Collaboratively Learning the Best Option, Using Bounded Memory

TL;DR

This paper demonstrates that social interaction enables groups of individuals with limited memory to learn the best option in multi-armed bandit problems, a task impossible for isolated individuals, using mean-field approximation techniques.

Contribution

It introduces a novel mean-field analysis approach to study social learning dynamics with bounded memory, proving group convergence to the best option.

Findings

As group size increases, all individuals learn the best option with probability approaching 1.

The fraction of individuals preferring the best option grows exponentially fast over time.

The mean-field approximation simplifies analysis by modeling stochastic dynamics with deterministic ODEs.

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 $t \to \infty$) it pulls only the arm with the highest average reward. While this goal is provably impossible for an isolated individual, we show that, in social groups, this goal can be achieved easily with the aid of social persuasion, i.e., communication. Specifically, we study the learning dynamics wherein an individual sequentially decides on which arm to pull next based on not only its private reward feedback but also the suggestions provided by randomly chosen peers. Our learning dynamics are hard to analyze via explicit probabilistic calculations due to the stochastic dependency induced by social interaction. Instead, we employ the mean-field approximation method from statistical physics and we show: (1) With probability $\to 1$ as the social group size $N \to \infty $, every individual in the social group learns the best option. (2) Over an arbitrary finite time horizon $[0, T]$, with high probability (in $N$), the fraction of individuals that prefer the best option grows to 1 exponentially fast as $t$ increases ($t\in [0, T]$). A major innovation of our mean-filed analysis is a simple yet powerful technique to deal with absorbing states in the interchange of limits $N \to \infty$ and $t \to \infty $. The mean-field approximation method allows us to approximate the probabilistic sample paths of our learning dynamics by a deterministic and smooth trajectory that corresponds to the unique solution of a well-behaved system of ordinary differential equations (ODEs). Such an approximation is desired because the analysis of a system of ODEs is relatively easier than that of the original stochastic system.