# Approximating high-dimensional infinite-order $U$-statistics: statistical and computational guarantees

## TL;DR

This paper develops statistical and computational methods for approximating high-dimensional infinite-order U-statistics, enabling uncertainty quantification in ensemble methods like random forests with guarantees on accuracy and efficiency.

## Contribution

It introduces non-asymptotic Gaussian approximation bounds and data-driven bootstrap methods for incomplete IOUS, addressing computational challenges in high dimensions.

## Key findings

- Derived non-asymptotic Gaussian approximation error bounds.
- Established statistical guarantees for bootstrap inference.
- Provided computational efficiency results for incomplete IOUS.

## Abstract

We study the problem of distributional approximations to high-dimensional non-degenerate $U$-statistics with random kernels of diverging orders. Infinite-order $U$-statistics (IOUS) are a useful tool for constructing simultaneous prediction intervals that quantify the uncertainty of ensemble methods such as subbagging and random forests. A major obstacle in using the IOUS is their computational intractability when the sample size and/or order are large. In this article, we derive non-asymptotic Gaussian approximation error bounds for an incomplete version of the IOUS with a random kernel. We also study data-driven inferential methods for the incomplete IOUS via bootstraps and develop their statistical and computational guarantees.