Universal Private Estimators

TL;DR

This paper introduces universal differentially private estimators for mean, variance, and scale that work across arbitrary continuous distributions, removing prior boundedness assumptions and matching or surpassing specialized estimators.

Contribution

The paper presents the first universal pure differential privacy estimators for statistical parameters applicable to any continuous distribution, removing boundedness constraints.

Findings

Estimators perform well on Gaussian and heavy-tailed distributions.

They match or improve existing estimators under certain conditions.

Boundedness assumptions are eliminated for pure differential privacy.

Abstract

We present \textit{universal} estimators for the statistical mean, variance, and scale (in particular, the interquartile range) under pure differential privacy. These estimators are universal in the sense that they work on an arbitrary, unknown continuous distribution $\mathcal{P}$ over $\mathbb{R}$, while yielding strong utility guarantees except for ill-behaved $\mathcal{P}$. For certain distribution families like Gaussians or heavy-tailed distributions, we show that our universal estimators match or improve existing estimators, which are often specifically designed for the given family and under \textit{a priori} boundedness assumptions on the mean and variance of $\mathcal{P}$. This is the first time these boundedness assumptions are removed under pure differential privacy. The main technical tools in our development are instance-optimal empirical estimators for the mean and quantiles over the unbounded integer domain, which can be of independent interest.