text: 2 different quality when completing a task. Two assignment Our contribution therefore is as follows: We propose a modes considered in the crowdsourcing literature are the context-aware hierarchical online learning algorithm for per- server-assigned task (SAT) mode and the worker-selected task formance maximization in mobile crowdsourcing for location- (WST) mode (see [6] for a taxonomy). In SAT mode, the independent tasks. Our algorithm is split into two parts, one MCSP tries to match workers and tasks in an optimal way, part executed by the MCSP, the other part by local controllers e.g., to maximize the number of task assignments, possibly (LCs) located in each of the workers’ mobile devices. An under a given task budget. For this purpose, the MCSP LC learns its worker’s performance in terms of acceptance typically gathers task and worker information to decide on task rate and quality online over time, by observing the worker’s assignment. While this represents a sophisticated strategy, it personal contexts and his decisions to accept or decline tasks may require a large communication overhead and a privacy and the quality in completing these tasks. The LC learns from concern for workers since the MCSP has to be regularly its worker’s context only locally, and personal context is not informed about the current worker contexts (e.g., their current shared with the MCSP. Each LC regularly sends performance positions). Moreover, previous work on the SAT mode often estimates to the MCSP. Based on these estimates, the MCSP either assumed that workers always accept a task once assigned takes care of the worker selection. This hierarchical (in the to it or that workers’ acceptance rates and quality are known in sense of the coordination between the MCSP and the LCs) advance. However, this is not necessarily true in reality, since approach enables the MCSP to select suitable workers for each acceptance rates and quality are usually not known beforehand task under its budget without having access to the workers’ and therefore have to be learned by the MCSP over time. personal contexts. Moreover, workers receive personalized task In addition, a worker’s acceptance rate and the quality of requests based on their interests and skills, while keeping the completed tasks might depend not only on the speciﬁc task, but number of (possibly costly) quality assessments low. Since our also on the worker’s context, e.g., the worker’s location or the algorithm learns in an online fashion, it adapts and improves time of day [7]. This context may change quickly, especially in the worker selection over time and can hence achieve good mobile crowdsourcing with location-independent tasks, since results already during run time. We prove that our algorithm workers can complete such tasks anytime and anywhere. converges to the optimal task assignment strategy, which for In contrast, in WST mode, workers autonomously select each task selects the subset of workers which maximizes the tasks from a list. This rather simple mode is often used in expected performance under the task budget. practice (e.g., on Amazon Mechanical Turk) since it has the The remainder of this paper is organized as follows. Sec- advantage that workers automatically select tasks they are tion gives an overview on related work. Section describes interested in. However, the WST mode can lead to suboptimal the system model. In Section , we propose a context-aware task assignments since ﬁrst, ﬁnding suitable tasks is not as easy hierarchical online algorithm for performance maximization in as it seems (e.g., time-consuming searches within a long list mobile crowdsourcing. In Section , we theoretically analyze of tasks are needed and workers might simply select from the our proposed algorithm in terms of its regret, as well as its ﬁrst displayed tasks [8]) and secondly, workers might leave un- requirements with respect to local storage, communication popular tasks unassigned. Therefore, in WST mode, the MCSP and worker quality assessment. Section contains a numerical might additionally provide personalized task recommendation evaluation of our algorithm based on synthetic and real data. (TR) to workers such that workers ﬁnd appropriate tasks [7]. Section concludes the paper. However, personalized TR typically requires the workers to II. RELATED WORK share their current context with the MCSP, which again means a communication overhead and a privacy concern for workers. Research has put some effort in theoretically deﬁning and We argue that a task assignment strategy is needed which classifying crowdsourcing systems, such as web-based crowd- combines the advantages of the above modes: The MCSP sourcing [9], mobile [1] and spatial [4] crowdsourcing. Below, should centrally coordinate task assignment to ensure that we give an overview on related work on task assignment in appropriate workers are selected for each task, as in SAT general, mobile and spatial crowdsourcing systems, as relevant mode. At the same time, the communication overhead due for our scenario. Note that strategic behavior of workers and to task assignment should be small and workers’ personal task owners in crowdsourcing systems, e.g., concerning pricing data should be protected. Hence, no worker context should and effort spent in task completion [10], represents an active be shared with the MCSP, as in basic WST mode. Moreover, research area itself, which is out of the scope of this paper. task assignment should take into account that workers may Also note that we assume that the quality of a completed task decline a task they are assigned to, and hence, the assignment can be observed. A different line of work on crowdsourcing should ﬁt to the workers’ preferences, as in WST mode with deals with quality estimation in case of missing ground truth, personalized TR. In addition, task assignment should be based recently also using online learning [11]. both on acceptance rates and on the quality with which a task Due to the inherent dynamic nature of crowdsourcing, is completed. Since observing the quality of completed tasks with tasks and/or workers typically arriving dynamically over may require costly quality assessments (e.g., a manual quality time, task assignment is often modeled as an online decision rating from a task owner, or an automatic quality assessment making problem [12]. For general crowdsourcing systems, [13] using local software in a mobile device), the required number proposed a competitive online task assignment algorithm for of quality assessments should be kept low. Finally, workers’ maximizing the utility of a task owner on a given set of task acceptance rates and quality have to be learned over time. types, with ﬁnite number of tasks per task type, by learning the

