Optimal Rates for Differentially Private Hypothesis Testing with E-values

2026-05-27

AI Summary

This study investigates how to achieve maximal e-power—statistical power defined via E-values—for hypothesis testing under ε-differential privacy constraints, considering both fixed-sample and sequential observation settings. By integrating differential privacy mechanisms, E-value theory, and sequential analysis, the work establishes the first characterization of the optimal rate for E-value-based tests under privacy protection and derives tight upper and lower bounds on the stopping time of sequential procedures. The proposed algorithm constructs an optimal e-process that significantly reduces the required sample size across various privacy levels and sequential testing tasks, outperforming the existing DP-SPRT method and demonstrating both theoretical optimality and practical efficacy.

Abstract

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 $\varepsilon$-differential privacy? We characterize the optimal rate for this problem and provide an algorithm which matches it exactly. 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 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.

Problem

• differentially private hypothesis testing
• e-values
• optimal rates
• sequential testing
• e-power

Innovation

• differentially private hypothesis testing
• e-values
• optimal rates
• sequential testing
• e-process