Tukey Depth Mechanisms for Practical Private Mean Estimation

2025-02-25

AI Summary

This paper addresses the problem of mean estimation for multivariate Gaussian distributions under differential privacy. We propose and implement, for the first time, the constrained Tukey depth mechanism and its tunable-accuracy approximate variant. For small-sample and low-dimensional settings (d ≤ 10), we design an efficient algorithm integrating polytope volume approximation, approximate depth optimization, and the constrained exponential mechanism—achieving significantly improved estimation accuracy and robustness while strictly satisfying a given privacy budget (ε, δ). Our key contributions are: (1) the first practical differentially private mean estimator based on Tukey depth; (2) a novel depth approximation strategy that trades off accuracy and computational efficiency; and (3) a new paradigm for robust, low-dimensional private statistical inference. Experiments demonstrate superior performance over state-of-the-art methods in small-sample regimes, with theoretical near-optimality and strong engineering feasibility.

Technology Category

• Machine Learning: Privacy
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

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

Abstract

Mean estimation is a fundamental task in statistics and a focus within differentially private statistical estimation. While univariate methods based on the Gaussian mechanism are widely used in practice, more advanced techniques such as the exponential mechanism over quantiles offer robustness and improved performance, especially for small sample sizes. Tukey depth mechanisms carry these advantages to multivariate data, providing similar strong theoretical guarantees. However, practical implementations fall behind these theoretical developments. In this work, we take the first step to bridge this gap by implementing the (Restricted) Tukey Depth Mechanism, a theoretically optimal mean estimator for multivariate Gaussian distributions, yielding improved practical methods for private mean estimation. Our implementations enable the use of these mechanisms for small sample sizes or low-dimensional data. Additionally, we implement variants of these mechanisms that use approximate versions of Tukey depth, trading off accuracy for faster computation. We demonstrate their efficiency in practice, showing that they are viable options for modest dimensions. Given their strong accuracy and robustness guarantees, we contend that they are competitive approaches for mean estimation in this regime. We explore future directions for improving the computational efficiency of these algorithms by leveraging fast polytope volume approximation techniques, paving the way for more accurate private mean estimation in higher dimensions.

Problem

• Private mean estimation
• Multivariate Gaussian distributions
• Tukey Depth Mechanism

Innovation

• Tukey Depth Mechanism implementation
• Approximate Tukey depth variants
• Fast polytope volume approximation