Distributed Learning with Infinitely Many Hypotheses

TL;DR

This paper studies distributed learning over networks with infinitely many hypotheses, providing bounds on belief convergence and connecting update rules to stochastic mirror descent.

Contribution

It extends analysis to countably and continuum hypotheses, offering non-asymptotic bounds and a new perspective on distributed belief updates as stochastic mirror descent.

Findings

Non-asymptotic concentration bounds derived.

Beliefs converge around the true hypothesis under certain conditions.

Distributed stochastic mirror descent framework is validated.

Abstract

We consider a distributed learning setup where a network of agents sequentially access realizations of a set of random variables with unknown distributions. The network objective is to find a parametrized distribution that best describes their joint observations in the sense of the Kullback-Leibler divergence. Apart from recent efforts in the literature, we analyze the case of countably many hypotheses and the case of a continuum of hypotheses. We provide non-asymptotic bounds for the concentration rate of the agents' beliefs around the correct hypothesis in terms of the number of agents, the network parameters, and the learning abilities of the agents. Additionally, we provide a novel motivation for a general set of distributed Non-Bayesian update rules as instances of the distributed stochastic mirror descent algorithm.