Private Quantiles Estimation in the Presence of Atoms

TL;DR

This paper investigates differentially private estimation of multiple quantiles, revealing limitations with peaked distributions and proposing a heuristic smoothing method to improve performance in such cases.

Contribution

It introduces a new heuristic smoothing approach, HSJointExp, that enhances quantile estimation under differential privacy for challenging distributions.

Findings

The IS-based method is computationally similar to JointExp.

Both algorithms are statistically consistent for continuous distributions.

The smoothed version significantly improves performance on peaked distributions.

Abstract

We consider the differentially private estimation of multiple quantiles (MQ) of a distribution from a dataset, a key building block in modern data analysis. We apply the recent non-smoothed Inverse Sensitivity (IS) mechanism to this specific problem. We establish that the resulting method is closely related to the recently published ad hoc algorithm JointExp. In particular, they share the same computational complexity and a similar efficiency. We prove the statistical consistency of these two algorithms for continuous distributions. Furthermore, we demonstrate both theoretically and empirically that this method suffers from an important lack of performance in the case of peaked distributions, which can degrade up to a potentially catastrophic impact in the presence of atoms. Its smoothed version (i.e. by applying a max kernel to its output density) would solve this problem, but remains an open challenge to implement. As a proxy, we propose a simple and numerically efficient method called Heuristically Smoothed JointExp (HSJointExp), which is endowed with performance guarantees for a broad class of distributions and achieves results that are orders of magnitude better on problematic datasets.