The Sample Complexity of Distributed Simple Binary Hypothesis Testing under Information Constraints

TL;DR

This paper establishes tight bounds on the sample complexity of distributed simple binary hypothesis testing under communication constraints, showing that interaction does not reduce sample complexity and providing new technical tools.

Contribution

It provides the first definitive answer that interaction does not help reduce sample complexity and derives optimally tight bounds for communication-constrained hypothesis testing.

Findings

Interaction does not reduce sample complexity.

Derived tight bounds for communication-constrained testing.

Introduced a reverse data-processing inequality for Hellinger divergences.

Abstract

This paper resolves two open problems from a recent paper, arXiv:2403.16981, concerning the sample complexity of distributed simple binary hypothesis testing under information constraints. The first open problem asks whether interaction reduces the sample complexity of distributed simple binary hypothesis testing. In this paper, we show that sequential interaction does not help. The second problem suggests tightening existing sample complexity bounds for communication-constrained simple binary hypothesis testing. We derive optimally tight bounds for this setting and resolve this problem. Our main technical contributions are: (i) a one-shot lower bound on the Bayes error in simple binary hypothesis testing that satisfies a crucial tensorisation property; (ii) a streamlined proof of the formula for the sample complexity of simple binary hypothesis testing without constraints, first established in arXiv:2403.16981; and (iii) a reverse data-processing inequality for Hellinger-$\lambda$ divergences, generalising the results from arXiv:1812.03031 and arXiv:2206.02765.