Minimax Optimal Procedures for Locally Private Estimation

TL;DR

This paper establishes the fundamental limits and develops optimal privacy-preserving estimation procedures under local differential privacy constraints across various statistical models, supported by theoretical bounds and experiments.

Contribution

It introduces new minimax optimal estimation methods under local privacy constraints, extending classical bounds and providing practical algorithms for diverse statistical problems.

Findings

Derived tight lower and upper bounds for private estimation.

Proposed new privacy-preserving mechanisms matching theoretical bounds.

Validated methods through experiments on sensitive data.

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 risk of the resulting statistical estimators. We develop private versions of classical information-theoretic bounds, in particular those due to Le Cam, Fano, and Assouad. These inequalities allow for a precise characterization of statistical rates under local privacy constraints and the development of provably (minimax) optimal estimation procedures. We provide a treatment of several canonical families of problems: mean estimation and median estimation, generalized linear models, 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. Additionally, we present a variety of experimental results for estimation problems involving sensitive data, including salaries, censored blog posts and articles, and drug abuse; these experiments demonstrate the importance of deriving optimal procedures.