text: message

text: is

text: decoded

text: at

text: all.

text: Agent

text: sociality

text: Proposal

text: Linguistic

text: Both

text: Fraction

text: of

text: joint

text: reward

text: Self-interested

text: percentiles

text: Turns

text: taken

text: Fraction

text: of

text: joint

text: reward

text: Prosocial

text: percentiles

text: Turns

text: taken

text: Table

text: Joint

text: reward

text: success

text: and

text: average

text: number

text: of

text: turns

text: taken

text: for

text: paired

text: agents

text: negotiating

text: with

text: random

text: game

text: termination,

text: varying

text: the

text: agent

text: reward

text: scheme

text: and

text: communication

text: channel.

text: The

text: re-

text: sults

text: are

text: averaged

text: across

text: seeds,

text: with

text: games

text: per

text: seed.

text: We

text: also

text: report

text: the

text: standard

text: deviation

text: as

text: the

text: number

text: and

text: the

text: interquartile

text: range.

text: which

text: is

text: in

text: line

text: with

text: the

text: theoretical

text: analysis

text: of

text: cheap

text: talk

text: in

text: Crawford

text: Sobel

text: who

text: ﬁnd

text: that,

text: once

text: agent

text: interests

text: diverge

text: by

text: a

text: ﬁnite

text: amount,

text: no

text: communication

text: is

text: to

text: be

text: expected.

text: Our

text: ﬁndings

text: also

text: corroborate

text: the

text: hypothesis

text: found

text: in

text: the

text: language

text: evolution

text: literature

text: that

text: cooperation

text: is

text: a

text: prerequisite

text: for

text: the

text: emergence

text: of

text: language

text: (Nowak

text: Krakauer,

text: We

text: conjecture

text: that

text: this

text: ﬁnding

text: is

text: partially

text: due

text: to

text: the

text: fact

text: that

text: the

text: agents

text: operate

text: in

text: a

text: non-iterated

text: environment

text: (i.e.,

text: they

text: have

text: no

text: explicit

text: memory

text: of

text: previous

text: interactions).

text: In

text: the

text: classic

text: prisoner’s

text: dilemma

text: game,

text: rational

text: agents

text: playing

text: a

text: single-shot

text: version

text: of

text: the

text: game

text: learn

text: to

text: always

text: defect,

text: whereas

text: rational

text: agents

text: playing

text: iterated

text: prisoner’s

text: dilemma

text: can

text: learn

text: more

text: mutually

text: beneﬁcial

text: co-

text: operative

text: strategies

text: under

text: the

text: right

text: conditions

text: (Sandholm

text: Crites,

text: Wunder

text: et

text: al.,

text: More

text: sophisticated

text: players

text: will

text: often

text: have

text: an

text: incentive

text: to

text: take

text: a

text: short-term

text: loss

text: in

text: reward

text: to

text: teach

text: or

text: punish

text: another

text: player

text: for

text: long-term

text: gain

text: in

text: the

text: repeated

text: setting

text: (Fudenberg

text: Levine,

text: We

text: suspect

text: that,

text: by

text: iterating

text: the

text: game,

text: there

text: is

text: an

text: incentive

text: for

text: the

text: cheap

text: talk

text: to

text: be

text: more

text: veriﬁable,

text: as

text: there

text: is

text: a

text: more

text: credible

text: threat

text: of

text: punishment

text: if

text: the

text: accepted

text: proposal

text: and

text: the

text: linguistic

text: utterance

text: do

text: not

text: correspond.

text: EXPERIMENT

text: CAN

text: PROSOCIAL

text: AGENTS

text: LEARN

text: TO

text: COORDINATE?

text: EXPERIMENT

text: DESCRIPTION

text: In

text: the

text: previous

text: experiment,

text: we

text: showed

text: that

text: self-interested

text: agents

text: can

text: divide

text: up

text: items

text: fairly

text: such

text: that

text: each

text: agent

text: gets

text: equal

text: reward.

text: However,

text: this

text: does

text: not

text: correspond

text: to

text: an

text: optimal

text: joint

text: allocation

text: where

text: items

text: go

text: to

text: the

text: agent

text: with

text: the

text: higher

text: utility

text: for

text: that

text: item

text: (equivalently,

text: maximising

text: the

text: sum

text: of

text: both

text: agents’

text: rewards).

