Collaboratively Learning the Best Option on Graphs, Using Bounded Local Memory
Name
1811.03968.pdf
Description
Submitted version
Size
396.79 KB
Format
Adobe PDF
Checksum (MD5)
f4200ed5e94073808f2734917c7ae6d4
Author(s)
Su, Lili
Zubeldia, Martin
Lynch, Nancy A.
Date Issued
2019
Journal
Proceedings of the ACM on Measurement and Analysis of Computing Systems
Publisher
Association for Computing Machinery (ACM)
Version
Original manuscript
Abstract
We consider multi-armed bandit problems in social groups wherein each individual has bounded memory and shares the common goal of learning the best arm/option. We say an individual learns the best option if eventually (as $t\diverge$) it pulls only the arm with the highest expected reward. While this goal is provably impossible for an isolated individual due to bounded memory, we show that, in social groups, this goal can be achieved easily with the aid of social persuasion (i.e., communication) as long as the communication networks/graphs satisfy some mild conditions. In this work, we model and analyze a type of learning dynamics which are well-observed in social groups. Specifically, under the learning dynamics of interest, an individual sequentially decides on which arm to pull next based on not only its private reward feedback but also the suggestion provided by a randomly chosen neighbor. To deal with the interplay between the randomness in the rewards and in the social interaction, we employ the \em mean-field approximation method. Considering the possibility that the individuals in the networks may not be exchangeable when the communication networks are not cliques, we go beyond the classic mean-field techniques and apply a refined version of mean-field approximation:
Using coupling we show that, if the communication graph is connected and is either regular or has doubly-stochastic degree-weighted adjacency matrix, with probability → 1 as the social group size N → ∞, every individual in the social group learns the best option. If the minimum degree of the graph diverges as N → ∞, over an arbitrary but given finite time horizon, the sample paths describing the opinion evolutions of the individuals are asymptotically independent. In addition, the proportions of the population with different opinions converge to the unique solution of a system of ODEs. Interestingly, the obtained system of ODEs are invariant to the structures of the communication graphs. In the solution of the obtained ODEs, the proportion of the population holding the correct opinion converges to 1 exponentially fast in time. Notably, our results hold even if the communication graphs are highly sparse.
Using coupling we show that, if the communication graph is connected and is either regular or has doubly-stochastic degree-weighted adjacency matrix, with probability → 1 as the social group size N → ∞, every individual in the social group learns the best option. If the minimum degree of the graph diverges as N → ∞, over an arbitrary but given finite time horizon, the sample paths describing the opinion evolutions of the individuals are asymptotically independent. In addition, the proportions of the population with different opinions converge to the unique solution of a system of ODEs. Interestingly, the obtained system of ODEs are invariant to the structures of the communication graphs. In the solution of the obtained ODEs, the proportion of the population holding the correct opinion converges to 1 exponentially fast in time. Notably, our results hold even if the communication graphs are highly sparse.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.1145/3322205.3311082
