Locally differentially private two-sample testing

SUMMARY We consider the problem of two-sample testing under a local differential privacy constraint using a permutation procedure. We develop testing procedures that are optimal up to logarithmic factors for general discrete distributions and continuous distributions subject to a smoothness constraint. Both noninteractive and interactive tests are considered, and we show that allowing interactivity results in an improvement in the minimax separation rates. Our results show that permutation procedures remain feasible in practice under local privacy constraints, despite the inability to perturb the nonprivate data directly and the ability to perturb only the private views. Furthermore, through a tighter theoretical analysis of the permutation procedure, we are able to relax a balanced-sample-size assumption imposed in the permutation-testing literature, regardless of the presence of the privacy constraint. Lastly, we conduct numerical experiments that demonstrate the performance of our proposed test and verify the theoretical findings, especially the improved performance enabled by allowing interactivity.