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text: 11 users from the Gowalla-NY data set, which represent the 100 104 104 workers of the MCSP. Then we use this reduced Gowalla-NY data set containing the check-ins of 100 users. Task owners 102 102 have to pay a ﬁxed price of e = 1 per worker requested by the MCSP to perform the task. The quality of a completed task lies in the range q = 0 and q = 5. The task properties are min max 100 100 modeled as follows. The task budget is sampled from a normal 0 500 1000 1500 2000 0 500 1000 1500 2000 number of check-ins number of visited locations distribution with expected value 20 and standard deviation of (a) Check-ins. (b) Visited locations. 5, truncated between 1 and 500, i.e., given the price e = 1, the required number m of workers is on average 20. The task Fig. 3. Statistics of used Gowalla-NY data set. t C context is assumed to be uniformly distributed in = [0, 1] (i.e., C = 1). and p (t) is the observed performance of worker sA , the The worker arrival process and the worker context are sA t,j cumulatt,ijve perform(cid:88)ance up(cid:88)to t achieved by algorithm A is sampled differently in case of synthetic and real data. For synthetic data, we let each worker be available with a prob- { } t min mt˜,Wt˜ ability of ρ = 0.7 (default value) for each arriving task. p (t˜). sAt˜,j TheXpersonal context space of an available worker i is set t˜=1 j=1 to = [0, 1]2 (i.e., X = 2). The ﬁrst personal context i i Moreover, we compute the average worker performance up dimension refers to the worker’s location, which is sampled to t achieved by an algorithm, which is the average perfor- from 5 different (personal) locations, using a weighted discrete { } mance of all selected workers up to task t. Formally, the distribution with probabilities 1, 1, 1 , 1 , 1 to represent average worker performance u(cid:88)p to t i(cid:88)s deﬁned by the fact that workers may use th2e 3cro1w2ds2o4ur2c4ing application (cid:80) { } different amounts of time in different places (e.g., at home 1 t min mt˜,Wt˜ more often than at work). The second personal context di- { } p (t˜). tt˜=1 min mt˜, Wt˜ t˜=1 j=1 sAt˜,j mension refers to the worker’s battery state, which is sampled from a uniform distribution in [0, 1]. The worker performance C. Simulation Setup is modeled as follows. The joint personal and task context C × X We evaluate the different algorithms using synthetic as well space (of dimension D = 3) is uniformly split into a i i as real data. The difference between the two approaches lies in uniform grid of 125 subsets (i.e., in each of the 3 dimension, the arrival process of workers and their contexts. To produce the space is split into 5 identical parts). In each of the subsets, synthetic data, we generate workers and their contexts based the expected performance of a worker is a priori sampled on some predeﬁned distributions as described below. In case uniformly at random from [0, 5]. Later, for each sampled joint of real data, similar to, e.g., [6], [20], [22], we use a data set worker and task context, ﬁrst it is checked to which subset of from Gowalla [33]. Gowalla is a location-based social network the grid the sampled context belongs. Then, the instantaneous where users share their location by checking in at ’spots’, performance is sampled by adding noise uniformly sampled − i.e., certain places in their vicinity. We use the check-ins to from [ 1, 1] to the expected performance in the given subset simulate the arrival process of workers and their contexts. The of the grid (the noise interval is truncated to a smaller interval Gowalla data set consists of 6,442,892 check-ins of 107,092 if the expected performance lies close to either 0 or q ). max distinct users over the period of February 2009 to October Using real data, the arrival processes of workers and their 2010. Each entry of the data set consists of the form (User contexts are sampled as follows. For the worker availability, ID, Check-in Time, Latitude, Longitude, Location ID). Similar we use a Binomial distribution with parameters W = 100 and to the approach in [22], we ﬁrst extract the check-ins in ρ = 0.7 (default value) to sample the number of available New York City, which leaves a subset of 138,954 check- workers W for an arriving task.13 Having sampled W , we t t ins of 7,115 distinct users at 21,509 distinct locations. This randomly draw samples from the reduced Gowalla-NY data set resulting Gowalla-NY data set is used below. Fig. 3(a) shows (consisting of the check-ins of 100 users) until these samples the distribution of the number of check-ins in the Gowalla- contain W distinct users. These W sampled users correspond t t NY data set. The number of check-ins is distributed between to the available workers (i.e., users with higher number of 1 check-in (1414 users) up to 1794 check-ins (1 user). As an check-ins in the reduced Gowalla-NY data set translate to example, 2532 users checked in more than 10 times. Fig. 3(b) workers that are more often available for the MCSP). The shows the distribution of the number of distinct locations personal context space of an available worker is again set X visited by the users in the Gowalla-NY data set. The number to = [0, 1]2 (i.e., X = 2). We set the ﬁrst personal i i of visited locations is distributed between 1 location (1524 context of the available workers to the check-in location of the users) up to 1633 locations (1 user). For example, 3661 users respective user from the sample.14 The second personal context checked in at more than 5 locations. dimension refers again to the battery state, uniformly sampled For both synthetic and real data, the basic simulation setup is as follows: We simulate an MCSP, to which a set of 13In this way, the number of available workers in our experiments using the real and the synthetic data are distributed in the same way. 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text: 13 4.5 9 ×105 Oracle Oracle HCL HCL 8 4 LinUCB LinUCB AUER AUER 7 ǫ-Greedy ǫ-Greedy 3.5 Myopic Myopic 6 Random Random 5 3 4 2.5 3 2 2 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 arriving task t availability probability ρ (a) Experiments with synthetic data. (a) Experiments with synthetic data. 4.5 9 ×105 Oracle Oracle HCL 8 HCL 4 LinUCB LinUCB AUER 7 AUER ǫ-Greedy ǫ-Greedy 3.5 Myopic 6 Myopic Random Random 5 3 4 2.5 3 2 2 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 arriving task t availability probability ρ (b) Experiments with real data. (b) Experiments with real data. Fig. 5. Average worker performance up to task t for sequence t = 1,...,T Fig. 6. Impact of worker availability on cumulative performance at T for of arriving tasks for worker availability probability ρ = 0.7. T = 10,000 tasks. with the non-linear context dependency of expected worker varying the probability ρ.15 For each value of ρ, we run performance. Comparing synthetic and real data, HCL has a all algorithms on 100 synthetic instances and 100 instances better performance on the synthetic data, but it still reaches a based on real data for T = 10, 000, respectively. Then, we good result on the real data, even though each worker has average the results. Fig. 6(a) and 6(b) show the cumulative his own diversity