Social learning and distributed hypothesis testing

Abstract

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.

Large deviation analysis for learning rate in distributed hypothesis testing

“…This assumption is technical assumption one and relaxes the assumption of bounded support for the likelihood ratio random variable in prior work [1], [3]- [6]. Next, we provide families of distributions which satisfy Assumption 2 even though they might have unbounded support.…”

“…Even though each individual cannot identify the parameter through local observations alone, the parameter may be collectively identifiable. In this paper we study the learning rule proposed in [1], which is based on local Bayesian updating followed by consensus averaging on a reweighting of the log beliefs of nodes. Under the assumptions of network-wide observability and connectivity, in [1], we characterized the rate of convergence.…”

“…In this paper we study the learning rule proposed in [1], which is based on local Bayesian updating followed by consensus averaging on a reweighting of the log beliefs of nodes. Under the assumptions of network-wide observability and connectivity, in [1], we characterized the rate of convergence. Similar, learning rules were discussed in [2] and [3], for which the rate of rejecting wrong hypothesis and rate of convergence to the true hypothesis were characterized.…”

“…Furthermore, under the assumption that log-likelihood ratios are bounded, [4], [3] and [5] provide finite time lower bound on the rate of concentration of the rejection rate. In [6], under a new assumption for aperiodic networks we provided tight and matching asymptotic upper and lower bound on the rate of decay of wrong hypothesis rejection rates with small deviations from expected rejection rate.…”

“…In [4], we have provided a comprehensive discussion of those most relevant to this paper [2], [3], [7]- [11]. In the interest of brevity, we refer the interested the reader to [4].…”