In the misspecified spectral algorithms 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 spectral algorithms are optimal for all $s\in (0,1)$ is an outstanding problem lasting for years. In this paper, we show that spectral algorithms are minimax optimal for any $\alpha_{0}-\frac{1}{\beta} < s < 1$, where $\beta$ is the eigenvalue decay rate of $\mathcal{H}$. We also give several classes of RKHSs whose embedding index satisfies $ \alpha_0 = \frac{1}{\beta} $. Thus, the spectral algorithms are minimax optimal for all $s\in (0,1)$ on these RKHSs.

翻译：在误设定谱算法问题中，研究者通常假设潜在的真实函数$f_{\rho}^{*} \in [\mathcal{H}]^{s}$，其中$[\mathcal{H}]^{s}$是再生核希尔伯特空间（RKHS）$\mathcal{H}$的一个较不光滑的插值空间，参数$s\in (0,1)$。现有的极小化最优结果要求$\|f_{\rho}^{*}\|_{L^{\infty}}<\infty$，这隐含地要求$s > \alpha_{0}$，其中$\alpha_{0}\in (0,1)$是嵌入指数，一个依赖于$\mathcal{H}$的常数。谱算法对于所有$s\in (0,1)$是否最优是一个持续多年的悬而未决的问题。在本文中，我们证明了对于任意$\alpha_{0}-\frac{1}{\beta} < s < 1$，谱算法是极小化最优的，其中$\beta$是$\mathcal{H}$的特征值衰减率。我们还给出了几类RKHS，其嵌入指数满足$\alpha_{0} = \frac{1}{\beta}$。因此，在这些RKHS上，谱算法对于所有$s\in (0,1)$都是极小化最优的。