Minimax Optimal Two-Sample Testing under Local Differential Privacy

TL;DR

This paper develops minimax optimal two-sample tests under local differential privacy for multinomial and continuous data, balancing privacy constraints with statistical utility, and introduces adaptive methods for unknown smoothness.

Contribution

It introduces private permutation tests for multinomial data, extends to continuous data via binning, and proposes adaptive testing methods for unknown smoothness, all achieving minimax rates under LDP.

Findings

Tests rigorously control type I error under LDP

Achieve minimax separation rates under privacy constraints

Adaptive test performs well without prior smoothness knowledge

Abstract

We explore the trade-off between privacy and statistical utility in private two-sample testing under local differential privacy (LDP) for both multinomial and continuous data. We begin by addressing the multinomial case, where we introduce private permutation tests using practical privacy mechanisms such as Laplace, discrete Laplace, and Google's RAPPOR. We then extend our multinomial approach to continuous data via binning and study its uniform separation rates under LDP over H\"older and Besov smoothness classes. The proposed tests for both discrete and continuous cases rigorously control the type I error for any finite sample size, strictly adhere to LDP constraints, and achieve minimax separation rates under LDP. The attained minimax rates reveal inherent privacy-utility trade-offs that are unavoidable in private testing. To address scenarios with unknown smoothness parameters in density testing, we propose an adaptive test based on a Bonferroni-type approach that ensures robust performance without prior knowledge of the smoothness parameters. We validate our theoretical findings with extensive numerical experiments and demonstrate the practical relevance and effectiveness of our proposed methods.