On the optimality of misspecified spectral algorithms
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In the misspecified spectral algorithms 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 spectral algorithms are optimal for all s∈ (0,1) is an outstanding problem lasting for years. In this paper, we show that spectral algorithms are minimax optimal for any α_0-1/β < s < 1, where β is the eigenvalue decay rate of ℋ. We also give several classes of RKHSs whose embedding index satisfies α_0 = 1/β. Thus, the spectral algorithms are minimax optimal for all s∈ (0,1) on these RKHSs.
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