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：在本文中，我们研究了一种基于鲁棒损失函数 $\mathcal{L}_{\sigma}$ 的在线学习算法，用于重现核希尔伯特空间 (RKHS) 上的回归问题。涉及缩放参数 $\sigma>0$ 的损失函数 $\mathcal{L}_{\sigma}$ 可以涵盖广泛使用的鲁棒损失。所提出的算法是在线最小二乘回归的鲁棒替代品，旨在估计条件均值函数。对于适当选择的 $\sigma$ 和步长，我们证明了该在线算法的最后迭代能够在均方距离上实现容量独立的最优收敛。此外，如果已知基础函数空间的附加信息，则我们还建立了在 RKHS 强收敛方面的最优容量相关率。就我们所知，这两个结果在在线学习的现有文献中都是新的。