Priv'IT: Private and Sample Efficient Identity Testing

We develop differentially private hypothesis testing methods for the small sample regime. Given a sample D from a categorical distribution p over some domain Σ, an explicitly described distribution q over Σ, some privacy parameter ε, accuracy parameter α, and requirements β_ I and β_ II for the type I and type II errors of our test, the goal is to distinguish between p=q and d_TV(p,q) ≥α. We provide theoretical bounds for the sample size | D| so that our method both satisfies (ε,0)-differential privacy, and guarantees β_ I and β_ 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 χ^2-test with noisy counts, or standard approaches such as repetition for endowing non-private χ^2-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.