Distributed Hypothesis Testing with Variable-Length Coding

TL;DR

This paper characterizes the optimal error exponent in distributed hypothesis testing with variable-length coding under average communication constraints, revealing that the strong converse does not hold and the exponent depends solely on channel capacity.

Contribution

It provides a novel analysis of distributed hypothesis testing with variable-length coding and stop-feedback under average communication constraints, extending previous fixed-length results.

Findings

Optimal type-II error exponent matches that of a scaled rate under average load.

Strong converse does not hold under average communication constraints.

Error exponent depends only on channel capacity, not the detailed transition law.

Abstract

The problem of distributed testing against independence with variable-length coding is considered when the \emph{average} and not the \emph{maximum} communication load is constrained as in previous works. The paper characterizes the optimum type-II error exponent of a single sensor single decision center system given a maximum type-I error probability when communication is either over a noise-free rate-$R$ link or over a noisy discrete memoryless channel (DMC) with stop-feedback. Specifically, let $\epsilon$ denote the maximum allowed type-I error probability. Then the optimum exponent of the system with a rate-$R$ link under a constraint on the average communication load coincides with the optimum exponent of such a system with a rate $R/(1-\epsilon)$ link under a maximum communication load constraint. A strong converse thus does not hold under an average communication load constraint. A similar observation holds also for testing against independence over DMCs. With variable-length coding and stop-feedback and under an average communication load constraint, the optimum type-II error exponent over a DMC of capacity $C$ equals the optimum exponent under fixed-length coding and a maximum communication load constraint when communication is over a DMC of capacity $C(1-\epsilon)^{-1}$. In particular, under variable-length coding over a DMC with stop feedback a strong converse result does not hold and the optimum error exponent depends on the transition law of the DMC only through its capacity.