We tackle estimating sparse coefficients in a linear regression when the covariates are sampled from an $L$-subexponential random vector. This vector belongs to a class of distributions that exhibit heavier tails than Gaussian random vector. Previous studies have established error bounds similar to those derived for Gaussian random vectors. However, these methods require stronger conditions than those used for Gaussian random vectors to derive the error bounds. In this study, we present an error bound identical to the one obtained for Gaussian random vectors up to constant factors without imposing stronger conditions, when the covariates are drawn from an $L$-subexponential random vector. Interestingly, we employ an $\ell_1$-penalized Huber regression, which is known for its robustness against heavy-tailed random noises rather than covariates. We believe that this study uncovers a new aspect of the $\ell_1$-penalized Huber regression method.

翻译：我们研究了当协变量从$L$-次指数随机向量中采样时，线性回归中稀疏系数的估计问题。该向量属于一类分布，其尾部比高斯随机向量更重。先前的研究已建立了与高斯随机向量相似的误差界。然而，这些方法在推导误差界时需要比高斯随机向量更强的条件。在本研究中，我们证明当协变量来自$L$-次指数随机向量时，无需施加更强的条件，即可获得与高斯随机向量在常数因子内完全一致的误差界。有趣的是，我们采用了$\ell_1$惩罚Huber回归方法，该方法以其对重尾随机噪声(而非协变量)的稳健性而闻名。我们认为本研究揭示了$\ell_1$惩罚Huber回归方法的一个新应用维度。