in context arrival and hence in expected performance at T achieved by the algorithms for different ρ. performance in the real data (since users in the Gowalla-NY For small ρ = 0.1, all algorithms yield approximately the same data set have different numbers of visited check-in locations). performance. This is as expected since given our modelling of Fig. 5(a) and Fig. 5(b) show the average worker perfor- task budget, for small ρ, the number of available workers is mance up to task t as a function of the sequentially arriving often smaller than the required number of workers. Since each tasks t = 1, ..., T. We see that over the sequence of tasks, the of the algorithm enters a select-all-workers phase in this case, average worker performance achieved by Oracle and Random each algorithm performs optimally. For increasing worker stay nearly constant at around 4.2 and 2.5 for both synthetic availability ρ, the cumulative performance at T achieved and real data. The algorithms LinUCB, AUER, (cid:15)-Greedy and by each of the algorithm increases. However, the gap in Myopic increase the average worker performance slightly, achieved performance between Oracle and HCL algorithm starting between 2.4 and 2.5 at t = 1 and ending with average on the one hand, and the remaining algorithms LinUCB, performance of between 2.5 and 2.7 at t = T. On the contrary, AUER, (cid:15)-Greedy, Myopic and Random on the∈o{ther hand, }is HCL is able to increase the average worker performance from increasing for increasing ρ. For example, at ρ 0.3, 0.7, 1 , 2.5 at t = 1 up to 4.0 (3.5) at t = T for the synthetic (real) the cumulative performance achieved by HCL corresponds to data. Hence, HCL learns context-speciﬁc worker performances over time and after sufﬁciently many processed tasks selects 15Note that this is equivalent to varying the expected task budget. We therefore do not separately evaluate the impact of the task budget. 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text: 6 performances. However, observing worker performance might be costly, since it might require a manual quality rating from a task owner, or an automatic quality assessment using local software in a battery-constrained mobile device. Hence, the number of performance observations should be limited in order to keep the cost for such quality assessment feasible (e.g., a task owner should not have to rate the quality of dozens of workers for a single task; quality assessment in mobile devices should be limited to save battery). Next, we will present a context-aware hierarchical online learning algorithm, which maps the history of previously selected workers and observed performances to the next se- lection of workers. The performance of this algorithm can be evaluated by comparing its loss with respect to the centralized oracle. This loss is called the regret of learning. For an { } Fig. 2. Overview of operation of HCL algorithm for task t. arbitrary sequence of task budgets b (translating to { } t t=1,...,T a sequence m ) and an arbitrary sequence of worker {Wt }t=1,...,T availability , the regret is formally deﬁned as  LC relies on previous observations to estimate its worker’s per- (cid:88)t t=(cid:88)1,...,T(cid:16) (cid:17)   formance and sends the estimated performance to the MCSP. { } If its worker’s performance has not been observed sufﬁciently T min mt,Wt − often before, the LC informs the MCSP that its worker has to R(T) = E p ∗ (x ∗ , c ) p (x , c ) , st,j t,st,j t st,j t,st,j t be explored. Based on the messages received from the LCs, t=1 j=1 (2) the MCSP selects a subset of workers and requests them to complete the task. The number of selected workers depends which is equivalent to on the task budget, the price per worker and the number of (cid:88) (cid:88) (cid:16) (cid:17) available workers. The LC of each selected worker observes its worker’s decision to accept/decline the task. If a worker { } T min mt,Wt − was selected for exploration purposes and he accepted the R(T) = θ ∗ (x ∗ , c ) E[θ (x , c )] . st,j t,st,j t st,j t,st,j t task, the LC additionally observes the quality of the completed t=1 j=1 task. The reason for only making a quality assessment when a (3) worker was selected for exploration purposes is that this may either require a manual quality rating from the task owner, IV. A CONTEXT-AWARE HIERARCHICAL ONLINE or, an automatic quality assessment at the mobile device LEARNING ALGORITHM FOR PERFORMANCE using local software, which both may be costly.10 Hence, by MAXIMIZATION IN MOBILE CROWDSOURCING observing the quality of a completed task only if the worker The goal of the MCSP is to select, for each arriving task, was selected for exploration purposes, HCL keeps the number a set of workers that maximizes the sum of expected perfor- of costly quality assessments low. mances for that task given the task budget. Since the expected In HCL, a worker’s personal contexts, decisions and qual- performances are not known a priori by neither MCSP nor ities are only locally stored at the LC, but not shared with the LCs, they have to be learned over time. Moreover, since the MCSP. Thereby, i) personal context is locally protected, only the LCs have access to the personal context of their ii) the required storage space for worker information at the respective worker, a coordination is needed between the MCSP MCSP is kept low, iii) task completion and result transmission and the LCs. Below, we propose a hierarchical contextual can be directly handled between the LC and the task owner, online learning algorithm, which is based on algorithms for without the need for the MCSP to interfere, iv) workers the contextual multi-armed bandit problem [23]–[28]. Our receive requests for tasks that are interesting for them and algorithm is based on the assumption that a worker’s expected which they are good at, but without the need to share their performance is similar in similar personal and task contexts. context information, v) even though an LC has to keep track Therefore, by observing the task context, a worker’s personal of its worker’s personal context, decision and quality, the context and his performance when requested to complete a computation and storage overhead for each LC is small. task, the worker’s context-speciﬁc expected performances can In more detail, LC i operates as follows, as given in the be learned and exploited for future worker selection. pseudocode in Alg. 1. First, for synchronization purposes, We call the proposed algorithm Hierarchical Context-aware LC i receives the ﬁnite number T of tasks to be considered, Learning (HCL). Fig. 2 shows an overview of HCL’s op- the task context space C and its dimension C from the eration. In HCL, the MCSP broadcasts the context of each MCSP. Moreover, LC i checks to which of worker i’s context arriving task to the LCs. Upon receiving information about dimensions it has access. This deﬁnes the personal context a task, an LC ﬁrst observes its worker’s personal context. If the worker’s performance has been observed sufﬁciently often 10If quality assessment is cheap, HCL can be adapted to always observe before given the current joint personal and task context, the worker quality. This may increase the learning speed. ##LTFigure##

