On Differentially Private U Statistics

TL;DR

This paper develops a new private estimation method for U-statistics, achieving near-optimal error rates by using local H"ajek projections, addressing limitations of existing private mean estimation techniques.

Contribution

It introduces a thresholding-based approach with local H"ajek projections for differentially private U-statistics, improving error bounds over previous methods.

Findings

Achieves nearly optimal private error for non-degenerate U-statistics

Provides strong evidence of near-optimality for degenerate U-statistics

Addresses limitations of existing private mean estimation algorithms

Abstract

We consider the problem of privately estimating a parameter $\mathbb{E}[h(X_1,\dots,X_k)]$, where $X_1$, $X_2$, $\dots$, $X_k$ are i.i.d. data from some distribution and $h$ is a permutation-invariant function. Without privacy constraints, standard estimators are U-statistics, which commonly arise in a wide range of problems, including nonparametric signed rank tests, symmetry testing, uniformity testing, and subgraph counts in random networks, and can be shown to be minimum variance unbiased estimators under mild conditions. Despite the recent outpouring of interest in private mean estimation, privatizing U-statistics has received little attention. While existing private mean estimation algorithms can be applied to obtain confidence intervals, we show that they can lead to suboptimal private error, e.g., constant-factor inflation in the leading term, or even $\Theta(1/n)$ rather than $O(1/n^2)$ in degenerate settings. To remedy this, we propose a new thresholding-based approach using \emph{local H\'ajek projections} to reweight different subsets of the data. This leads to nearly optimal private error for non-degenerate U-statistics and a strong indication of near-optimality for degenerate U-statistics.