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

• URL: http://arxiv.org/abs/2503.20120v1
• Date: Wed, 26 Mar 2025 00:00:53 GMT
• Title: On the Robustness of Kernel Ridge Regression Using the Cauchy Loss Function
• Abstract summary: Robust regression aims to develop methods for estimating an unknown regression function in the presence of outliers, heavy-tailed distributions, or contaminated data.<n>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 noise.<n>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.
• Score: 3.4956406636452626
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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 \textit{kernel Cauchy ridge regressor} (\textit{KCRR}), which minimizes a regularized empirical Cauchy risk to achieve robustness. To derive the $L_2$-risk bound for KCRR, we establish a connection between the excess Cauchy risk and $L_2$-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 $L_2$-risk, we establish the almost minimax-optimal convergence rate for KCRR in terms of $L_2$-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.

Related papers

• RE-SAC: Disentangling aleatoric and epistemic risks in bus fleet control: A stable and robust ensemble DRL approach [19.34848029403215]
  Bus holding control is challenging due to traffic and passenger demand.<n>Standard actor-critic algorithms suffer from Q-value instability in volatile environments.<n>We propose a robust ensemble soft actor-critic framework to disentangle uncertainties.
  arXiv  Detail & Related papers  (2026-03-19T01:37:45Z)
• On the Generalization and Robustness in Conditional Value-at-Risk [12.253712889424584]
  We develop a learning-theoretic analysis of Conditional Value-at-Risk (CVaR)-based empirical risk minimization under heavy-tailed and contaminated data.<n>We establish sharp, high-probability generalization and excess risk bounds under minimal moment assumptions.<n>We show that CVaR decisions themselves can be intrinsically unstable under heavy tails.
  arXiv  Detail & Related papers  (2026-02-20T08:10:11Z)
• Robust Bayesian Dynamic Programming for On-policy Risk-sensitive Reinforcement Learning [4.71677151409532]
  We propose a novel framework for risk-sensitive reinforcement learning that incorporates robustness against transition uncertainty.<n>Our framework unifies and generalizes most existing RL frameworks by permitting general coherent risk measures for both inner and outer risk measures.
  arXiv  Detail & Related papers  (2025-12-31T03:13:22Z)
• Understanding Robust Machine Learning for Nonparametric Regression with Heavy-Tailed Noise [10.844819221753042]
  We use Huber regression as a close-up example within Tikhonov-regularized risk minimization.<n>We address two central challenges: (i) the breakdown of standard concentration tools under weak moment assumptions, and (ii) the analytical difficulties introduced by unbounded hypothesis spaces.<n>Our study delivers principled rules, extends beyond Huber to other robust losses, and highlights prediction error, not excess risk, as the fundamental lens for analyzing robust learning.
  arXiv  Detail & Related papers  (2025-10-10T21:57:18Z)
• Stochastic Weakly Convex Optimization Under Heavy-Tailed Noises [55.43924214633558]
  In this paper, we focus on two types of noises: one is sub-Weibull noises, and the other is SsBC noises.<n>Under these two noise assumptions, the in-expectation and high-probability convergence of SFOMs have been studied in the contexts of convex optimization and smooth optimization.
  arXiv  Detail & Related papers  (2025-07-17T16:48:45Z)
• Convergence of Clipped-SGD for Convex $(L_0,L_1)$-Smooth Optimization with Heavy-Tailed Noise [60.17850744118546]
  First-order methods with clipping, such as Clip-SGD, exhibit stronger convergence guarantees than SGD under the $(L_$1)$-smoothness assumption.<n>We establish the first high-probability convergence bounds for Clip-SGD applied to convex $(L_$1)$-smooth optimization with heavytailed noise.
  arXiv  Detail & Related papers  (2025-05-27T07:23:42Z)
• Provable Risk-Sensitive Distributional Reinforcement Learning with General Function Approximation [54.61816424792866]
  We introduce a general framework on Risk-Sensitive Distributional Reinforcement Learning (RS-DisRL), with static Lipschitz Risk Measures (LRM) and general function approximation. We design two innovative meta-algorithms: textttRS-DisRL-M, a model-based strategy for model-based function approximation, and textttRS-DisRL-V, a model-free approach for general value function approximation.
  arXiv  Detail & Related papers  (2024-02-28T08:43:18Z)
• Optimal Rates and Saturation for Noiseless Kernel Ridge Regression [4.585021053685196]
  We present a comprehensive study of Kernel ridge regression (KRR) in the noiseless regime. KRR is a fundamental method for learning functions from finite samples. We introduce a refined notion of degrees of freedom, which we believe has broader applicability in the analysis of kernel methods.
  arXiv  Detail & Related papers  (2024-02-24T04:57:59Z)
• Risk-Sensitive Diffusion: Robustly Optimizing Diffusion Models with Noisy Samples [58.68233326265417]
  Non-image data are prevalent in real applications and tend to be noisy. Risk-sensitive SDE is a type of differential equation (SDE) parameterized by the risk vector. We conduct systematic studies for both Gaussian and non-Gaussian noise distributions.
