Large deviation analysis for learning rate in distributed hypothesis testing

Abstract

This paper considers a problem of distributed hypothesis testing and cooperative 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. We consider a social ("non-Bayesian") learning rule from previous literature, 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. For this learning rule, we know that under mild assumptions, the belief of any node in any incorrect parameter converges to zero exponentially fast, and the exponential rate of learning is a characterized by the network structure and the divergences between the observations' distributions. Furthermore, any (large) deviation from this nominal rate of rejecting wrong hypothesis is known to occur with an exponentially small probability when the bounded likelihood functions is uniformly bounded. While relaxing the assumption of uniformly bounded log-likelihood functions, we show that the rates of rejecting wrong hypotheses, collectively, satisfy a large deviation principle under a mild technical condition on the log-moment generating functions of log-likelihood random variables.

A New Approach for Distributed Hypothesis Testing with Extensions to Byzantine-Resilience

“…Thus, based on (13), we must have µ i,t(ω) (θ ) > 0, yielding the desired contradiction. With η(ω) min{γ 1 − δ, γ 2 (ω)} > 0, one can easily verify the following by referring to (13):…”

“…The key point of distinction among such rules stems from the specific manner in which neighboring opinions are aggregated. Specifically, linear opinion pooling is studied in [4][5][6], whereas log-linear opinion pooling is studied in [7][8][9][10][11][12][13][14]. Under appropriate conditions on the observation model and the network structure, each of these approaches enable every agent to learn the true state exponentially fast, with probability 1.…”

“…We point out that beliefconsensus algorithms on graphs have been studied prior to [4] as well as in [16,17]. The model in [16,17] differs from that in [4][5][6][7][8][9][10][11][12][13][14] in one key aspect: while in the former each agent has access to only one observation, the latter allows for influx of new information into the network in the form of a time-series of observations at every agent.…”

“…In particular, for the linear consensus protocol introduced in [4], the limiting rate at which a particular false hypothesis is eliminated is almost surely upper-bounded by a quantity that depends on the relative entropies and centralities of the agents [5]. The log-linear rules in [9][10][11][12][13] improve upon such a rate: with probability 1, the asymptotic rate of rejection of a false hypothesis under such rules is a convex combination of the agents' relative entropies, where the convex weights correspond to the eigenvector centralities of the agents. In contrast, based on our approach, each false hypothesis is rejected by every agent exponentially fast, at a rate that is almost surely lower-bounded by the best relative entropy (between the true state and the false hypothesis) among all agents, provided the underlying network is static and strongly-connected.…”

Distributed Hypothesis Testing and Social Learning in Finite Time with a Finite Amount of Communication

“…This class of problems has been studied for several decades, initially for scenarios involving a centralized fusion center [1], and more recently in fully distributed settings where agents are interconnected over a network [2][3][4][5][6][7][8][9][10][11][12][13]. The distributed algorithms provided in these latter papers require each agent to iteratively combine belief vectors obtained from their neighbors with Bayesian updates involving their local signals [2][3][4][5][6][7][8][9][10][11][12][13]. These rules ensure that all agents asymptotically learn the true state of the world, with the main differences being in the rate of learning.…”