We consider the problem of two-sample testing under a local differential privacy constraint where a permutation procedure is used to calibrate the tests. 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 permute the non-private data directly and only the private views. Further, through a refined theoretical analysis of the permutation procedure, we are able to avoid an equal sample size assumption which has been made 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.

翻译：我们研究了在局部差分隐私约束下的双样本检验问题，其中使用置换过程来校准检验。我们针对一般离散分布和满足光滑性约束的连续分布，构建了在至多对数因子范围内最优的检验方法。同时考虑了非交互式和交互式检验，并证明允许交互性可改善极小极大分离率。我们的结果表明，尽管无法直接置换原始非隐私数据而只能置换其隐私化视图，置换过程在局部隐私约束下仍具有实际可行性。此外，通过对置换过程的理论分析进行精细化改进，我们避免了以往置换检验文献中（无论是否存在隐私约束）普遍采用的等样本量假设。最后，我们通过数值实验验证了所提出检验方法的性能，并证实了理论结论，特别是允许交互性带来的性能提升。