In this paper, we study an online learning algorithm with a robust loss function $\mathcal{L}_{\sigma}$ for regression over a reproducing kernel Hilbert space (RKHS). The loss function $\mathcal{L}_{\sigma}$ involving a scaling parameter $\sigma>0$ can cover a wide range of commonly used robust losses. The proposed algorithm is then a robust alternative for online least squares regression aiming to estimate the conditional mean function. For properly chosen $\sigma$ and step size, we show that the last iterate of this online algorithm can achieve optimal capacity independent convergence in the mean square distance. Moreover, if additional information on the underlying function space is known, we also establish optimal capacity dependent rates for strong convergence in RKHS. To the best of our knowledge, both of the two results are new to the existing literature of online learning.

翻译：本文研究了一种基于再生核希尔伯特空间（RKHS）回归的鲁棒损失函数$\mathcal{L}_{\sigma}$的在线学习算法。该损失函数$\mathcal{L}_{\sigma}$包含缩放参数$\sigma>0$，可覆盖多种常用鲁棒损失函数。所提出的算法旨在估计条件均值函数，是用于在线最小二乘回归的鲁棒替代方案。通过适当选择$\sigma$和步长，我们证明了该在线算法的最后一步迭代能在均方距离意义下达到最优容量无关收敛。此外，若已知函数空间的额外信息，我们还建立了RKHS中强收敛的最优容量依赖速率。据我们所知，这两个结果在现有在线学习文献中均为首次提出。