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Optimality of Robust Online Learning
Foundations of Computational Mathematics ( IF 2.5 ) Pub Date : 2023-07-26 , DOI: 10.1007/s10208-023-09616-9
Zheng-Chu Guo , Andreas Christmann , Lei Shi

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.



中文翻译:

鲁棒在线学习的最优性

在本文中,我们研究了一种具有鲁棒损失函数\(\mathcal {L}_{\sigma }\)的在线学习算法,用于在再生核希尔伯特空间(RKHS)上进行回归。涉及缩放参数\(\sigma >0\)的损失函数\(\mathcal {L}_{\sigma }\ )可以覆盖广泛的常用鲁棒损失。因此,所提出的算法是在线最小二乘回归的稳健替代方案,旨在估计条件均值函数。对于正确选择的\(\sigma \)和步长,我们表明该在线算法的最后一次迭代可以在均方距离上实现最佳容量独立收敛。此外,如果知道有关底层函数空间的附加信息,我们还可以建立 RKHS 强收敛的最佳容量相关速率。据我们所知,这两个结果对于现有的在线学习文献来说都是新的。

更新日期:2023-07-26
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