# Optimal Confidence Band for Kernel Gradient Flow Estimator

arXiv: https://arxiv.org/abs/2605.05768 · HTML: https://mlsanity.com/paper/2605.05768/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 57 references that are in the map.
- First posted on arXiv 7 May 2026.

## Builds on (most cited first, 10 of 10)

- [Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences](https://mlsanity.com/paper/1807.02582/) (2018)
- [On the Eigenvalue Decay Rates of a Class of Neural-Network Related Kernel Functions Defined on General Domains](https://mlsanity.com/paper/2305.02657/) (2023)
- [Generalization Ability of Wide Neural Networks on ℝ](https://mlsanity.com/paper/2302.05933/) (2023)
- [Frequentist coverage and sup-norm convergence rate in Gaussian process regression](https://mlsanity.com/paper/1708.04753/) (2017)
- [On the Saturation Effect of Kernel Ridge Regression](https://mlsanity.com/paper/2405.09362/) (2024)
- [Generalization Error Curves for Analytic Spectral Algorithms under Power-law Decay](https://mlsanity.com/paper/2401.01599/) (2024)
- [Sobolev Spaces, Kernels and Discrepancies over Hyperspheres](https://mlsanity.com/paper/2211.09196/) (2022)
- [Kernel Ridge Regression Inference](https://mlsanity.com/paper/2302.06578/) (2023)
- [On Kernel Regression with Data-Dependent Kernels](https://mlsanity.com/paper/2209.01691/) (2022)
- [The L^∞ Learnability of Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2306.02833/) (2023)

## Built on by (most cited first, 1 of 1)

- [Estimating Population-Risk Curves Along Nonconvex Gradient Flows from the Training Sample](https://mlsanity.com/paper/2608.30261/) (2026)

## Abstract (arXiv)

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>α0, where α0∈(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.