Priv'IT: Private and Sample Efficient Identity Testing

We develop differentially private hypothesis testing methods for the small sample regime. Given a sample cal D from a categorical distribution p over some domain Sigma, an explicitly described distribution q over Sigma, some privacy parameter varepsilon, accuracy parameter α, and requirements βrm I and βrm II for the type I and type II errors of our test, the goal is to distinguish between p=q and d_rmTV(p,q) ≥ α. We provide theoretical bounds for the sample size |cal D| so that our method both satisfies (varepsilon,0)-differential privacy, and guarantees βrm I and βrm II type I and type II errors. We show that differential privacy may come for free in some regimes of parameters, and we always beat the sample complexity resulting from running the chi2-test with noisy counts, or standard approaches such as repetition for endowing non-private chi2-style statistics with differential privacy guarantees. We experimentally compare the sample complexity of our method to that of recently proposed methods for private hypothesis testing.