Optimal high-dimensional and nonparametric distributed testing under communication constraints

TL;DR

This paper establishes fundamental limits and develops optimal algorithms for high-dimensional and nonparametric distributed hypothesis testing under communication constraints, revealing unique phenomena not seen in estimation tasks.

Contribution

It introduces minimax bounds and optimal testing procedures for distributed nonparametric testing with communication limits, including adaptive strategies and shared randomness effects.

Findings

Distributed testing can be performed with minimal communication, even 1-bit, outperforming local tests.

Shared randomness enhances testing performance in certain regimes.

Theoretical lower bounds match the performance of proposed algorithms.

Abstract

We derive minimax testing errors in a distributed framework where the data is split over multiple machines and their communication to a central machine is limited to $b$ bits. We investigate both the $d$- and infinite-dimensional signal detection problem under Gaussian white noise. We also derive distributed testing algorithms reaching the theoretical lower bounds. Our results show that distributed testing is subject to fundamentally different phenomena that are not observed in distributed estimation. Among our findings, we show that testing protocols that have access to shared randomness can perform strictly better in some regimes than those that do not. We also observe that consistent nonparametric distributed testing is always possible, even with as little as $1$-bit of communication and the corresponding test outperforms the best local test using only the information available at a single local machine. Furthermore, we also derive adaptive nonparametric distributed testing strategies and the corresponding theoretical lower bounds.