Kernel Ridge Regression over Hölder Classes: Function and Derivative Losses
Abstract
We study ordinary kernel ridge regression over full Hölder-Zygmund balls, measuring errors in Sobolev norms and in pointwise and uniform losses for mixed derivatives. For targets of smoothness s, every RKHS norm equivalent to Hs+d/2 gives the optimal Hr risk for 0≤ r<s; integer choices of r include integrated squared derivative losses. Pointwise and uniform guarantees require additional kernel structure. A localized wavelet kernel attains the sharp derivative rates with the same regularization orders as for function estimation, whereas equivalent changes of the RKHS norm can destroy the population bias rates. For the ordinary Gaussian kernel with a shrinking bandwidth, we prove a uniform derivative risk bound with the optimal polynomial rate and an explicit, unoptimized logarithmic factor. The results allow bounded measurable design densities bounded away from zero and conditionally sub-Gaussian noise.
Related
- On Lipschitz Spaces, Semigroup Transference, and Heat Kernel Derivative Estimates in Dunkl--Schrödinger Analysis
- Kernel Density Regression Estimation with Covariates on Riemannian Manifolds
- Robust Error Bounds for Vector-Valued Kernel Ridge Regression
- Infinite-Dimensional Spherical Kernel ridge Regression
- Semi-Supervised Kernel Ridge Fréchet Regression
BibTeX
@misc{hou2026kernel,
title = {{Kernel Ridge Regression over Hölder Classes: Function and Derivative Losses}},
author = {Yuxuan Hou and Yicheng Li},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2607.26065},
eprint = {2607.26065},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.26065},
}
RIS
TY - UNPB
TI - Kernel Ridge Regression over Hölder Classes: Function and Derivative Losses
AU - Yuxuan Hou
AU - Yicheng Li
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2607.26065
UR - https://arxiv.org/abs/2607.26065
AB - We study ordinary kernel ridge regression over full Hölder-Zygmund balls, measuring errors in Sobolev norms and in pointwise and uniform losses for mixed derivatives. For targets of smoothness s, every RKHS norm equivalent to H^s+d/2 gives the optimal H^r risk for 0≤ r<s; integer choices of r include integrated squared derivative losses. Pointwise and uniform guarantees require additional kernel structure. A localized wavelet kernel attains the sharp derivative rates with the same regularization orders as for function estimation, whereas equivalent changes of the RKHS norm can destroy the population bias rates. For the ordinary Gaussian kernel with a shrinking bandwidth, we prove a uniform derivative risk bound with the optimal polynomial rate and an explicit, unoptimized logarithmic factor. The results allow bounded measurable design densities bounded away from zero and conditionally sub-Gaussian noise.
ER -