Social Learning and Distributed Hypothesis Testing

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 marginals of the joint observation distribution conditioned on each hypothesis 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 each hypothesis based on their local observations, communicate these updates to their neighbors, and then perform a "non-Bayesian" linear consensus using the log-beliefs of their neighbors. Under mild assumptions, we show that the belief of any node on a wrong hypothesis converges to zero exponentially fast, and the exponential rate of learning is characterized by the nodes' influence of the network and the divergences between the observations' distributions. For a broad class of observation statistics which includes distributions with unbounded support such as Gaussian mixtures, we show that rate of rejection of wrong hypothesis satisfies a large deviation principle i.e., the probability of sample paths on which the rate of rejection of wrong hypothesis deviates from the mean rate vanishes exponentially fast and we characterize the rate function in terms of the nodes' influence of the network and the local observation models.This paper was presented in part in [1]- [3]. A. Lalitha and T. Javidi are with the

Distributed estimation and learning over heterogeneous networks

“…Geometric averaging and logarithmic opinion pools have a long history in Bayesian analysis and behavioral decision models [40], [41] and they can be also justified under specific behavioral assumptions [42]. The are also quite popular as a non-Bayesian update rule in distributed detection and estimation litrature [43], [44], [10], [14], [45]. In [45] the authors use a logarithmic opinion pool to combine the estimated posterior probability distributions in a Bayesian consensus filter; and show that as a result: the sum of KullbackLeibler divergences between the consensual probability distribution and the local posterior probability distributions is minimized.…”

“…Other relevant results investigate the formation and evolution of beliefs in social networks and subsequent shaping of the individual and mass behavior through social learning [12], [13], [14]. The archetype of such models is the one due to DeGroot [15], where agents update their beliefs to a convex combination of their neighbor's beliefs and the coefficients correspond to the level of confidence that each agent puts in each of her neighbors.…”

Social Learning with Partial Information Sharing

“…Proof of Theorem 1. When θ TX = θ0, if the inequality in (12) holds, the convergence rate from Lemma 1 will be strictly negative, which implies that, since µ k,i (θ0) is bounded by 1, for any θ = θ0, limi→∞ µ k,i (θ) a.s. = 0, resulting in (12). When θ TX = θ0, the convergence behavior will depend on the sign of the RHS of (6).…”

“…Many algorithmic approaches have been conceived for this purpose [1][2][3][4], including the non-Bayesian approach, in which agents update their beliefs or opinions by using local streaming observations and by combining information shared by their neighbors. Some of the main studies along these lines rely on consensus and diffusion strategies [5,6], both with linear and log-exponential belief combinations (see, e.g., [7][8][9][10][11][12]). In all of these works, it is assumed that agents share with their neighbors their entire belief vectors.…”

Learning Graph Influence from Social Interactions

“…However, with the exception [6][7][8], most prior works focus mainly on strongly-connected networks (i.e., graphs where there is a direct and reverse path between any two agents, in addition to some agents having a self-loop as a sign of confidence in their own information). Under this setting, the limiting (as time elapses) evolution of the individual agents' belief has been shown to converge collectively to the same opinion, which can be the true underlying hypothesis [5,6,10], or a hypothesis minimizing a suitable objective function [9].…”