# Optimal Rates for Differentially Private Hypothesis Testing with E-values

arXiv: https://arxiv.org/abs/2605.28952 · HTML: https://mlsanity.com/paper/2605.28952/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 48 references that are in the map.
- First posted on arXiv 27 May 2026.

## Builds on (most cited first, 29 of 29)

- [Privacy Amplification by Subsampling: Tight Analyses via Couplings and Divergences](https://mlsanity.com/paper/1807.01647/) (2018)
- [Game-theoretic statistics and safe anytime-valid inference](https://mlsanity.com/paper/2210.01948/) (2022)
- [Safe Testing](https://mlsanity.com/paper/1906.07801/) (2019)
- [Hypothesis testing with e-values](https://mlsanity.com/paper/2410.23614/) (2024)
- [There are natural scores: Full comment on Shafer, "Testing by betting: A strategy for statistical and scientific communication"](https://mlsanity.com/paper/2102.05569/) (2021)
- [Admissible anytime-valid sequential inference must rely on nonnegative martingales](https://mlsanity.com/paper/2009.03167/) (2020)
- [Anytime-valid off-policy inference for contextual bandits](https://mlsanity.com/paper/2210.10768/) (2022)
- [E-detectors: a nonparametric framework for sequential change detection](https://mlsanity.com/paper/2203.03532/) (2022)
- [The Structure of Optimal Private Tests for Simple Hypotheses](https://mlsanity.com/paper/1811.11148/) (2018)
- [Auditing Fairness by Betting](https://mlsanity.com/paper/2305.17570/) (2023)
- [Unbiased Statistical Estimation and Valid Confidence Intervals Under Differential Privacy](https://mlsanity.com/paper/2110.14465/) (2021)
- [Online multiple testing with e-values](https://mlsanity.com/paper/2311.06412/) (2023)
- [Improved Differentially Private Analysis of Variance](https://mlsanity.com/paper/1903.00534/) (2019)
- [Quantile Multi-Armed Bandits: Optimal Best-Arm Identification and a Differentially Private Scheme](https://mlsanity.com/paper/2006.06792/) (2020)
- [The Test of Tests: A Framework For Differentially Private Hypothesis Testing](https://mlsanity.com/paper/2302.04260/) (2023)
- [Differentially Private Change-Point Detection](https://mlsanity.com/paper/1808.10056/) (2018)
- [Instance-Optimal Differentially Private Estimation](https://mlsanity.com/paper/2210.15819/) (2022)
- [RiLACS: Risk-Limiting Audits via Confidence Sequences](https://mlsanity.com/paper/2107.11323/) (2021)
- [Nonparametric extensions of randomized response for private confidence sets](https://mlsanity.com/paper/2202.08728/) (2022)
- [Differentially Private Best-Arm Identification](https://mlsanity.com/paper/2406.06408/) (2024)
- [Sequentially Auditing Differential Privacy](https://mlsanity.com/paper/2509.07055/) (2025)
- [Vector-valued self-normalized concentration inequalities beyond sub-Gaussianity](https://mlsanity.com/paper/2511.03606/) (2025)
- [Private Sequential Hypothesis Testing for Statisticians: Privacy, Error Rates, and Sample Size](https://mlsanity.com/paper/2204.04597/) (2022)
- [Accumulation Bias in Meta-Analysis: The Need to Consider Time in Error Control](https://mlsanity.com/paper/1905.13494/) (2019)
- [DP-SPRT: Differentially Private Sequential Probability Ratio Tests](https://mlsanity.com/paper/2508.06377/) (2025)
- [Differentially Private E-Values](https://mlsanity.com/paper/2510.18654/) (2025)
- [E-values for Adaptive Clinical Trials: Anytime-Valid Monitoring in Practice](https://mlsanity.com/paper/2602.06379/) (2026)
- [E-values as statistical evidence: A comparison to Bayes factors, likelihoods, and p-values](https://mlsanity.com/paper/2603.24421/) (2026)
- [Optimal e-variables under constraints](https://mlsanity.com/paper/2604.21680/) (2026)

## Built on by (most cited first, 1 of 1)

- [Private Anytime Selective-Risk Certification for Federated Retrieval-Augmented Generation: Guarantees and Empirical Limits](https://mlsanity.com/paper/2608.07913/) (2026)

## Abstract (arXiv)

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 mathbbP and mathbbQ, what is the maximum achievable e-power when testing X~ mathbbPn against X~mathbbQn 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.