On the Arithmetic and Geometric Fusion of Beliefs for Distributed Inference

TL;DR

This paper analyzes how different belief fusion methods in distributed hypothesis testing affect learning speed, showing that log-linear fusion leads to faster exponential learning rates, with insights into network effects.

Contribution

It provides a comparative analysis of linear and log-linear belief combination rules, including closed-form solutions for specific network architectures.

Findings

Agents learn the true hypothesis exponentially fast under both methods.

Log-linear fusion achieves a faster learning rate than linear fusion.

Network connectivity and information diversity influence the learning rates.

Abstract

We study the asymptotic learning rates under linear and log-linear combination rules of belief vectors in a distributed hypothesis testing problem. We show that under both combination strategies, agents are able to learn the truth exponentially fast, with a faster rate under log-linear fusion. We examine the gap between the rates in terms of network connectivity and information diversity. We also provide closed-form expressions for special cases involving federated architectures and exchangeable networks.