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

arXiv: https://arxiv.org/abs/1910.03861 · HTML: https://mlsanity.com/paper/1910.03861/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Oct 2019.

## Built on by (most cited first, 7 of 7)

- [Federated Calibration and Evaluation of Binary Classifiers](https://mlsanity.com/paper/2210.12526/) (2022)
- [On Differentially Private U Statistics](https://mlsanity.com/paper/2407.04945/) (2024)
- [PAPAYA Federated Analytics Stack: Engineering Privacy, Scalability and Practicality](https://mlsanity.com/paper/2412.02340/) (2024)
- [SoK: Verifiable Cross-Silo FL](https://mlsanity.com/paper/2410.09124/) (2024)
- [Federated Computation of ROC and PR Curves](https://mlsanity.com/paper/2510.04979/) (2025)
- [Accurate, private, secure, federated U-statistics with higher degree](https://mlsanity.com/paper/2603.01986/) (2026)
- [On Gossip Algorithms for Machine Learning with Pairwise Objectives](https://mlsanity.com/paper/2603.24128/) (2026)

## Abstract (arXiv)

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 ε-LDP estimate with a Mean Squared Error (MSE) of O(1/√nε) 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(α3/nε2) for arbitrary data distribution. We also show that 2-party secure computation allows to design a protocol with MSE of O(1/nε2), 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.