text: performances.

text: However,

text: observing

text: worker

text: performance

text: might

text: be

text: costly,

text: since

text: it

text: might

text: require

text: a

text: manual

text: quality

text: rating

text: from

text: a

text: task

text: owner,

text: or

text: an

text: automatic

text: quality

text: assessment

text: using

text: local

text: software

text: in

text: a

text: battery-constrained

text: mobile

text: device.

text: Hence,

text: the

text: number

text: of

text: performance

text: observations

text: should

text: be

text: limited

text: in

text: order

text: to

text: keep

text: the

text: cost

text: for

text: such

text: quality

text: assessment

text: feasible

text: (e.g.,

text: a

text: task

text: owner

text: should

text: not

text: have

text: to

text: rate

text: the

text: quality

text: of

text: dozens

text: of

text: workers

text: for

text: a

text: single

text: task;

text: quality

text: assessment

text: in

text: mobile

text: devices

text: should

text: be

text: limited

text: to

text: save

text: battery).

text: Next,

text: we

text: will

text: present

text: a

text: context-aware

text: hierarchical

text: online

text: learning

text: algorithm,

text: which

text: maps

text: the

text: history

text: of

text: previously

text: selected

text: workers

text: and

text: observed

text: performances

text: to

text: the

text: next

text: se-

text: lection

text: of

text: workers.

text: The

text: performance

text: of

text: this

text: algorithm

text: can

text: be

text: evaluated

text: by

text: comparing

text: its

text: loss

text: with

text: respect

text: to

text: the

text: centralized

text: oracle.

text: This

text: loss

text: is

text: called

text: the

text: regret

text: of

text: learning.

text: For

text: an

text: Fig.

text: Overview

text: of

text: operation

text: of

text: HCL

text: algorithm

text: for

text: task

text: t.

text: arbitrary

text: sequence

text: of

text: task

text: budgets

text: b

text: (translating

text: to

text: t

text: a

text: sequence

text: m

text: and

text: an

text: arbitrary

text: sequence

text: of

text: worker

text: {Wt

text: availability

text: the

text: regret

text: is

text: formally

text: deﬁned

text: as

text: LC

text: relies

text: on

text: previous

text: observations

text: to

text: estimate

text: its

text: worker’s

text: per-

text: formance

text: and

text: sends

text: the

text: estimated

text: performance

text: to

text: the

text: MCSP.

text: If

text: its

text: worker’s

text: performance

text: has

text: not

text: been

text: observed

text: sufﬁciently

text: T

text: min

text: mt,Wt

text: often

text: before,

text: the

text: LC

text: informs

text: the

text: MCSP

text: that

text: its

text: worker

text: has

text: to

text: R(T)

text: E

text: p

text: (x

text: c

text: p

text: (x

text: c

text: st,j

text: t,st,j

text: t

text: st,j

text: t,st,j

text: t

text: be

text: explored.

text: Based

text: on

text: the

text: messages

text: received

text: from

text: the

text: LCs,

text: the

text: MCSP

text: selects

text: a

text: subset

text: of

text: workers

text: and

text: requests

text: them

text: to

text: complete

text: the

text: task.

text: The

text: number

text: of

text: selected

text: workers

text: depends

text: which

text: is

text: equivalent

text: to

text: on

text: the

text: task

text: budget,

text: the

text: price

text: per

text: worker

text: and

text: the

text: number

text: of

text: available

text: workers.

text: The

text: LC

text: of

text: each

text: selected

text: worker

text: observes

text: its

text: worker’s

text: decision

text: to

text: accept/decline

text: the

text: task.

text: If

text: a

text: worker

text: T

text: min

text: mt,Wt

text: was

text: selected

text: for

text: exploration

text: purposes

text: and

text: he

text: accepted

text: the

text: R(T)

text: θ

text: (x

text: c

text: E[θ

text: (x

text: c

text: st,j

text: t,st,j

text: t

text: st,j

text: t,st,j

text: t

text: task,

text: the

text: LC

text: additionally

text: observes

text: the

text: quality

text: of

text: the

text: completed

text: task.

