Finite-time Analysis of the Distributed Detection Problem

TL;DR

This paper provides a finite-time analysis of distributed detection algorithms in fixed and random networks, introducing new bounds based on network properties and signal structures, with exponential convergence guarantees.

Contribution

It introduces a finite-time analysis framework for distributed detection, including the Kullback-Leibler cost and convergence bounds in both fixed and random network settings.

Findings

Bounded the detection cost in fixed networks using spectral gap and entropy.

Proved exponential convergence in random networks with high probability.

Non-asymptotic convergence rate scales inversely with spectral gap.

Abstract

This paper addresses the problem of distributed detection in fixed and switching networks. A network of agents observe partially informative signals about the unknown state of the world. Hence, they collaborate with each other to identify the true state. We propose an update rule building on distributed, stochastic optimization methods. Our main focus is on the finite-time analysis of the problem. For fixed networks, we bring forward the notion of Kullback-Leibler cost to measure the efficiency of the algorithm versus its centralized analog. We bound the cost in terms of the network size, spectral gap and relative entropy of agents' signal structures. We further consider the problem in random networks where the structure is realized according to a stationary distribution. We then prove that the convergence is exponentially fast (with high probability), and the non-asymptotic rate scales inversely in the spectral gap of the expected network.