Communication-constrained hypothesis testing: Optimality, robustness, and reverse data processing inequalities

TL;DR

This paper investigates hypothesis testing with communication limits, showing near-optimal sample complexity bounds, developing efficient algorithms, and extending results to robust and multi-hypothesis scenarios using novel inequalities.

Contribution

It introduces tight bounds on sample complexity under communication constraints, develops polynomial-time algorithms, and establishes new reverse data processing inequalities for robust testing.

Findings

Sample complexity under communication constraints is at most logarithmically larger than unconstrained case.

Developed a polynomial-time algorithm achieving near-optimal sample complexity.

Communication constraints can cause exponential increase in sample complexity for multi-hypothesis testing.

Abstract

We study hypothesis testing under communication constraints, where each sample is quantized before being revealed to a statistician. Without communication constraints, it is well known that the sample complexity of simple binary hypothesis testing is characterized by the Hellinger distance between the distributions. We show that the sample complexity of simple binary hypothesis testing under communication constraints is at most a logarithmic factor larger than in the unconstrained setting and this bound is tight. We develop a polynomial-time algorithm that achieves the aforementioned sample complexity. Our framework extends to robust hypothesis testing, where the distributions are corrupted in the total variation distance. Our proofs rely on a new reverse data processing inequality and a reverse Markov inequality, which may be of independent interest. For simple $M$-ary hypothesis testing, the sample complexity in the absence of communication constraints has a logarithmic dependence on $M$. We show that communication constraints can cause an exponential blow-up leading to $\Omega(M)$ sample complexity even for adaptive algorithms.