Average-Case Averages: Private Algorithms for Smooth Sensitivity and Mean Estimation

TL;DR

This paper introduces instance-dependent noise scaling methods for differential privacy, improving accuracy in mean estimation by leveraging smooth sensitivity and proposing new noise distributions, especially for the trimmed mean estimator.

Contribution

It develops new methods for private mean estimation using smooth sensitivity and introduces three novel noise distributions for better accuracy.

Findings

Trimmed mean reduces sensitivity and improves privacy-accuracy trade-off.

New noise distributions outperform existing methods in experiments.

Methods are effective under average-case distributional assumptions.

Abstract

The simplest and most widely applied method for guaranteeing differential privacy is to add instance-independent noise to a statistic of interest that is scaled to its global sensitivity. However, global sensitivity is a worst-case notion that is often too conservative for realized dataset instances. We provide methods for scaling noise in an instance-dependent way and demonstrate that they provide greater accuracy under average-case distributional assumptions. Specifically, we consider the basic problem of privately estimating the mean of a real distribution from i.i.d.~samples. The standard empirical mean estimator can have arbitrarily-high global sensitivity. We propose the trimmed mean estimator, which interpolates between the mean and the median, as a way of attaining much lower sensitivity on average while losing very little in terms of statistical accuracy. To privately estimate the trimmed mean, we revisit the smooth sensitivity framework of Nissim, Raskhodnikova, and Smith (STOC 2007), which provides a framework for using instance-dependent sensitivity. We propose three new additive noise distributions which provide concentrated differential privacy when scaled to smooth sensitivity. We provide theoretical and experimental evidence showing that our noise distributions compare favorably to others in the literature, in particular, when applied to the mean estimation problem.