On the Statistical Complexity of Estimation and Testing under Privacy Constraints

TL;DR

This paper investigates the fundamental limits of statistical estimation and testing under differential privacy constraints, providing new bounds, insights into utility degradation, and demonstrating the effectiveness of DP-SGLD for private maximum likelihood estimation.

Contribution

It introduces a transport-based characterization of test power under differential privacy and derives new inequalities, enhancing understanding of privacy-utility trade-offs in statistical procedures.

Findings

Privacy can cause significant utility loss in some problems at high privacy levels.

In other scenarios, modest privacy levels only slightly impact performance.

DP-SGLD achieves near-optimal private maximum likelihood estimates across various models.

Abstract

The challenge of producing accurate statistics while respecting the privacy of the individuals in a sample is an important area of research. We study minimax lower bounds for classes of differentially private estimators. In particular, we show how to characterize the power of a statistical test under differential privacy in a plug-and-play fashion by solving an appropriate transport problem. With specific coupling constructions, this observation allows us to derive Le Cam-type and Fano-type inequalities not only for regular definitions of differential privacy but also for those based on Renyi divergence. We then proceed to illustrate our results on three simple, fully worked out examples. In particular, we show that the problem class has a huge importance on the provable degradation of utility due to privacy. In certain scenarios, we show that maintaining privacy results in a noticeable reduction in performance only when the level of privacy protection is very high. Conversely, for other problems, even a modest level of privacy protection can lead to a significant decrease in performance. Finally, we demonstrate that the DP-SGLD algorithm, a private convex solver, can be employed for maximum likelihood estimation with a high degree of confidence, as it provides near-optimal results with respect to both the size of the sample and the level of privacy protection. This algorithm is applicable to a broad range of parametric estimation procedures, including exponential families.