On the Optimality of Misspecified Kernel Ridge Regression

In the misspecified kernel ridge regression problem, researchers usually assume the underground true function $f_{\rho}^{\star} \in [\mathcal{H}]^{s}$, 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 $\left\Vert f_{\rho}^{\star} \right \Vert_{L^{\infty}} < \infty$ 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.