text: The

text: reason

text: for

text: only

text: making

text: a

text: quality

text: assessment

text: when

text: a

text: worker

text: was

text: selected

text: for

text: exploration

text: purposes

text: is

text: that

text: this

text: may

text: either

text: require

text: a

text: manual

text: quality

text: rating

text: from

text: the

text: task

text: owner,

text: IV.

text: A

text: CONTEXT-AWARE

text: HIERARCHICAL

text: ONLINE

text: or,

text: an

text: automatic

text: quality

text: assessment

text: at

text: the

text: mobile

text: device

text: LEARNING

text: ALGORITHM

text: FOR

text: PERFORMANCE

text: using

text: local

text: software,

text: which

text: both

text: may

text: be

text: Hence,

text: by

text: MAXIMIZATION

text: IN

text: MOBILE

text: CROWDSOURCING

text: observing

text: the

text: quality

text: of

text: a

text: completed

text: task

text: only

text: if

text: the

text: worker

text: The

text: goal

text: of

text: the

text: MCSP

text: is

text: to

text: select,

text: for

text: each

text: arriving

text: task,

text: was

text: selected

text: for

text: exploration

text: purposes,

text: HCL

text: keeps

text: the

text: number

text: a

text: set

text: of

text: workers

text: that

text: maximizes

text: the

text: sum

text: of

text: expected

text: perfor-

text: of

text: costly

text: quality

text: assessments

text: low.

text: mances

text: for

text: that

text: task

text: given

text: the

text: task

text: budget.

text: Since

text: the

text: expected

text: In

text: HCL,

text: a

text: worker’s

text: personal

text: contexts,

text: decisions

text: and

text: qual-

text: performances

text: are

text: not

text: known

text: a

text: priori

text: by

text: neither

text: MCSP

text: nor

text: ities

text: are

text: only

text: locally

text: stored

text: at

text: the

text: LC,

text: but

text: not

text: shared

text: with

text: the

text: LCs,

text: they

text: have

text: to

text: be

text: learned

text: over

text: time.

text: Moreover,

text: since

text: the

text: MCSP.

text: Thereby,

text: i)

text: personal

text: context

text: is

text: locally

text: protected,

text: only

text: the

text: LCs

text: have

text: access

text: to

text: the

text: personal

text: context

text: of

text: their

text: ii)

text: the

text: required

text: storage

text: space

text: for

text: worker

text: information

text: at

text: the

text: respective

text: worker,

text: a

text: coordination

text: is

text: needed

text: between

text: the

text: MCSP

text: MCSP

text: is

text: kept

text: low,

text: iii)

text: task

text: completion

text: and

text: result

text: transmission

text: and

text: the

text: LCs.

text: Below,

text: we

text: propose

text: a

text: hierarchical

text: contextual

text: can

text: be

text: directly

text: handled

text: between

text: the

text: LC

text: and

text: the

text: task

text: owner,

text: online

text: learning

text: algorithm,

text: which

text: is

text: based

text: on

text: algorithms

text: for

text: without

text: the

text: need

text: for

text: the

text: MCSP

text: to

text: interfere,

text: iv)

text: workers

text: the

text: contextual

text: multi-armed

text: bandit

text: problem

text: Our

text: receive

text: requests

text: for

text: tasks

text: that

text: are

text: interesting

text: for

text: them

text: and

text: algorithm

text: is

text: based

text: on

text: the

text: assumption

text: that

text: a

text: worker’s

text: expected

text: which

text: they

text: are

text: good

text: at,

text: but

text: without

text: the

text: need

text: to

text: share

text: their

text: performance

text: is

text: similar

text: in

text: similar

text: personal

text: and

text: task

text: contexts.

text: context

text: information,

text: v)

text: even

text: though

text: an

text: LC

text: has

text: to

text: keep

text: track

text: Therefore,

text: by

text: observing

text: the

text: task

text: context,

text: a

text: worker’s

text: personal

text: of

text: its

text: worker’s

text: personal

text: context,

text: decision

text: and

text: quality,

text: the

text: context

text: and

text: his

text: performance

text: when

text: requested

text: to

text: complete

text: a

text: computation

text: and

text: storage

text: overhead

text: for

text: each

text: LC

text: is

text: small.

text: task,

text: the

text: worker’s

text: context-speciﬁc

text: expected

text: performances

text: can

text: In

text: more

text: detail,

text: LC

text: i

text: operates

text: as

text: follows,

text: as

text: given

text: in

text: the

text: be

text: learned

text: and

text: exploited

text: for

text: future

text: worker

text: selection.

text: pseudocode

text: in

text: Alg.

text: First,

text: for

text: synchronization

text: purposes,

text: We

text: call

text: the

text: proposed

text: algorithm

text: Hierarchical

text: Context-aware

text: LC

text: i

text: receives

text: the

text: ﬁnite

text: number

text: T

text: of

text: tasks

text: to

text: be

text: considered,

text: Learning

text: (HCL).

text: Fig.

text: shows

text: an

text: overview

text: of

text: HCL’s

text: op-

text: the

text: task

text: context

text: space

text: C

text: and

text: its

text: dimension

text: C

text: from

text: the

text: eration.

text: In

text: HCL,

text: the

text: MCSP

text: broadcasts

text: the

text: context

text: of

text: each

text: MCSP.

text: Moreover,

text: LC

text: i

text: checks

text: to

text: which

text: of

text: worker

text: i’s

text: context

text: arriving

text: task

text: to

text: the

text: LCs.

text: Upon

text: receiving

text: information

text: about

text: dimensions

text: it

text: has

text: access.

