Minimax Optimal Two-Sample Testing under Local Differential Privacy
This work addresses privacy-preserving statistical testing for data analysts, offering rigorous methods but is incremental as it builds on existing LDP frameworks.
The paper tackles the problem of two-sample testing under local differential privacy (LDP) for multinomial and continuous data, achieving minimax separation rates that reveal unavoidable privacy-utility trade-offs, with validation through numerical experiments.
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 Hölder 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.