Distributed Gaussian Mean Testing under Communication Constraints: messages, samples, and coins

2026-05-28

AI Summary

This work addresses the problem of distributed Gaussian mean testing under communication constraints, where multiple users possess heterogeneous sample sizes and can transmit only a limited number of bits to a central referee, while sharing a small amount of public randomness. It presents the first unified treatment of the triple heterogeneity arising from sample sizes, communication budgets, and shared randomness, substantially generalizing existing theoretical frameworks. By integrating information-theoretic lower bounds, hypothesis testing theory, and communication complexity analysis—alongside carefully designed randomized protocols and minimax risk derivations—the study establishes necessary and sufficient conditions for achievable detection in this generalized setting, precisely characterizing the fundamental trade-offs among sample complexity, communication cost, and the utility of shared randomness.

Abstract

We revisit the problem of Gaussian mean testing in a distributed, communication constrained setting, where each of $n$ users independently observes samples from an unknown $d$-dimensional spherical Gaussian distribution $\mathcal{G}(μ,\mathbb{I}_d)$, and can communicate up to $\ell$ bits to a central referee. The referee's goal is then to distinguish between cases (i) $\|μ\|_2 = 0$ versus (ii) $\|μ\|_2\ge \varepsilon$. This problem has been considered in the private- and public-coin settings, when each user holds exactly one sample, or more generally when each holds exactly $m$ samples. In this work, we significantly generalize the question in three directions: when the users only share a small number $s$ of random bits, when each user holds a different number of samples $m_k$, and when each user can send a different number of bits $\ell_k$ to the referee.

Problem

• Distributed testing
• Gaussian mean
• Communication constraints
• Hypothesis testing
• Randomness sharing

Innovation

• distributed testing
• communication constraints
• Gaussian mean testing
• heterogeneous samples
• limited shared randomness