标题： 一种易于调节的鲁棒稀疏高维线性回归方法 显示英文标题

标题： An Easily Tunable Approach to Robust and Sparse High-Dimensional Linear Regression

摘要： 稀疏线性回归方法（如 Lasso）需要一个依赖于噪声方差的调整参数，而噪声方差通常是未知且难以在实践中估计的。 在重尾噪声或对抗性异常值存在的情况下，这个问题变得更加困难。 本文提出了一种用于鲁棒和稀疏线性回归的估计器，消除了对噪声尺度显式先验知识的需求。 我们的方法基于 Huber 损失，并结合了一种迭代方案，该方案交替进行系数估计和通过均值中位数的自适应噪声校准。 该方法具有理论基础，在子高斯噪声和重尾噪声假设下均能实现尖锐的非渐近误差界。 此外，所提出的方法能够容纳响应中的任意异常值污染，而无需事先知道异常值的数量或稀疏水平。 虽然先前的鲁棒估计器避免了与噪声尺度或稀疏性相关的调整参数，但当异常值数量未知时，我们的程序达到了可比的误差界；当异常值数量已知时，达到了改进的误差界。 特别是，改进的误差界在常数因子范围内匹配已知的 minimax 下界。 显示英文摘要

摘要： Sparse linear regression methods such as Lasso require a tuning parameter that depends on the noise variance, which is typically unknown and difficult to estimate in practice. In the presence of heavy-tailed noise or adversarial outliers, this problem becomes more challenging. In this paper, we propose an estimator for robust and sparse linear regression that eliminates the need for explicit prior knowledge of the noise scale. Our method builds on the Huber loss and incorporates an iterative scheme that alternates between coefficient estimation and adaptive noise calibration via median-of-means. The approach is theoretically grounded and achieves sharp non-asymptotic error bounds under both sub-Gaussian and heavy-tailed noise assumptions. Moreover, the proposed method accommodates arbitrary outlier contamination in the response without requiring prior knowledge of the number of outliers or the sparsity level. While previous robust estimators avoid tuning parameters related to the noise scale or sparsity, our procedure achieves comparable error bounds when the number of outliers is unknown, and improved bounds when it is known. In particular, the improved bounds match the known minimax lower bounds up to constant factors.