Optimal Private Median Estimation under Minimal Distributional Assumptions

TL;DR

This paper investigates the problem of estimating the median under differential privacy with minimal assumptions on the distribution, providing tight bounds and an optimal polynomial-time algorithm.

Contribution

It introduces nearly-tight bounds for median estimation under minimal distributional assumptions and presents a polynomial-time algorithm that achieves these bounds.

Findings

Exact statistical rate of median estimation derived

Optimal differentially private algorithm designed

Results applicable to distributions with unbounded values

Abstract

We study the fundamental task of estimating the median of an underlying distribution from a finite number of samples, under pure differential privacy constraints. We focus on distributions satisfying the minimal assumption that they have a positive density at a small neighborhood around the median. In particular, the distribution is allowed to output unbounded values and is not required to have finite moments. We compute the exact, up-to-constant terms, statistical rate of estimation for the median by providing nearly-tight upper and lower bounds. Furthermore, we design a polynomial-time differentially private algorithm which provably achieves the optimal performance. At a technical level, our results leverage a Lipschitz Extension Lemma which allows us to design and analyze differentially private algorithms solely on appropriately defined "typical" instances of the samples.