Differentially Private Testing of Identity and Closeness of Discrete Distributions

We study the fundamental problems of identity testing (goodness of fit), and closeness testing (two sample test) of distributions over $k$ elements, under differential privacy. While the problems have a long history in statistics, finite sample bounds for these problems have only been established recently. In this work, we derive upper and lower bounds on the sample complexity of both the problems under $\varepsilon$-differential privacy. Our results improve over the best known algorithms for identity testing, and are the first results for differentially private closeness testing. Our bounds are tight up to a constant factor whenever the number of samples is $O(k)$, a regime which has garnered much attention over the last decade since it allows property testing even when only a fraction of the domain is observed. Our upper bounds are obtained by converting (and sometimes combining) the known non-private testing algorithms into differentially private algorithms. We propose a simple, yet general procedure based on coupling of distributions, to establish sample complexity lower bounds for differentially private algorithms.