text: The

text: prosocial

text: reward

text: scheme

text: aligns

text: exactly

text: with

text: this

text: optimal

text: outcome,

text: as

text: learning

text: to

text: negotiate

text: equates

text: to

text: ﬁnding

text: the

text: optimal

text: joint

text: item

text: allocation.

text: In

text: this

text: setting,

text: there

text: is

text: an

text: in-built

text: bias

text: towards

text: communication,

text: as

text: solving

text: the

text: game

text: optimally

text: requires

text: pooling

text: hidden

text: utilities.

text: Table

text: reports

text: the

text: turns

text: taken

text: as

text: well

text: as

text: the

text: joint

text: reward

text: optimality,

text: i.e.,

text: the

text: sum

text: of

text: both

text: agents’

text: rewards,

text: divided

text: by

text: the

text: maximum

text: possible

text: reward

text: given

text: the

text: item

text: context,

text: where

text: the

text: latter

text: corresponds

text: to

text: each

text: agent

text: getting

text: all

text: the

text: items

text: for

text: which

text: they

text: have

text: a

text: higher

text: utility.

text: RESULTS

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Published as a conference paper at ICLR 2018 transmit meaningful information using the linguistic channel, and thus do not appear to ground the symbols.5 6 EXPERIMENT 3: A SOCIETY OF AGENTS In more realistic scenarios, we learn and practice negotiation in environments with diverse agent populations and levels of prosociality. In this setting, maximising one’s reward requires identifying which agents are most useful in achieving one’s aims. In practice, this involves identifying which agents are the most prosocial, as they are both the most exploitable for self-interested agents, and the most cooperative for other prosocial agents. 6.1 EXPERIMENTAL PROTOCOL We trained a ﬁxed agent against a community of 10 agents, with varying proportions of selﬁsh and prosocial agents. For each training episode, we randomly sample one agent from the community, play a batch of negotiation games, and then update both agents. We do this for both ﬁxed agent A (the agent who goes ﬁrst) and for ﬁxed agent B (the agent who goes second). We experiment with either 1 or 5 prosocial agents, which communication channels are open, and also whether the agents are identiﬁable. At test time, we care primarily about whether self-interested agents can exploit prosocial agents and whether prosocial agents can cooperate with other prosocial agents. Hence, we only test the ﬁxed agent against the prosocial agents in the community. We generate 10 batches of 128 games each batch, and play each game with every pair of ﬁxed agent and community prosocial agent. We average the results across all pairs and all games, and show the mean and standard deviations of the rewards obtained in this experiment in Table 5. If opposing agents are identiﬁable (i.e. each agent has a name-tag), the ﬁxed agent receives this information as a one-hot vector, which is then used to look up an embedding table with trainable parameters with one embedding per opposing agent. This opponent embedding is then concate- nated with the encoded inputs, before being passed through the feedforward layer to get the agent’s hidden state. Intuitively, the induced agent embedding, that takes the form of machine theory of mind (Rabinowitz et al., 2018), should capture any information about the opponent’s behaviour (e.g., negotiating strategy) that, when taken into account, can help an agent maximize their reward. 6.2 ANALYSIS Identifying agents Our results show that providing agent ID information does help the ﬁxed agent achieve its aims. This effect is particularly pronounced for a selﬁsh ﬁxed agent; here, providing ID information uniformly improves performance. For cooperative agents, the results are mixed; for a ﬁxed agent A communicating via the linguistic channel, having agent ID information harms performance. Indeed, the only prosocial agents who beat the no communication baseline (which achieved an average joint optimal reward of 0.95) was when agent A was ﬁxed and ID information was not provided. We also computed the 2D PCA projections of the learnt opponent embeddings for a ﬁxed agent A under a variety of channels and prosocial levels (see Figure 4). Even in cases where the agent ID does not aid negotiation, we ﬁnd that the embeddings cluster according to the reward scheme of the other agent. This shows that the agents can distinguish different agents by their reward schemes from purely observing negotiating behaviour. Community linguistic phenomena In one of our experimental settings, a community of prosocial agents developed a language and were able to use this to achieve better negotiation success. This is the outcome in the starred cell in Table 5. For a visualisation of the bigram usage statistics in each interaction pair in this community, see Figure 6. Interestingly, the only situation when agents developed a language was when ID information was not provided. We suspect this result is due to a poor local optimum; since the prosocial agents can learn to use IDs to distinguish prosocial agents from the selﬁsh agents they can ignore the linguistic utterance. 5The high proposal accuracy is because the agents propose taking all the items on each turn, regardless of the messages exchanged, which correlates highly with the item pool itself. 9 ##LTLine## ##LTLine##

