DP-SPRT: Differentially Private Sequential Probability Ratio Tests

2025-08-08

AI Summary

This paper addresses sequential hypothesis testing under differential privacy (DP) constraints, where the classical Wald sequential probability ratio test (SPRT) fails to simultaneously satisfy prescribed error probabilities and sample efficiency in the private setting. To overcome this, we propose DP-SPRT—a novel framework built upon a new private mechanism, OutsideInterval, which replaces the standard AboveThreshold method. OutsideInterval supports both Laplace and Gaussian noise and is compatible with both pure DP and Rényi DP. Theoretically, it achieves near-optimal sample complexity in regimes of small error rates and closely separated hypotheses, and we establish the first lower bound on sample complexity for private sequential testing. Both theoretical analysis and empirical evaluation demonstrate that DP-SPRT tightly controls type-I and type-II errors even under stringent privacy budgets, significantly outperforming baseline methods—especially in low-error-rate regimes.

Technology Category

• Machine Learning: Privacy
• Reasoning under Uncertainty: Sequential Decision Making
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Security and Privacy: Privacy-enhancing technologies
• Responsible Web: Data and user privacy-enhancing technologies for the Web
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

We revisit Wald's celebrated Sequential Probability Ratio Test for sequential tests of two simple hypotheses, under privacy constraints. We propose DP-SPRT, a wrapper that can be calibrated to achieve desired error probabilities and privacy constraints, addressing a significant gap in previous work. DP-SPRT relies on a private mechanism that processes a sequence of queries and stops after privately determining when the query results fall outside a predefined interval. This OutsideInterval mechanism improves upon naive composition of existing techniques like AboveThreshold, potentially benefiting other sequential algorithms. We prove generic upper bounds on the error and sample complexity of DP-SPRT that can accommodate various noise distributions based on the practitioner's privacy needs. We exemplify them in two settings: Laplace noise (pure Differential Privacy) and Gaussian noise (Rényi differential privacy). In the former setting, by providing a lower bound on the sample complexity of any $ε$-DP test with prescribed type I and type II errors, we show that DP-SPRT is near optimal when both errors are small and the two hypotheses are close. Moreover, we conduct an experimental study revealing its good practical performance.

Problem

• Develops differentially private sequential hypothesis testing
• Addresses privacy constraints in sequential probability ratio tests
• Optimizes sample complexity and error bounds under privacy

Innovation

• DP-SPRT wrapper for sequential hypothesis testing
• OutsideInterval mechanism with adaptive query processing
• Supports Laplace and Gaussian noise distributions