  arXiv  Detail & Related papers  (2024-02-03T08:41:51Z)
• Model-Based Epistemic Variance of Values for Risk-Aware Policy Optimization [59.758009422067]
  We consider the problem of quantifying uncertainty over expected cumulative rewards in model-based reinforcement learning. We propose a new uncertainty Bellman equation (UBE) whose solution converges to the true posterior variance over values. We introduce a general-purpose policy optimization algorithm, Q-Uncertainty Soft Actor-Critic (QU-SAC) that can be applied for either risk-seeking or risk-averse policy optimization.
  arXiv  Detail & Related papers  (2023-12-07T15:55:58Z)
• High-dimensional robust regression under heavy-tailed data: Asymptotics and Universality [7.416689632227865]
  We investigate the high-dimensional properties of robust regression estimators in the presence of heavy-tailed noise. We show that, despite being consistent, the Huber loss with optimally tuned location parameter $delta$ is suboptimal in the high-dimensional regime. We derive the decay rates for the excess risk of ridge regression.
  arXiv  Detail & Related papers  (2023-09-28T14:39:50Z)
• Optimally tackling covariate shift in RKHS-based nonparametric regression [43.457497490211985]
  We show that a kernel ridge regression estimator with a carefully chosen regularization parameter is minimax rate-optimal. We also show that a naive estimator, which minimizes the empirical risk over the function class, is strictly sub-optimal. We propose a reweighted KRR estimator that weights samples based on a careful truncation of the likelihood ratios.
  arXiv  Detail & Related papers  (2022-05-06T02:33:24Z)
• On the Double Descent of Random Features Models Trained with SGD [78.0918823643911]
  We study properties of random features (RF) regression in high dimensions optimized by gradient descent (SGD) We derive precise non-asymptotic error bounds of RF regression under both constant and adaptive step-size SGD setting. We observe the double descent phenomenon both theoretically and empirically.
  arXiv  Detail & Related papers  (2021-10-13T17:47:39Z)
• Exploring the Training Robustness of Distributional Reinforcement Learning against Noisy State Observations [7.776010676090131]
  State observations that an agent observes may contain measurement errors or adversarial noises, misleading the agent to take suboptimal actions or even collapse while training. In this paper, we study the training robustness of distributional Reinforcement Learning (RL), a class of state-of-the-art methods that estimate the whole distribution, as opposed to only the expectation, of the total return.
  arXiv  Detail & Related papers  (2021-09-17T22:37:39Z)
• The Interplay Between Implicit Bias and Benign Overfitting in Two-Layer Linear Networks [51.1848572349154]
  neural network models that perfectly fit noisy data can generalize well to unseen test data. We consider interpolating two-layer linear neural networks trained with gradient flow on the squared loss and derive bounds on the excess risk.
  arXiv  Detail & Related papers  (2021-08-25T22:01:01Z)
• Interpolation can hurt robust generalization even when there is no noise [76.3492338989419]
  We show that avoiding generalization through ridge regularization can significantly improve generalization even in the absence of noise. We prove this phenomenon for the robust risk of both linear regression and classification and hence provide the first theoretical result on robust overfitting.
  arXiv  Detail & Related papers  (2021-08-05T23:04:15Z)
• Risk Bounds and Rademacher Complexity in Batch Reinforcement Learning [36.015585972493575]
  This paper considers batch Reinforcement Learning (RL) with general value function approximation. The excess risk of Empirical Risk Minimizer (ERM) is bounded by the Rademacher complexity of the function class. Fast statistical rates can be achieved by using tools of local Rademacher complexity.
  arXiv  Detail & Related papers  (2021-03-25T14:45:29Z)
• Noisy Linear Convergence of Stochastic Gradient Descent for CV@R Statistical Learning under Polyak-{\L}ojasiewicz Conditions [4.721069729610892]
  Conditional Value-at-Risk ($mathrmCV@R$) is one of the most popular measures of risk. We prove that $mathrmCV@R$ can be used as a performance criterion in supervised statistical learning.
  arXiv  Detail & Related papers  (2020-12-14T18:22:53Z)
• Large-Scale Methods for Distributionally Robust Optimization [53.98643772533416]
  We prove that our algorithms require a number of evaluations gradient independent of training set size and number of parameters. Experiments on MNIST and ImageNet confirm the theoretical scaling of our algorithms, which are 9--36 times more efficient than full-batch methods.
  arXiv  Detail & Related papers  (2020-10-12T17:41:44Z)
• Multiplicative noise and heavy tails in stochastic optimization [62.993432503309485]
  empirical optimization is central to modern machine learning, but its role in its success is still unclear. We show that it commonly arises in parameters of discrete multiplicative noise due to variance. A detailed analysis is conducted in which we describe on key factors, including recent step size, and data, all exhibit similar results on state-of-the-art neural network models.
  arXiv  Detail & Related papers  (2020-06-11T09:58:01Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.