Optimal Rates for Differentially Private Hypothesis Testing with E-values

E-values have attracted considerable interest in recent years as flexible tools for enabling anytime-valid and adaptive data analysis. Hypothesis testing is at the core of many of these applications, which can often involve private or sensitive data. In this work, we answer a simple but important question: given two distributions $\mathbb{P}$ and $\mathbb{Q}$, what is the maximum achievable e-power when testing $X\sim {\mathbb{P}}^n$ against $X\sim {\mathbb{Q}}^n$ with e-values that satisfy $\epsilon$-differential privacy? We characterize the optimal rate for this problem and provide an algorithm which matches it asymptotically. In the sequential setting, when observations arrive one-by-one and the analyst chooses when to halt, we give matching upper and lower bounds on the stopping times of any differentially private e-process. Numerical experiments confirm the practicality of our algorithms, which require less data than the recently proposed DP-SPRT across a range of sequential testing problems and privacy levels.