Local Privacy, Data Processing Inequalities, and Statistical Minimax Rates

TL;DR

This paper investigates the fundamental limits of statistical estimation under local privacy constraints, establishing tight bounds and proposing optimal privacy-preserving mechanisms across various statistical problems.

Contribution

It introduces new information-theoretic bounds linking privacy guarantees to statistical utility, and develops optimal mechanisms and estimators for multiple canonical problems.

Findings

Derived tight bounds on mutual information and divergence under privacy constraints.

Established optimal privacy-preserving mechanisms matching theoretical bounds.

Provided computationally efficient estimators achieving the bounds.

Abstract

Working under a model of privacy in which data remains private even from the statistician, we study the tradeoff between privacy guarantees and the utility of the resulting statistical estimators. We prove bounds on information-theoretic quantities, including mutual information and Kullback-Leibler divergence, that depend on the privacy guarantees. When combined with standard minimax techniques, including the Le Cam, Fano, and Assouad methods, these inequalities allow for a precise characterization of statistical rates under local privacy constraints. We provide a treatment of several canonical families of problems: mean estimation, parameter estimation in fixed-design regression, multinomial probability estimation, and nonparametric density estimation. For all of these families, we provide lower and upper bounds that match up to constant factors, and exhibit new (optimal) privacy-preserving mechanisms and computationally efficient estimators that achieve the bounds.