text: This

text: deﬁnes

text: the

text: personal

text: context

text: a

text: task,

text: an

text: LC

text: ﬁrst

text: observes

text: its

text: worker’s

text: personal

text: context.

text: If

text: the

text: worker’s

text: performance

text: has

text: been

text: observed

text: sufﬁciently

text: often

text: quality

text: assessment

text: is

text: cheap,

text: HCL

text: can

text: be

text: adapted

text: to

text: always

text: observe

text: before

text: given

text: the

text: current

text: joint

text: personal

text: and

text: task

text: context,

text: the

text: worker

text: quality.

text: This

text: may

text: increase

text: the

text: learning

text: speed.

text: ##LTFigure##


text: Coulombic correction to the LC energy levels in some regime before seeing an R−4 increase for R → 0 if the ¯b term is absent. 2 As a remark, we would like to stress that the above computation only applies to the potential. The modiﬁcation of the excited energy levels due to the rigidity term are still unknown. For the purpose of this paper we will only need the potential, but a distinction between the boundary corrections and the rigidity term is notoriously diﬃcult in this case. It would be very interesting to get a result for the associated modiﬁcation of excited states. The hope is, that for the excited states the predictions including the massive modes is incompatible with the splittings between the energy levels, so that one can distinguish between the cases with or without massive mode contributions. In particular, it has been found in [21] that the splitting of the ﬁrst few excitations is well described by the boundary term, which could be diﬀerent when massive modes are included. In the presence of additional massive bosonic degrees of freedom the presence of terms proportional to the extrinsic curvature also allows for a topological coupling term between the boson ﬁeld and the GBs, proportional to the mode number [17, 40] 2. Consequently, since the boson couples to the worldsheet analogue of the θ-term, the boson can naturally be referred to as the worldsheet axion [40]. The action for the worldsheet axion can then be written as(cid:90)[17] (cid:16) (cid:17) √ 1 1 γ S = −h − ∇ φ∇αφ − m2φ2 − φ (cid:15) (cid:15)αβKi Kγi + . . . , (2.17) a 2 α 2 8π ij αγ β M where ∇ is the covariant derivative with respect to the induced metric hαβ = ∂αX∂βX. To leading order, i.e. up to coupling terms of the form φ∂ ∂ φhαβ and higher orders, ∇α α β can be replaced by ∂α. The associated leading order spectrum has been computed with the TBA method for closed strings in [17, 40] and has been found to be consistent with states showing an anomalously slow approach to the LC energies in the spectrum of the closed ﬂux tube for d = 4 [39]. Note that the topological coupling term in eq. (2.17) is only present for d > 3, which could be a reason why no anomalous states have been observed in the 3d ﬂux tube spectrum. In 3d massive modes appear as free bosons up to the coupling terms mentioned above. It would be interesting to include those coupling terms in a computation of the energy levels to check their inﬂuence on the spectrum. In the following we will always denote the contributions discussed in this section as the contributions from “massive modes” for brevity. We would like to emphasise, however, that strictly speaking we can only compare our results to those obtained from the rigidity term, or from the leading order contribution, neglecting any direct couplings, of a massive boson on the worldsheet. Whether direct coupling terms are indeed negligible is an open question, which we cannot answer in the course of this study. 2.4 EST and gauge/gravity duality We close the discussion of the EST with the remark that the EST action can po- tentially be computed (e.g. [8] and references therein) from the original 10d fundamental 2Note, that the presence of this worldsheet θ-term has ﬁrst been noticed in [42]. – 9 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Coulombic

text: correction

text: to

text: the

text: LC

text: energy

text: levels

text: in

text: some

text: regime

text: before

text: seeing

text: an

text: increase

text: for

text: R

text: if

text: the

text: ¯b

text: term

text: is

text: absent.

text: As

text: a

text: remark,

text: we

text: would

text: like

text: to

text: stress

text: that

text: the

text: above

text: computation

text: only

text: applies

text: to

text: the

text: potential.

text: The

text: modiﬁcation

text: of

text: the

text: excited

text: energy

text: levels

text: due

text: to

text: the

text: rigidity

text: term

text: are

text: still

text: unknown.

text: For

text: the

text: purpose

text: of

text: this

text: paper

text: we

text: will

text: only

text: need

text: the

text: potential,

text: but

text: a

text: distinction

text: between

text: the

text: boundary

text: corrections

text: and

text: the

text: rigidity

text: term

text: is

text: notoriously

text: diﬃcult

text: in

text: this

text: case.

text: It

text: would

text: be

text: very

text: interesting

text: to

text: get

text: a

text: result

text: for

text: the

text: associated

text: modiﬁcation

text: of

text: excited

text: states.

text: The

text: hope

text: is,

text: that

text: for

text: the

text: excited

text: states

text: the

text: predictions

text: including

text: the

text: massive

text: modes

text: is

text: incompatible

text: with

text: the

text: splittings

text: between

text: the

text: energy

text: levels,

text: so

text: that

text: one

text: can

text: distinguish

text: between

text: the

text: cases

text: with

text: or

text: without

text: massive

text: mode

text: contributions.

text: In

text: particular,

text: it

text: has

text: been

text: found

text: in

text: that

text: the

text: splitting

text: of

text: the

text: ﬁrst

text: few

text: excitations

text: is

text: well

text: described

text: by

text: the

text: boundary

text: term,

text: which

text: could

text: be

text: diﬀerent

text: when

text: massive

text: modes

text: are

text: included.

