On the Robustness of Kernel Ridge Regression Using the Cauchy Loss Function

Robust regression aims to develop methods for estimating an unknown regression function in the presence of outliers, heavy-tailed distributions, or contaminated data, which can severely impact performance. Most existing theoretical results in robust regression assume that the noise has a finite absolute mean, an assumption violated by certain distributions, such as Cauchy and some Pareto noise. In this paper, we introduce a generalized Cauchy noise framework that accommodates all noise distributions with finite moments of any order, even when the absolute mean is infinite. Within this framework, we study the textitkernel Cauchy ridge regressor (textitKCRR), which minimizes a regularized empirical Cauchy risk to achieve robustness. To derive the L2-risk bound for KCRR, we establish a connection between the excess Cauchy risk and L2-risk for sufficiently large scale parameters of the Cauchy loss, which reveals that these two risks are equivalent. Furthermore, under the assumption that the regression function satisfies H"older smoothness, we derive excess Cauchy risk bounds for KCRR, showing improved performance as the scale parameter decreases. By considering the twofold effect of the scale parameter on the excess Cauchy risk and its equivalence with the L2-risk, we establish the almost minimax-optimal convergence rate for KCRR in terms of L2-risk, highlighting the robustness of the Cauchy loss in handling various types of noise. Finally, we validate the effectiveness of KCRR through experiments on both synthetic and real-world datasets under diverse noise corruption scenarios.