Private Protocols for U-Statistics in the Local Model and Beyond

TL;DR

This paper develops new privacy-preserving protocols for computing U-statistics in the local differential privacy model, enabling accurate statistical estimates like AUC and Gini difference with theoretical error bounds and practical efficiency.

Contribution

It introduces novel LDP protocols for U-statistics, including a specialized AUC protocol using hierarchical histograms, and demonstrates secure computation methods with optimal error bounds.

Findings

LDP protocol achieves MSE of O(1/√(n)ε) for general U-statistics.

Hierarchical histogram protocol attains MSE of O(α^3/nε^2) for AUC.

Secure two-party computation protocol reaches MSE of O(1/nε^2) with linear communication.

Abstract

In this paper, we study the problem of computing $U$-statistics of degree $2$, i.e., quantities that come in the form of averages over pairs of data points, in the local model of differential privacy (LDP). The class of $U$-statistics covers many statistical estimates of interest, including Gini mean difference, Kendall's tau coefficient and Area under the ROC Curve (AUC), as well as empirical risk measures for machine learning problems such as ranking, clustering and metric learning. We first introduce an LDP protocol based on quantizing the data into bins and applying randomized response, which guarantees an $\epsilon$-LDP estimate with a Mean Squared Error (MSE) of $O(1/\sqrt{n}\epsilon)$ under regularity assumptions on the $U$-statistic or the data distribution. We then propose a specialized protocol for AUC based on a novel use of hierarchical histograms that achieves MSE of $O(\alpha^3/n\epsilon^2)$ for arbitrary data distribution. We also show that 2-party secure computation allows to design a protocol with MSE of $O(1/n\epsilon^2)$, without any assumption on the kernel function or data distribution and with total communication linear in the number of users $n$. Finally, we evaluate the performance of our protocols through experiments on synthetic and real datasets.