Social Learning and Distributed Hypothesis Testing

TL;DR

This paper analyzes a distributed hypothesis testing method where nodes update beliefs based on local observations and neighbor communication, proving exponential convergence to the true hypothesis under mild conditions.

Contribution

It introduces a belief update rule combining Bayesian and non-Bayesian consensus, and characterizes the exponential convergence rate in networked social learning.

Findings

Beliefs in incorrect hypotheses decay exponentially fast.

Convergence rate depends on network structure and distribution divergences.

Main result is the concentration property of the convergence rate.

Abstract

This paper considers a problem of distributed hypothesis testing and social learning. Individual nodes in a network receive noisy local (private) observations whose distribution is parameterized by a discrete parameter (hypotheses). The conditional distributions are known locally at the nodes, but the true parameter/hypothesis is not known. An update rule is analyzed in which nodes first perform a Bayesian update of their belief (distribution estimate) of the parameter based on their local observation, communicate these updates to their neighbors, and then perform a "non-Bayesian" linear consensus using the log-beliefs of their neighbors. In this paper we show that under mild assumptions, the belief of any node in any incorrect hypothesis converges to zero exponentially fast, and we characterize the exponential rate of learning which is given in terms of the network structure and the divergences between the observations' distributions. Our main result is the concentration property established on the rate of convergence.