Accurate, private, secure, federated U-statistics with higher degree

We study the problem of computing a U-statistic with a kernel function $f$ of degree $k \geq 2$, i.e., the average of some function $f$ over all $k$-tuples of instances, in a federated learning setting. U-statistics of degree $2$ include several useful statistics such as Kendall's $\tau$ coefficient, the Area under the Receiver-Operator Curve and the Gini mean difference. Existing methods provide solutions only under the lower-utility local differential privacy model and/or scale poorly in the size of the domain discretization. In this work, we propose a protocol that securely computes U-statistics of degree $k \geq 2$ under central differential privacy by leveraging Multi Party Computation (MPC). Our method substantially improves accuracy when compared to prior solutions. We provide a detailed theoretical analysis of its accuracy, communication and computational properties. We evaluate its performance empirically, obtaining favorable results, e.g., for Kendall's $\tau$ coefficient, our approach reduces the Mean Squared Error by up to four orders of magnitude over existing baselines.