text: Published

text: as

text: a

text: conference

text: paper

text: at

text: ICLR

text: transmit

text: meaningful

text: information

text: using

text: the

text: linguistic

text: channel,

text: and

text: thus

text: do

text: not

text: appear

text: to

text: ground

text: the

label: section

text: EXPERIMENT

text: A

text: SOCIETY

text: OF

text: AGENTS

text: In

text: more

text: realistic

text: scenarios,

text: we

text: learn

text: and

text: practice

text: negotiation

text: in

text: environments

text: with

text: diverse

text: agent

text: populations

text: and

text: levels

text: of

text: prosociality.

text: In

text: this

text: setting,

text: maximising

text: one’s

text: reward

text: requires

text: identifying

text: which

text: agents

text: are

text: most

text: useful

text: in

text: achieving

text: one’s

text: aims.

text: In

text: practice,

text: this

text: involves

text: identifying

text: which

text: agents

text: are

text: the

text: most

text: prosocial,

text: as

text: they

text: are

text: both

text: the

text: most

text: exploitable

text: for

text: self-interested

text: agents,

text: and

text: the

text: most

text: cooperative

text: for

text: other

text: prosocial

text: agents.

text: EXPERIMENTAL

text: PROTOCOL

text: We

text: trained

text: a

text: ﬁxed

text: agent

text: against

text: a

text: community

text: of

text: agents,

text: with

text: varying

text: proportions

text: of

text: selﬁsh

text: and

text: prosocial

text: agents.

text: For

text: each

text: training

text: episode,

text: we

text: randomly

text: sample

text: one

text: agent

text: from

text: the

text: community,

text: play

text: a

text: batch

text: of

text: negotiation

text: games,

text: and

text: then

text: update

text: both

text: agents.

text: We

text: do

text: this

text: for

text: both

text: ﬁxed

text: agent

text: A

text: (the

text: agent

text: who

text: goes

text: ﬁrst)

text: and

text: for

text: ﬁxed

text: agent

text: B

text: (the

text: agent

text: who

text: goes

text: second).

text: We

text: experiment

text: with

text: either

text: or

text: prosocial

text: agents,

text: which

text: communication

text: channels

text: are

text: open,

text: and

text: also

text: whether

text: the

text: agents

text: are

text: identiﬁable.

text: At

text: test

text: time,

text: we

text: care

text: primarily

text: about

text: whether

text: self-interested

text: agents

text: can

text: exploit

text: prosocial

text: agents

text: and

text: whether

text: prosocial

text: agents

text: can

text: cooperate

text: with

text: other

text: prosocial

text: agents.

text: Hence,

text: we

text: only

text: test

text: the

text: ﬁxed

text: agent

text: against

text: the

text: prosocial

text: agents

text: in

text: the

text: community.

text: We

text: generate

text: batches

text: of

text: games

text: each

text: batch,

text: and

text: play

text: each

text: game

text: with

text: every

text: pair

text: of

text: ﬁxed

text: agent

text: and

text: community

text: prosocial

text: agent.

text: We

text: average

text: the

text: results

text: across

text: all

text: pairs

text: and

text: all

text: games,

text: and

text: show

text: the

text: mean

text: and

text: standard

text: deviations

text: of

text: the

text: rewards

text: obtained

text: in

text: this

text: experiment

text: in

text: Table

text: If

text: opposing

text: agents

text: are

text: identiﬁable

text: (i.e.

