On the Optimality of Misspecified Kernel Ridge Regression
(2305.07241)Abstract
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.
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