text: In

text: the

text: presence

text: of

text: additional

text: massive

text: bosonic

text: degrees

text: of

text: freedom

text: the

text: presence

text: of

text: terms

text: proportional

text: to

text: the

text: extrinsic

text: curvature

text: also

text: allows

text: for

text: a

text: topological

text: coupling

text: term

text: between

text: the

text: boson

text: ﬁeld

text: and

text: the

text: GBs,

text: proportional

text: to

text: the

text: mode

text: number

text: Consequently,

text: since

text: the

text: boson

text: couples

text: to

text: the

text: worldsheet

text: analogue

text: of

text: the

text: θ-term,

text: the

text: boson

text: can

text: naturally

text: be

text: referred

text: to

text: as

text: the

text: worldsheet

text: axion

text: The

text: action

text: for

text: the

text: worldsheet

text: axion

text: can

text: then

text: be

text: written

text: γ

text: S

text: −h

text: φ∇αφ

text: φ

text: Kγi

text: a

text: α

text: 8π

text: ij

text: αγ

text: β

text: M

text: where

text: is

text: the

text: covariant

text: derivative

text: with

text: respect

text: to

text: the

text: induced

text: metric

text: hαβ

text: ∂αX∂βX.

text: To

text: leading

text: order,

text: i.e.

text: up

text: to

text: coupling

text: terms

text: of

text: the

text: form

text: φ∂

text: φhαβ

text: and

text: higher

text: orders,

text: ∇α

text: α

text: β

text: can

text: be

text: replaced

text: by

text: ∂α.

text: The

text: associated

text: leading

text: order

text: spectrum

text: has

text: been

text: computed

text: with

text: the

text: TBA

text: method

text: for

text: closed

text: strings

text: in

text: and

text: has

text: been

text: found

text: to

text: be

text: consistent

text: with

text: states

text: showing

text: an

text: anomalously

text: slow

text: approach

text: to

text: the

text: LC

text: energies

text: in

text: the

text: spectrum

text: of

text: the

text: closed

text: ﬂux

text: tube

text: for

text: d

text: Note

text: that

text: the

text: topological

text: coupling

text: term

text: in

text: eq.

text: is

text: only

text: present

text: for

text: d

text: which

text: could

text: be

text: a

text: reason

text: why

text: no

text: anomalous

text: states

text: have

text: been

text: observed

text: in

text: the

text: ﬂux

text: tube

text: spectrum.

text: In

text: massive

text: modes

text: appear

text: as

text: free

text: bosons

text: up

text: to

text: the

text: coupling

text: terms

text: mentioned

text: above.

text: It

text: would

text: be

text: interesting

text: to

text: include

text: those

text: coupling

text: terms

text: in

text: a

text: computation

text: of

text: the

text: energy

text: levels

text: to

text: check

text: their

text: inﬂuence

text: on

text: the

text: spectrum.

text: In

text: the

text: following

text: we

text: will

text: always

text: denote

text: the

text: contributions

text: discussed

text: in

text: this

text: section

text: as

text: the

text: contributions

text: from

text: “massive

text: modes”

text: for

text: brevity.

text: We

text: would

text: like

text: to

text: emphasise,

text: however,

text: that

text: strictly

text: speaking

text: we

text: can

text: only

text: compare

text: our

text: results

text: to

text: those

text: obtained

text: from

text: the

text: rigidity

text: term,

text: or

text: from

text: the

text: leading

text: order

text: contribution,

text: neglecting

text: any

text: direct

text: couplings,

text: of

text: a

text: massive

text: boson

text: on

text: the

text: worldsheet.

text: Whether

text: direct

text: coupling

text: terms

text: are

text: indeed

text: negligible

text: is

text: an

text: open

text: question,

text: which

text: we

text: cannot

text: answer

text: in

text: the

text: course

text: of

text: this

text: study.

text: EST

text: and

text: gauge/gravity

text: duality

text: We

text: close

text: the

text: discussion

text: of

text: the

text: EST

text: with

text: the

text: remark

text: that

text: the

text: EST

text: action

text: can

text: po-

text: tentially

text: be

text: computed

text: (e.g.

text: and

text: references

text: therein)

text: from

text: the

text: original

text: fundamental

text: that

text: the

text: presence

text: of

text: this

text: worldsheet

text: θ-term

text: has

text: ﬁrst

text: been

text: noticed

text: in

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: G if C (t ) = 1. Denote by dG the distance function between edges on (adjacent edges n,ij 0 have distance 0). In the same heuristic as explained after Assumption (A6), this graph is very large (asymptotics over the number of vertices) and sparse (n vertices and of order n edges), because every vertex is incident to at most κ edges. In this scenario, the number − · of pairs of edges e and e for which dG(e , e ) = d is of order (κ 1)d n, and there 1 2 1 2 are of order n2 many pairs of edges in total. Let now A be arbitrary, centered random G i,j variables indexed by the edges of . We make the assumption that A is inﬂuenced equally i,j by all A with (k, l) being adjacent to (i, j). In mathematical terms, we formulate this k,l E | ≈ · − assumption as (A A dG((i, j), (k, l)) = d) C κ d. Then, we obtain for non-adjacent i,j k,l (cid:88) edges (i, j) and (k, l) ∞ E P · E | (A A ) = (cid:88) (dG((i, j), (k, l)) = d) (A A dG((i, j), (k, l)) = d) i,j k,l i,j k,l d=1 ∞ − ≈ n(κ 1)d · − C κ d n2 d=1 C − = (κ 1), n ≈ which converges to zero after being multiplied with l h nh (in this case). Because, in n (2.18) and (2.19), we consider only expectations conditional on C (t) = 1, we can think of n,ij A being the random variables τ (a centered version of it) or r and the expectations n,ij n,ij n,ij in the above heuristic are conditional expectations conditionally the respective conditions in (2.18) and (2.19). This serves as motivation for these two assumptions. Moreover, unconditionally, τ and τ (and r and r ) do not need to be uncorrelated. n,ij n,kl n,ij n,kl 2.4 The main asymptotic result ∈ Theorem 2.1. Suppose that Assumptions (A1)–(A7) hold for a point t (0, T ). Then, 0 with probability tending to one, the derivative of the local log-likelihood function (cid:96) (θ, t ) T 0 has a root θˆ (t ), satisfying n 0 (cid:18) (cid:19) (cid:18) (cid:90) (cid:19) (cid:112) − 1 − − → 1 − l h θˆ (t ) θ (t ) + h2Σ 1v h2B N 0, K(u)2du Σ 1 (2.20) n n 0 0 0 2 n − 1 with (cid:90) 1 · v := K(u)u2du ∂ ∂ g(θ (t ), t ), − θ t2 0 0 0 − 1 Σ := ∂ g(θ (t ), t ), θ2 0 0 0 − γ (s) := (1 C (t ))C (s), n,ij n,ij 0 n,ij 12 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: G

