Tukey Depth Mechanisms for Practical Private Mean Estimation

TL;DR

This paper implements and evaluates practical versions of Tukey depth mechanisms for differentially private mean estimation in multivariate data, improving robustness and accuracy especially for small samples and low dimensions.

Contribution

It bridges the gap between theoretical optimal multivariate private mean estimators and practical implementations, including approximate variants for efficiency.

Findings

Implemented the (Restricted) Tukey Depth Mechanism for practical use.

Demonstrated efficiency of approximate Tukey depth variants in low dimensions.

Showed these methods are competitive for small sample sizes and modest dimensions.

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.