In this paper, we study regression problems over a separable Hilbert space with the square loss, covering non-parametric regression over a reproducing kernel Hilbert space. We investigate a class of spectral/regularized algorithms, including ridge regression, principal component regression, and gradient methods. We prove optimal, high-probability convergence results in terms of variants of norms for the studied algorithms, considering a capacity assumption on the hypothesis space and a general source condition on the target function. Consequently, we obtain almost sure convergence results with optimal rates. Our results improve and generalize previous results, filling a theoretical gap for the non-attainable cases.

在本文中，我们研究具有平方损失的可分离希尔伯特空间上的回归问题，涵盖了再生核希尔伯特空间上的非参数回归。我们研究了一类频谱/正则化算法，包括岭回归，主成分回归和梯度方法。考虑到假设空间上的容量假设和目标函数的一般源条件，我们针对所研究算法的规范变体证明了最佳的高概率收敛结果。因此，我们可以获得最佳速率的几乎确定的收敛结果。我们的结果改进并概括了以前的结果，填补了无法达到的情况的理论空白。