Optimal Confidence Band for Kernel Gradient Flow Estimator

TL;DR

This paper derives minimax optimal convergence rates and confidence bands for kernel gradient flow estimators in regression, advancing understanding of their uniform inference properties.

Contribution

It establishes the optimal convergence rates and confidence band widths for kernel gradient flows under the source condition framework.

Findings

Convergence rates match the minimax optimal rates.

Constructed confidence bands are width-optimal and nearly minimax.

Results apply to both continuous and discrete kernel gradient flows.

Abstract

In this paper, we investigate the supremum-norm generalization error and the uniform inference for a specific class of kernel regression methods, namely the kernel gradient flows. Under the widely adopted capacity-source condition framework in the kernel regression literature, we first establish convergence rates for the supremum norm generalization error of both continuous and discrete kernel gradient flows under the source condition $s>\alpha_0$, where $\alpha_0\in(0,1)$ denotes the embedding index of the kernel function. Moreover, we show that these rates match the minimax optimal rates. Building on this result, we then construct simultaneous confidence bands for both continuous and discrete kernel gradient flows. Notably, the widths of the proposed confidence bands are also optimal, in the sense that their shrinkage rates are greater than, while can be arbitrarily close to, the minimax optimal rates.