  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.
