Locally differentially private two-sample testing

Summary We consider the problem of two-sample testing under a local differential privacy constraint where a permutation procedure is used. We develop testing procedures which are optimal up to logarithmic factors, for general discrete distributions and continuous distributions subject to a smoothness constraint. Both non-interactive and interactive tests are considered, and we show 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 non-private data directly and only the private views. Further, through a tighter theoretical analysis of the permutation procedure, we are able to relax a balanced sample size assumption which is imposed in the permutation testing literature regardless of the presence of the privacy constraint. Lastly, we conduct numerical experiments which demonstrate the performance of our proposed test and verify the theoretical findings, especially the improved performance enabled by allowing interactivity.