A Berry-Esseen theorem for incomplete U-statistics with Bernoulli sampling

There has been a resurgence of interest in the asymptotic normality of incomplete U-statistics that only sum over roughly as many kernel evaluations as there are data samples, due to its computational efficiency and usefulness in quantifying the uncertainty for ensemble-based predictions. In this paper, we focus on the normal convergence of one such construction, the incomplete U-statistic with Bernoulli sampling, based on a raw sample of size $n$ and a computational budget $N$. Under minimalistic moment assumptions on the kernel, we offer accompanying Berry-Esseen bounds of the natural rate $1/\sqrt{\min(N, n)}$ that characterize the normal approximating accuracy involved when $n \asymp N$, i.e. $n$ and $N$ are of the same order in such a way that $n/N$ is lower-and-upper bounded by constants. Our key techniques include Stein's method specialized for the so-called Studentized nonlinear statistics, and an exponential lower tail bound for non-negative kernel U-statistics.