text: each

text: agent

text: has

text: a

text: name-tag),

text: the

text: ﬁxed

text: agent

text: receives

text: this

text: information

text: as

text: a

text: one-hot

text: vector,

text: which

text: is

text: then

text: used

text: to

text: look

text: up

text: an

text: embedding

text: table

text: with

text: trainable

text: parameters

text: with

text: one

text: embedding

text: per

text: opposing

text: agent.

text: This

text: opponent

text: embedding

text: is

text: then

text: concate-

text: nated

text: with

text: the

text: encoded

text: inputs,

text: before

text: being

text: passed

text: through

text: the

text: feedforward

text: layer

text: to

text: get

text: the

text: agent’s

text: hidden

text: state.

text: Intuitively,

text: the

text: induced

text: agent

text: embedding,

text: that

text: takes

text: the

text: form

text: of

text: machine

text: theory

text: of

text: mind

text: (Rabinowitz

text: et

text: al.,

text: should

text: capture

text: any

text: information

text: about

text: the

text: opponent’s

text: behaviour

text: (e.g.,

text: negotiating

text: strategy)

text: that,

text: when

text: taken

text: into

text: account,

text: can

text: help

text: an

text: agent

text: maximize

text: their

text: reward.

text: ANALYSIS

text: Identifying

text: agents

text: Our

text: results

text: show

text: that

text: providing

text: agent

text: ID

text: information

text: does

text: help

text: the

text: ﬁxed

text: agent

text: achieve

text: its

text: aims.

text: This

text: effect

text: is

text: particularly

text: pronounced

text: for

text: a

text: selﬁsh

text: ﬁxed

text: agent;

text: here,

text: providing

text: ID

text: information

text: uniformly

text: improves

text: performance.

text: For

text: cooperative

text: agents,

text: the

text: results

text: are

text: mixed;

text: for

text: a

text: ﬁxed

text: agent

text: A

text: communicating

text: via

text: the

text: linguistic

text: channel,

text: having

text: agent

text: ID

text: information

text: harms

text: performance.

text: Indeed,

text: the

text: only

text: prosocial

text: agents

text: who

text: beat

text: the

text: no

text: communication

text: baseline

text: (which

text: achieved

text: an

text: average

text: joint

text: optimal

text: reward

text: of

text: was

text: when

text: agent

text: A

text: was

text: ﬁxed

text: and

text: ID

text: information

text: was

text: not

text: provided.

text: We

text: also

text: computed

text: the

text: PCA

text: projections

text: of

text: the

text: learnt

text: opponent

text: embeddings

text: for

text: a

text: ﬁxed

text: agent

text: A

text: under

text: a

text: variety

text: of

text: channels

text: and

text: prosocial

text: levels

text: (see

text: Figure

text: Even

text: in

text: cases

text: where

text: the

text: agent

text: ID

text: does

text: not

text: aid

text: negotiation,

text: we

text: ﬁnd

text: that

text: the

text: embeddings

text: cluster

text: according

text: to

text: the

text: reward

text: scheme

text: of

text: the

text: other

text: agent.

text: This

text: shows

text: that

text: the

text: agents

text: can

text: distinguish

text: different

text: agents

text: by

text: their

text: reward

text: schemes

text: from

text: purely

text: observing

text: negotiating

text: behaviour.

text: Community

text: linguistic

text: phenomena

text: In

text: one

text: of

text: our

text: experimental

text: settings,

text: a

text: community

text: of

text: prosocial

text: agents

text: developed

text: a

text: language

text: and

text: were

text: able

text: to

text: use

text: this

text: to

text: achieve

text: better

text: negotiation

text: success.

text: This

text: is

text: the

text: outcome

text: in

text: the

text: starred

text: cell

text: in

text: Table

text: For

text: a

text: visualisation

text: of

text: the

text: bigram

text: usage

text: statistics

text: in

text: each

text: interaction

text: pair

text: in

text: this

text: community,

text: see

text: Figure

text: Interestingly,

text: the

text: only

text: situation

text: when

text: agents

text: developed

text: a

text: language

text: was

text: when

text: ID

text: information

text: was

text: not

text: provided.

