We address a task of estimating sparse coefficients in linear regression when the covariates are drawn from an $L$-subexponential random vector, which belongs to a class of distributions having heavier tails than a Gaussian random vector. Prior works have tackled this issue by assuming that the covariates are drawn from an $L$-subexponential random vector and have established error bounds that resemble those derived for Gaussian random vectors. However, these previous methods require stronger conditions to derive error bounds than those employed for Gaussian random vectors. In the present paper, we present an error bound identical to that obtained for Gaussian random vectors, up to constant factors, without requiring stronger conditions, even when the covariates are drawn from an $L$-subexponential random vector. Somewhat interestingly, we utilize an $\ell_1$-penalized Huber regression, that is recognized for its robustness to heavy-tailed random noises, not covariates. We believe that the present paper reveals a new aspect of the $\ell_1$-penalized Huber regression.

翻译：我们研究了当协变量来自$L$-次指数随机向量时的稀疏线性回归系数估计问题，该类分布比高斯随机向量具有更重的尾部特征。先前的工作通过假设协变量服从$L$-次指数随机向量来处理该问题，并建立了与高斯随机向量类似的误差界。然而，这些方法在推导误差界时需要比高斯随机向量更严格的条件。在本文中，我们提出了一种与高斯随机向量相同的误差界（仅相差常数因子），且无需更强条件，即使协变量来自$L$-次指数随机向量。有趣的是，我们使用了$\ell_1$惩罚Huber回归——该模型以对重尾随机噪声（而非协变量）的鲁棒性著称。我们相信本文揭示了$\ell_1$惩罚Huber回归的新特性。