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Near-Optimal Private Tests for Simple and MLR Hypotheses

Yu-Wei Chen, Raghu Pasupathy, Jordan Awan

TL;DR

This work develops near-optimal private tests for simple and MLR hypotheses under Gaussian differential privacy by constructing a data-driven private mean estimator with adaptive clamping. The core methods GDP-Quant and GDP-MeanEst deliver privacy guarantees while achieving GDP minimax-optimal rates and preserving asymptotic relative efficiency ($ARE=1$) with non-private tests. The framework yields unbiased, GDP-compliant tests for simple, one-sided MLR, and two-sided exponential-family hypotheses, with rigorous guarantees and practical calibration. Empirical results show that the private tests outperform competing DP methods and approach the power of non-private tests at moderate sample sizes and privacy budgets, highlighting potential impact for privacy-conscious statistical inference.

Abstract

We develop a near-optimal testing procedure under the framework of Gaussian differential privacy for simple as well as one- and two-sided tests under monotone likelihood ratio conditions. Our mechanism is based on a private mean estimator with data-driven clamping bounds, whose population risk matches the private minimax rate up to logarithmic factors. Using this estimator, we construct private test statistics that achieve the same asymptotic relative efficiency as the non-private, most powerful tests while maintaining conservative type I error control. In addition to our theoretical results, our numerical experiments show that our private tests outperform competing DP methods and offer comparable power to the non-private most powerful tests, even at moderately small sample sizes and privacy loss budgets.

Near-Optimal Private Tests for Simple and MLR Hypotheses

TL;DR

This work develops near-optimal private tests for simple and MLR hypotheses under Gaussian differential privacy by constructing a data-driven private mean estimator with adaptive clamping. The core methods GDP-Quant and GDP-MeanEst deliver privacy guarantees while achieving GDP minimax-optimal rates and preserving asymptotic relative efficiency () with non-private tests. The framework yields unbiased, GDP-compliant tests for simple, one-sided MLR, and two-sided exponential-family hypotheses, with rigorous guarantees and practical calibration. Empirical results show that the private tests outperform competing DP methods and approach the power of non-private tests at moderate sample sizes and privacy budgets, highlighting potential impact for privacy-conscious statistical inference.

Abstract

We develop a near-optimal testing procedure under the framework of Gaussian differential privacy for simple as well as one- and two-sided tests under monotone likelihood ratio conditions. Our mechanism is based on a private mean estimator with data-driven clamping bounds, whose population risk matches the private minimax rate up to logarithmic factors. Using this estimator, we construct private test statistics that achieve the same asymptotic relative efficiency as the non-private, most powerful tests while maintaining conservative type I error control. In addition to our theoretical results, our numerical experiments show that our private tests outperform competing DP methods and offer comparable power to the non-private most powerful tests, even at moderately small sample sizes and privacy loss budgets.
Paper Structure (26 sections, 23 theorems, 97 equations, 10 figures, 2 algorithms)

This paper contains 26 sections, 23 theorems, 97 equations, 10 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Hypothesis Testing
  4. Differential Privacy
  5. Problem Formulation and Basic Algorithms
  6. Problem Formulation
  7. Private Quantile Estimation
  8. Private Mean Estimation
  9. Near-Optimal Private Tests
  10. Simple Hypothesis
  11. One-sided Hypothesis with MLR
  12. Two-sided Hypothesis with Exponential Family
  13. Simulation
  14. Private Mean Estimation
  15. Simple Hypothesis
  16. ...and 11 more sections

Key Result

Proposition 2.2

If the type II error $-\frac{1}{n}\log P_{\theta_1}\left( \varphi_{\nu,n}^{(i)}=0 \right)\to c^{(i)}(\theta_1)$, then $c^{(i)}(\theta_1)$ is the Bahadur slope, and $\mathrm{ARE} = \frac{c^{(1)}(\theta_1)}{c^{(2)}(\theta_1)}$.

Figures (10)

  • Figure 1: GDP mean estimation comparison ($\epsilon=1$)
  • Figure 2: Simple hypothesis under $t$ data
  • Figure 3: One-sided hypothesis under Gaussian data ($\epsilon=1$)
  • Figure 4: Two-sided hypothesis under Logistic data ($\epsilon=1$)
  • Figure 5: GDP mean estimation comparison ($\epsilon=0.5$)
  • ...and 5 more figures

Theorems & Definitions (49)

  • Definition 2.1: Relative efficiency, van2000asymptotic
  • Proposition 2.2: Bahadur slope, bahadur1967rates
  • Definition 2.3: Monotone Likelihood Ratio
  • Definition 2.4: Gaussian Differential Privacy, dong2022gaussian
  • Proposition 2.5: Gaussian Mechanism, dong2022gaussian
  • Lemma 0: GDP-Quant
  • proof : Proof Sketch
  • Proposition 3.1: GDP Guarantee
  • Definition 3.2
  • Theorem 3.2: GDP-MeanEst Utility
  • ...and 39 more