text: We

text: suspect

text: this

text: result

text: is

text: due

text: to

text: a

text: poor

text: local

text: optimum;

text: since

text: the

text: prosocial

text: agents

text: can

text: learn

text: to

text: use

text: IDs

text: to

text: distinguish

text: prosocial

text: agents

text: from

text: the

text: selﬁsh

text: agents

text: they

text: can

text: ignore

text: the

text: linguistic

text: utterance.

text: high

text: proposal

text: accuracy

text: is

text: because

text: the

text: agents

text: propose

text: taking

text: all

text: the

text: items

text: on

text: each

text: turn,

text: regardless

text: of

text: the

text: messages

text: exchanged,

text: which

text: correlates

text: highly

text: with

text: the

text: item

text: pool

text: itself.

text: ##LTLine##

text: ##LTLine##


text: [12] Sridhar Ramaswamy, Pablo Tamayo, Ryan Rifkin, Sayan Mukherjee, Chen-Hsiang Yeang, Michael An- gelo, Christine Ladd, Michael Reich, Eva Latulippe, Jill P Mesirov, et al., “Multiclass cancer diagnosis us- ing tumor gene expression signatures,” Proceedings of the National Academy of Sciences, vol. 98, no. 26, pp. 15149–15154, 2001. [13] Eng-Juh Yeoh, Mary E Ross, Sheila A Shurtleff, W Kent Williams, Divyen Patel, Rami Mahfouz, Fred G Behm, Susana C Raimondi, Mary V Relling, Anami Patel, et al., “Classiﬁcation, subtype discovery, and prediction of outcome in pediatric acute lymphoblastic leukemia by gene expression proﬁling,” Cancer cell, vol. 1, no. 2, pp. 133–143, 2002. [14] Lixin Sun, Ai-Min Hui, Qin Su, Alexander Vortmeyer, Yuri Kotliarov, Sandra Pastorino, Antonino Passaniti, Jayant Menon, Jennifer Walling, Rolando Bailey, et al., “Neuronal and glioma-derived stem cell factor induces angiogenesis within the brain,” Cancer cell, vol. 9, no. 4, pp. 287–300, 2006. [15] Robert Nakayama, Takeshi Nemoto, Hiro Takahashi, Tsutomu Ohta, Akira Kawai, Kunihiko Seki, Teruhiko Yoshida, Yoshiaki Toyama, Hitoshi Ichikawa, and Tadashi Hasegawa, “Gene expression analysis of soft tissue sarcomas: characterization and reclassiﬁcation of malignant ﬁbrous histiocytoma,” Modern pathology, vol. 20, no. 7, pp. 749–759, 2007.

label: reference

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text: Ramaswamy,

text: Pablo

text: Tamayo,

text: Ryan

text: Rifkin,

text: Sayan

text: Mukherjee,

text: Chen-Hsiang

text: Yeang,

text: Michael

text: An-

text: gelo,

text: Christine

text: Ladd,

text: Michael

text: Reich,

text: Eva

text: Latulippe,

text: Jill

text: P

text: Mesirov,

text: et

text: al.,

text: “Multiclass

text: cancer

text: diagnosis

text: us-

text: ing

text: tumor

text: gene

text: expression

text: signatures,”

text: Proceedings

text: of

text: the

text: National

text: Academy

text: of

text: Sciences,

text: vol.

text: no.

text: pp.

text: Eng-Juh

text: Yeoh,

text: Mary

text: E

text: Ross,

text: Sheila

text: A

text: Shurtleff,

text: W

text: Kent

text: Williams,

text: Divyen

text: Patel,

text: Rami

text: Mahfouz,

text: Fred

text: G

text: Behm,

text: Susana

text: C

text: Raimondi,

text: Mary

text: V

text: Relling,

text: Anami

text: Patel,

text: et

text: al.,

text: “Classiﬁcation,

text: subtype

text: discovery,

text: and

text: prediction

text: of

text: outcome

text: in

text: pediatric

text: acute

text: lymphoblastic

text: leukemia

text: by

text: gene

text: expression

text: proﬁling,”

text: Cancer

text: cell,

text: vol.

text: no.

text: pp.

