On the Statistical Complexity of Estimation and Testing under Privacy Constraints
This work addresses the challenge of balancing statistical accuracy with privacy protection for researchers and practitioners in machine learning and statistics, providing foundational insights into the statistical complexity of private estimation and testing.
The paper tackles the problem of deriving minimax lower bounds for differentially private estimators and characterizing the power of statistical tests under privacy constraints, showing that the impact of privacy on performance varies significantly across problem classes, with some scenarios experiencing noticeable degradation only at high privacy levels while others suffer significant decreases even with modest protection.
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.