On the Optimality of Misspecified Kernel Ridge Regression

In the misspecified kernel ridge regression problem, researchers usually assume the underground true function $f_{\rho }^{\ast }\in [\mathcal{H}{]}^s$, 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 $\mathrm{\parallel }{f}_{\rho }^{\ast }{\mathrm{\parallel }}_{L^{\mathrm{\infty }}}<\mathrm{\infty }$ 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.