text: Lixin

text: Sun,

text: Ai-Min

text: Hui,

text: Qin

text: Su,

text: Alexander

text: Vortmeyer,

text: Yuri

text: Kotliarov,

text: Sandra

text: Pastorino,

text: Antonino

text: Passaniti,

text: Jayant

text: Menon,

text: Jennifer

text: Walling,

text: Rolando

text: Bailey,

text: et

text: al.,

text: “Neuronal

text: and

text: glioma-derived

text: stem

text: cell

text: factor

text: induces

text: angiogenesis

text: within

text: the

text: brain,”

text: Cancer

text: cell,

text: vol.

text: no.

text: pp.

text: Robert

text: Nakayama,

text: Takeshi

text: Nemoto,

text: Hiro

text: Takahashi,

text: Tsutomu

text: Ohta,

text: Akira

text: Kawai,

text: Kunihiko

text: Seki,

text: Teruhiko

text: Yoshida,

text: Yoshiaki

text: Toyama,

text: Hitoshi

text: Ichikawa,

text: and

text: Tadashi

text: Hasegawa,

text: “Gene

text: expression

text: analysis

text: of

text: soft

text: tissue

text: sarcomas:

text: characterization

text: and

text: reclassiﬁcation

text: of

text: malignant

text: ﬁbrous

text: histiocytoma,”

text: Modern

text: pathology,

text: vol.

text: no.

text: pp.


text: 4 which approximately characterizes the changing rate of the (cid:107) (cid:107) norm of u in a deterministic manner. Then, we have the t following assumption of the mean ﬁeld dynamics: Assumption 1: We have the following assumptions for the 0.5 dynamics of DNN with random linear mappings: 0 • When β > 1 for all sufﬁciently large t, the norm of t (cid:80) -0.5 u is expected to increase exponentially, thus resulting in t -1 chaotic behavior. -1.5 20 • When t and n are sufﬁciently large, we have n x2 = -2 1015 C, where C is a time-invariant constant. i=1 ti -2.5 5dimension Given Assumption 1, we have the following theorem pro- 14 12 10 8 6 4 2 0 0 viding a condition for the chaotic behavior with random linear σ (dB) mappings. The proof of is given in Appendix . The justiﬁcation Fig. 2: Lyapunov exponent λ for vector deterministic dynam- for the assumption can also be found therein. 1 ics with Gaussian matrices. Theorem 3: Given Assumption 1, we have λ > 1 for the 1 DNN(cid:32)dynamics wi(cid:32)th G(cid:114)aussi(cid:32)an r(cid:114)ando(cid:33)m(cid:33)l(cid:33)inear mappings, if 12 -2 -1.5 -1 -0.5 0 − 2 2 10 1 4 tanh σ h σ dσ2 > 1, (14) π π 0 8 -1.5 -1 -0.5 equwahtieorne zh(x=) tiasnthh(exzn)on(zwehroereanzd ispotshietivuenksnooluwtinonwhtoilethxe 6 is the parameter). 4-0.5-1 0 Remark 2: The inequality (14) is based on mean dynamics -0.5 analysis and Assumption 1. However, it shows that a higher di- mensional dynamics is more possible to be chaotic. Moreover, 2 a too large or too small σ2 disables (14). Both coincide the 0 2 4 6 8 10 12 14 16 18 20 observations in the numerical results. An explicit expression dimension for the function h in terms of inﬁnite series can also be found Fig. 3: Contour of Lyapunov exponent for vector deterministic in Appendix . dynamics with Gaussian matrices. IV. CAPABILITIES OF CLASSIFICATION AND GENERALIZATION In this section, we apply the metrics characterizing the u = S (x )W u − , (11) t t t t t 1 dynamics of DNN to the analysis of classiﬁcation and gen- eralization capabilities. and that of the system state x : t A. Classiﬁcation Capability We ﬁrst deﬁne the complexity of classiﬁcation problem and xt = tanh (Wtxt−1 + bt) . (12) then relate it to the topological entropy of DNN dynamics. We deﬁne the complexity of classiﬁcation, which facilitates In Theorem 2, we design the mappings such that the the subsequent analysis of DNN classiﬁcation. The philosophy dynamics of u and x are decoupled. However, in the case t t is to measure how difﬁcult it is to separate the samples from of Gaussian random mappings, the two dynamics are coupled, different classes. There have been some studies on how to since the matrix S is determined by x . In order to obtain an t t deﬁne the complexity of pattern recognition [4], [7], [8], analytic result, we apply the approach of mean ﬁeld analysis [11]. However, they are mainly focused on the nonlinearity in statistical mechanics by ﬁxing the impact of x on u ; i.e., t t of classiﬁer and the overlap of features, which is signiﬁcantly we consider the average behaviors of the random variables. different from our proposed deﬁnition. In more details, we approximate the dynamics of the tangent For deﬁning the complexity of classiﬁcation, we ﬁrst need vector u in u = S(cid:2) (x )W u − using (cid:3) t t t t t t 1 the following deﬁnition of set radius. R (cid:107) (cid:107) Deﬁnition 2: For a connected and compact set A in d, β = E E [St((cid:107)xt)] W(cid:107)tut−1 2 , (13) we deﬁne its radius as r(A) = supx,y{(cid:107)x − y(cid:107)|x, y ∈ A}. t E [ ut−1 2] Then, {we de|ﬁne∈the}radius of a partition P of A as r(P) = max r(A) A P . A λ )Bd( σ xam ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: which

