Priv’IT: Private and Sample Efficient Identity Testing

We develop differentially private hypothesis testing methods for the small sample regime. Given a sample $\mathcal{D}$ from a categorical distribution $p$ over some domain $\Sigma$, an explicitly described distribution $q$ over $\Sigma$, some privacy parameter $\epsilon$, accuracy parameter $\alpha$, and requirements $\beta_\mathrm{I}$ and $\beta_\mathrm{II}$ for the type I and type II errors of our test, the goal is to distinguish between $p=q$ and $d_\mathrm{tv}(p,q) \ge \alpha$. We provide theoretical bounds for the sample size $|\mathcal{D}|$ so that our method both satisfies $(\epsilon,0)$-differential privacy, and guarantees $\beta_\mathrm{I}$ and $\beta_\mathrm{II}$ 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 $\chi^2$-test with noisy counts, or standard approaches such as repetition for endowing non-private $\chi^2$-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.