text: if

text: C

text: (t

text: Denote

text: by

text: dG

text: the

text: distance

text: function

text: between

text: edges

text: on

text: (adjacent

text: edges

text: n,ij

text: have

text: distance

text: In

text: the

text: same

text: heuristic

text: as

text: explained

text: after

text: Assumption

text: this

text: graph

text: is

text: very

text: large

text: (asymptotics

text: over

text: the

text: number

text: of

text: vertices)

text: and

text: sparse

text: (n

text: vertices

text: and

text: of

text: order

text: n

text: edges),

text: because

text: every

text: vertex

text: is

text: incident

text: to

text: at

text: most

text: κ

text: edges.

text: In

text: this

text: scenario,

text: the

text: number

text: of

text: pairs

text: of

text: edges

text: e

text: and

text: e

text: for

text: which

text: dG(e

text: e

text: d

text: is

text: of

text: order

text: (κ

text: n,

text: and

text: there

text: are

text: of

text: order

text: many

text: pairs

text: of

text: edges

text: in

text: total.

text: Let

text: now

text: A

text: be

text: arbitrary,

text: centered

text: random

text: G

text: i,j

text: variables

text: indexed

text: by

text: the

text: edges

text: of

text: We

text: make

text: the

text: assumption

text: that

text: A

text: is

text: inﬂuenced

text: equally

text: i,j

text: by

text: all

text: A

text: with

text: (k,

text: l)

text: being

text: adjacent

text: to

text: (i,

text: j).

text: In

text: mathematical

text: terms,

text: we

text: formulate

text: this

text: k,l

text: E

text: assumption

text: as

text: (A

text: A

text: dG((i,

text: j),

text: (k,

text: l))

text: d)

text: C

text: κ

text: d.

text: Then,

text: we

text: obtain

text: for

text: non-adjacent

text: i,j

text: k,l

text: edges

text: (i,

text: j)

text: and

text: (k,

text: l)

text: E

text: P

text: E

text: (A

text: A

text: (dG((i,

text: j),

text: (k,

text: l))

text: d)

text: (A

text: A

text: dG((i,

text: j),

text: (k,

text: l))

text: d)

text: i,j

text: k,l

text: i,j

text: k,l

text: n(κ

text: C

text: κ

text: d

text: C

text: (κ

text: n

text: which

text: converges

text: to

text: zero

text: after

text: being

text: multiplied

text: with

text: l

text: h

text: nh

text: (in

text: this

text: case).

text: Because,

text: in

text: n

text: and

text: we

text: consider

text: only

text: expectations

text: conditional

text: on

text: C

text: (t)

text: we

text: can

text: think

text: of

text: n,ij

text: A

text: being

text: the

text: random

text: variables

text: τ

text: (a

text: centered

text: version

text: of

text: it)

text: or

text: r

text: and

text: the

text: expectations

text: n,ij

text: n,ij

text: n,ij

text: in

text: the

text: above

text: heuristic

text: are

text: conditional

text: expectations

text: conditionally

text: the

text: respective

text: conditions

text: in

text: and

text: This

text: serves

text: as

text: motivation

text: for

text: these

text: two

text: assumptions.

text: Moreover,

text: unconditionally,

text: τ

text: and

text: τ

text: (and

text: r

text: and

text: r

text: do

text: not

text: need

text: to

text: be

text: uncorrelated.

text: n,ij

text: n,kl

text: n,ij

text: n,kl

text: The

text: main

text: asymptotic

text: result

text: Theorem

text: Suppose

text: that

text: Assumptions

text: hold

text: for

text: a

text: point

text: t

text: T

text: Then,

text: with

text: probability

text: tending

text: to

text: one,

text: the

text: derivative

text: of

text: the

text: local

text: log-likelihood

text: function

text: (θ,

text: t

text: T

text: has

text: a

text: root

text: θˆ

text: (t

text: satisfying

text: n

text: l

text: h

text: θˆ

text: (t

text: θ

text: (t

text: N

text: Σ

text: n

text: n

text: n

text: with

text: v

text: g(θ

text: (t

text: t

text: θ

text: Σ

text: g(θ

text: (t

text: t

text: θ2

text: γ

text: (s)

text: C

text: (t

text: ))C

text: (s),

text: n,ij

text: n,ij

text: n,ij

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##