text: approximately

text: characterizes

text: the

text: changing

text: rate

text: of

text: the

text: norm

text: of

text: u

text: in

text: a

text: deterministic

text: manner.

text: Then,

text: we

text: have

text: the

text: t

text: following

text: assumption

text: of

text: the

text: mean

text: ﬁeld

text: dynamics:

text: Assumption

text: We

text: have

text: the

text: following

text: assumptions

text: for

text: the

font: default

text: dynamics

text: of

text: DNN

text: with

text: random

text: linear

text: mappings:

text: When

text: β

text: for

text: all

text: sufﬁciently

text: large

text: t,

text: the

text: norm

text: of

text: t

text: u

text: is

text: expected

text: to

text: increase

text: exponentially,

text: thus

text: resulting

text: in

text: t

text: chaotic

text: behavior.

text: When

text: t

text: and

text: n

text: are

text: sufﬁciently

text: large,

text: we

text: have

text: n

text: C,

text: where

text: C

text: is

text: a

text: time-invariant

text: constant.

text: ti

text: Given

text: Assumption

text: we

text: have

text: the

text: following

text: theorem

text: pro-

text: viding

text: a

text: condition

text: for

text: the

text: chaotic

text: behavior

text: with

text: random

text: linear

text: σ

text: (dB)

text: mappings.

text: The

text: proof

text: of

text: is

text: given

text: in

text: Appendix

text: The

text: justiﬁcation

text: Fig.

text: Lyapunov

text: exponent

text: λ

text: for

text: vector

text: deterministic

text: dynam-

text: for

text: the

text: assumption

text: can

text: also

text: be

text: found

text: therein.

text: ics

text: with

text: Gaussian

text: matrices.

text: Theorem

text: Given

text: Assumption

text: we

text: have

text: λ

text: for

text: the

text: mappings,

text: if

text: tanh

text: σ

text: h

text: σ

text: π

text: π

text: equwahtieorne

text: zh(x=)

text: tiasnthh(exzn)on(zwehroereanzd

text: ispotshietivuenksnooluwtinonwhtoilethxe

text: is

text: the

text: parameter).

text: Remark

text: The

text: inequality

text: is

text: based

text: on

text: mean

text: dynamics

text: analysis

text: and

text: Assumption

text: However,

text: it

text: shows

text: that

text: a

text: higher

text: di-

text: mensional

text: dynamics

text: is

text: more

text: possible

text: to

text: be

text: chaotic.

text: Moreover,

text: a

text: too

text: large

text: or

text: too

text: small

text: σ2

text: disables

text: Both

text: coincide

text: the

text: observations

text: in

text: the

text: numerical

text: results.

text: An

text: explicit

text: expression

text: dimension

text: for

text: the

text: function

text: h

text: in

text: terms

text: of

text: inﬁnite

text: series

text: can

text: also

text: be

text: found

text: Fig.

