In the misspecified kernel ridge regression problem, researchers usually assume the underground true function $f_{\rho}^{*} \in [\mathcal{H}]^{s}$, a less-smooth interpolation space of a reproducing kernel Hilbert space (RKHS) $\mathcal{H}$ for some $s\in (0,1)$. The existing minimax optimal results require $\|f_{\rho}^{*}\|_{L^{\infty}}<\infty$ which implicitly requires $s > \alpha_{0}$ where $\alpha_{0}\in (0,1)$ is the embedding index, a constant depending on $\mathcal{H}$. Whether the KRR is optimal for all $s\in (0,1)$ is an outstanding problem lasting for years. In this paper, we show that KRR is minimax optimal for any $s\in (0,1)$ when the $\mathcal{H}$ is a Sobolev RKHS.

翻译：在误设核岭回归问题中，研究者通常假设潜在真实函数$f_{\rho}^{*}$属于某个再生核希尔伯特空间(RKHS)$\mathcal{H}$的插值空间$[\mathcal{H}]^{s}$（其中$s\in (0,1)$），该空间的光滑性弱于$\mathcal{H}$。已有极小极大最优性结果要求$\|f_{\rho}^{*}\|_{L^{\infty}}<\infty$，这隐含着$s>\alpha_{0}$的条件，其中$\alpha_{0}\in (0,1)$为嵌入指数，是一个依赖于$\mathcal{H}$的常数。核岭回归是否对所有$s\in (0,1)$均达到最优是困扰学界多年的悬而未决问题。本文证明，当$\mathcal{H}$为Sobolev型再生核希尔伯特空间时，核岭回归对任意$s\in (0,1)$均达到极小极大最优性。