Order-Explicit Linearization of High-Dimensional U-Statistics

We give an order-explicit large deviation bound for the difference between a high-dimensional U-statistic and its H'ajek projection. In particular, we show that any U-statistic of order b on n observations, with a d-dimensional kernel whose coordinates have psi1-Orlicz norm at most φ, has a maximum deviation from its H'ajek projection of order Op(φ b n-1log2(dn)). The proof relies on the development of novel order-explicit moment inequalities for higher-order Hoeffding components. We show that this rate is unimprovable, up to the polynomial factor on the logarithmic term. As corollaries, we obtain new Bernstein-type concentration and Gaussian approximation results for high-dimensional U-statistics. We apply these results to establish the consistency of a set of resampling-based simultaneous confidence intervals built around a class of nonparametric regression estimators constructed with subsampled kernels. This class encompasses several forms of random forest regression, including Generalized Random Forests.