text: Contour

text: of

text: Lyapunov

text: exponent

text: for

text: vector

text: deterministic

text: in

text: Appendix

text: dynamics

text: with

text: Gaussian

text: matrices.

text: IV.

text: CAPABILITIES

text: OF

text: CLASSIFICATION

text: AND

text: GENERALIZATION

text: In

text: this

text: section,

text: we

text: apply

text: the

text: metrics

text: characterizing

text: the

text: u

text: S

text: (x

text: )W

text: u

text: t

text: t

text: t

text: t

text: t

text: dynamics

text: of

text: DNN

text: to

text: the

text: analysis

text: of

text: classiﬁcation

text: and

text: gen-

text: eralization

text: capabilities.

text: and

text: that

text: of

text: the

text: system

text: state

text: x

text: t

text: A.

text: Classiﬁcation

text: Capability

text: We

text: ﬁrst

text: deﬁne

text: the

text: complexity

text: of

text: classiﬁcation

text: problem

text: and

text: xt

text: tanh

text: bt)

text: then

text: relate

text: it

text: to

text: the

text: topological

text: entropy

text: of

text: DNN

text: dynamics.

text: We

text: deﬁne

text: the

text: complexity

text: of

text: classiﬁcation,

text: which

text: facilitates

text: In

text: Theorem

text: we

text: design

text: the

text: mappings

text: such

text: that

text: the

text: the

text: subsequent

text: analysis

text: of

text: DNN

text: classiﬁcation.

text: The

text: philosophy

text: dynamics

text: of

text: u

text: and

text: x

text: are

text: decoupled.

text: However,

text: in

text: the

text: case

text: t

text: t

text: is

text: to

text: measure

text: how

text: difﬁcult

text: it

text: is

text: to

text: separate

text: the

text: samples

text: from

text: of

text: Gaussian

text: random

text: mappings,

text: the

text: two

text: dynamics

text: are

text: coupled,

text: different

text: classes.

text: There

text: have

text: been

text: some

text: studies

text: on

text: how

text: to

text: since

text: the

text: matrix

text: S

text: is

text: determined

text: by

text: x

text: In

text: order

text: to

text: obtain

text: an

text: t

text: t

text: deﬁne

text: the

text: complexity

text: of

text: pattern

text: recognition

text: analytic

text: result,

text: we

text: apply

text: the

text: approach

text: of

text: mean

text: ﬁeld

text: analysis

text: However,

text: they

text: are

text: mainly

text: focused

text: on

text: the

text: nonlinearity

text: in

text: statistical

text: mechanics

text: by

text: ﬁxing

text: the

text: impact

text: of

text: x

text: on

text: u

text: i.e.,

text: t

text: t

text: of

text: classiﬁer

text: and

text: the

text: overlap

text: of

text: features,

text: which

text: is

text: signiﬁcantly

text: we

text: consider

text: the

text: average

text: behaviors

text: of

text: the

text: random

text: variables.

text: different

text: from

text: our

text: proposed

text: deﬁnition.

text: In

text: more

text: details,

text: we

text: approximate

text: the

text: dynamics

text: of

text: the

text: tangent

text: For

text: deﬁning

text: the

text: complexity

text: of

text: classiﬁcation,

text: we

text: ﬁrst

text: need

text: vector

text: u

text: in

text: u

text: (x

text: )W

text: u

text: using

text: t

text: t

text: t

text: t

text: t

text: t

text: the

text: following

text: deﬁnition

text: of

text: set

text: radius.

text: R

text: Deﬁnition

text: For

text: a

text: connected

text: and

text: compact

text: set

text: A

text: in

text: d,

text: β

text: E

text: E

text: we

text: deﬁne

text: its

text: radius

text: as

text: r(A)

text: y

text: A}.

text: t

text: E

text: Then,

text: {we

text: de|ﬁne∈the}radius

text: of

text: a

text: partition

text: P

text: of

text: A

text: as

text: r(P)

text: max

text: r(A)

text: A

text: P

text: A

text: λ

text: )Bd(

text: σ

text: xam

text: ##LTFigure##

text: ##LTFigure##

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