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On the Optimality of Misspecified Kernel Ridge Regression
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:41331-41353, 2023.
Abstract
In the misspecified kernel ridge regression problem, researchers usually assume the underground true function f⋆ρ∈[H]s, a less-smooth interpolation space of a reproducing kernel Hilbert space (RKHS) H for some s∈(0,1). The existing minimax optimal results require ‖ 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.