Convergence Rates for Differentially Private Statistical Estimation

TL;DR

This paper establishes theoretical bounds on the convergence rates of differentially private estimators, linking privacy guarantees to robust statistical measures like Gross Error Sensitivity, and highlights the necessity of bounded range conditions for privacy.

Contribution

It introduces upper and lower bounds on convergence rates for private estimators and connects differential privacy with robust statistics, providing new theoretical insights.

Findings

Convergence rate bounds depend on Gross Error Sensitivity.

Bounded range is necessary for strict differential privacy.

Established a formal link between privacy and robustness in statistics.

Abstract

Differential privacy is a cryptographically-motivated definition of privacy which has gained significant attention over the past few years. Differentially private solutions enforce privacy by adding random noise to a function computed over the data, and the challenge in designing such algorithms is to control the added noise in order to optimize the privacy-accuracy-sample size tradeoff. This work studies differentially-private statistical estimation, and shows upper and lower bounds on the convergence rates of differentially private approximations to statistical estimators. Our results reveal a formal connection between differential privacy and the notion of Gross Error Sensitivity (GES) in robust statistics, by showing that the convergence rate of any differentially private approximation to an estimator that is accurate over a large class of distributions has to grow with the GES of the estimator. We then provide an upper bound on the convergence rate of a differentially private approximation to an estimator with bounded range and bounded GES. We show that the bounded range condition is necessary if we wish to ensure a strict form of differential privacy.