When Kernel Ridge Regression Meets the Hölder-Zygmund Class: Minimax Optimality and Failure of Properness

Yuxuan Hou

When Kernel Ridge Regression Meets the Hölder-Zygmund Class: Minimax Optimality and Failure of Properness

Machine Learning 2026-06-19 v1 Machine Learning Statistics Theory

Authors: Yuxuan Hou

Abstract

We study kernel ridge regression for nonparametric regression over the H\"older-Zygmund class. Using an RKHS equivalent to a Sobolev space of smoothness s+d/2, we prove that misspecified KRR attains the minimax L2 rate n^{-2s/(2s+d)}. We also show that properness fails in the H\"older-Zygmund norm: even for the zero regression function with Gaussian noise, the expected squared H\"older-Zygmund norm of the KRR noise component grows as log n.

Keywords

kernel ridge regression Hölder-Zygmund class minimax optimality properness nonparametric regression

Cite

@article{arxiv.2607.26065,
  title  = {When Kernel Ridge Regression Meets the Hölder-Zygmund Class: Minimax Optimality and Failure of Properness},
  author = {Yuxuan Hou},
  journal= {arXiv preprint arXiv:2607.26065},
  year   = {2026}
}