We develop a technique to design efficiently computable estimators for sparse linear regression in the simultaneous presence of two adversaries: oblivious and adaptive. We design several robust algorithms that outperform the state of the art even in the special case when oblivious adversary simply adds Gaussian noise. In particular, we provide a polynomial-time algorithm that with high probability recovers the signal up to error $O(\sqrt{\varepsilon})$ as long as the number of samples $n \ge \tilde{O}(k^2/\varepsilon)$, only assuming some bounds on the third and the fourth moments of the distribution ${D}$ of the design. In addition, prior to this work, even in the special case of Gaussian design and noise, no polynomial time algorithm was known to achieve error $o(\sqrt{\varepsilon})$ in the sparse setting $n < d^2$. We show that under some assumptions on the fourth and the eighth moments of ${D}$, there is a polynomial-time algorithm that achieves error $o(\sqrt{\varepsilon})$ as long as $n \ge \tilde{O}(k^4 / \varepsilon^3)$. For Gaussian distribution, this algorithm achieves error $O(\varepsilon^{3/4})$. Moreover, our algorithm achieves error $o(\sqrt{\varepsilon})$ for all log-concave distributions if $\varepsilon \le 1/\text{polylog(d)}$. Our algorithms are based on the filtering of the covariates that uses sum-of-squares relaxations, and weighted Huber loss minimization with $\ell_1$ regularizer. We provide a novel analysis of weighted penalized Huber loss that is suitable for heavy-tailed designs in the presence of two adversaries. Furthermore, we complement our algorithmic results with Statistical Query lower bounds, providing evidence that our estimators are likely to have nearly optimal sample complexity.

本文研究了在同时存在不可知和自适应对手的情况下，稀疏线性回归的有效估计器设计。研究提出了几种稳健算法，在加入高斯噪声的特殊情况下仍超越现有技术，且能在多项式时间内高概率恢复信号，显示出在稀疏设置中具有近乎最优的样本复杂性。

具有非同质设计的稳健稀疏回归