  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.
