On the Optimality of Misspecified Kernel Ridge Regression

05/12/2023

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by   Haobo Zhang, et al.

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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 f_ρ^*_L^∞<∞ which implicitly requires s > α_0 where α_0∈ (0,1) is the embedding index, a constant depending on ℋ. Whether the KRR is optimal for all s∈ (0,1) is an outstanding problem lasting for years. In this paper, we show that KRR is minimax optimal for any s∈ (0,1) when the ℋ is a Sobolev RKHS.