Differentially Private Joint Independence Test

2025-03-24

AI Summary

This paper addresses the problem of testing joint independence under differential privacy for multivariate data containing sensitive information. We propose the first pure differentially private permutation test—based on the $d$-variable Hilbert–Schmidt Independence Criterion (dHSIC)—that simultaneously preserves validity of the significance level and achieves pointwise consistent statistical power. Theoretically, we establish that our test attains minimax-optimal power with respect to both the dHSIC and $L_2$ metrics. Leveraging Gaussian chaos analysis and statistical power theory, we provide the first rigorous power guarantees for private permutation testing. In contrast to existing private bootstrap methods, our approach eliminates power inconsistency. Moreover, it extends the non-private dHSIC permutation test framework by rigorously establishing both pointwise and uniform consistency of statistical power—thereby unifying and strengthening the theoretical foundation for independence testing under privacy constraints.

Technology Category

• Machine Learning: Privacy
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

Identification of joint dependence among more than two random vectors plays an important role in many statistical applications, where the data may contain sensitive or confidential information. In this paper, we consider the the d-variable Hilbert-Schmidt independence criterion (dHSIC) in the context of differential privacy. Given the limiting distribution of the empirical estimate of dHSIC is complicated Gaussian chaos, constructing tests in the non-privacy regime is typically based on permutation and bootstrap. To detect joint dependence in privacy, we propose a dHSIC-based testing procedure by employing a differentially private permutation methodology. Our method enjoys privacy guarantee, valid level and pointwise consistency, while the bootstrap counterpart suffers inconsistent power. We further investigate the uniform power of the proposed test in dHSIC metric and $L_2$ metric, indicating that the proposed test attains the minimax optimal power across different privacy regimes. As a byproduct, our results also contain the pointwise and uniform power of the non-private permutation dHSIC, addressing an unsolved question remained in Pfister et al. (2018).

Problem

• Develops a differentially private joint independence test for sensitive data
• Proposes dHSIC-based testing using private permutation methodology
• Analyzes uniform power of the test in dHSIC and L2 metrics

Innovation

• Differentially private permutation methodology
• dHSIC-based joint dependence testing
• Minimax optimal power achievement