Instance-Optimal Differentially Private Estimation

TL;DR

This paper develops locally minimax optimal estimators under differential privacy, adapting to easier instances and leveraging private hypothesis testing results for improved estimation in exponential families.

Contribution

It introduces locally minimax differentially private estimators for exponential families and tail rate estimation, connecting private hypothesis testing to estimation optimality.

Findings

Constructed locally minimax private estimators for exponential families

Demonstrated that private hypothesis testing informs estimator design

Achieved improved convergence rates for private estimation

Abstract

In this work, we study local minimax convergence estimation rates subject to $\epsilon$-differential privacy. Unlike worst-case rates, which may be conservative, algorithms that are locally minimax optimal must adapt to easy instances of the problem. We construct locally minimax differentially private estimators for one-parameter exponential families and estimating the tail rate of a distribution. In these cases, we show that optimal algorithms for simple hypothesis testing, namely the recent optimal private testers of Canonne et al. (2019), directly inform the design of locally minimax estimation algorithms.