Optimal Confidence Band for Kernel Gradient Flow Estimator

2026-05-07

AI Summary

This work addresses the absence of minimax optimal confidence bands for kernel gradient flow estimators in the uniform norm by systematically analyzing uniform generalization errors under both continuous and discrete settings within the capacity–source condition framework. By integrating reproducing kernel Hilbert space theory with extreme-value probabilistic analysis, the study establishes—for the first time—the minimax optimal convergence rate of kernel gradient flows in the uniform norm. Furthermore, it constructs simultaneous confidence bands whose width can be made arbitrarily close to this optimal rate, thereby achieving simultaneous optimality in both generalization error and statistical inference.

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 $s>α_0$, where $α_0\in(0,1)$ 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.

Problem

• confidence band
• kernel gradient flow
• supremum norm
• generalization error
• minimax optimality

Innovation

• kernel gradient flow
• supremum-norm generalization error
• minimax optimality
• simultaneous confidence band
• uniform inference