Private Statistical Estimation of Many Quantiles

TL;DR

This paper investigates methods for privately estimating many quantiles of a distribution using differential privacy, comparing empirical quantile estimation and density estimation approaches, and analyzing their tradeoffs.

Contribution

It introduces and compares two approaches for private quantile estimation, highlighting the tradeoff between direct quantile estimation and density-based methods.

Findings

Density estimation outperforms empirical quantile estimation for many quantiles.

The paper analyzes the statistical properties of recursive quantile estimation algorithms.

It identifies a tradeoff between the two estimation approaches.

Abstract

This work studies the estimation of many statistical quantiles under differential privacy. More precisely, given a distribution and access to i.i.d. samples from it, we study the estimation of the inverse of its cumulative distribution function (the quantile function) at specific points. For instance, this task is of key importance in private data generation. We present two different approaches. The first one consists in privately estimating the empirical quantiles of the samples and using this result as an estimator of the quantiles of the distribution. In particular, we study the statistical properties of the recently published algorithm introduced by Kaplan et al. 2022 that privately estimates the quantiles recursively. The second approach is to use techniques of density estimation in order to uniformly estimate the quantile function on an interval. In particular, we show that there is a tradeoff between the two methods. When we want to estimate many quantiles, it is better to estimate the density rather than estimating the quantile function at specific points.