We study the problem of high-dimensional robust linear regression where a learner is given access to $n$ samples from the generative model $Y = \langle X,w^* \rangle + \epsilon$ (with $X \in \mathbb{R}^d$ and $\epsilon$ independent), in which an $\eta$ fraction of the samples have been adversarially corrupted. We propose estimators for this problem under two settings: (i) $X$ is L4-L2 hypercontractive, $\mathbb{E} [XX^\top]$ has bounded condition number and $\epsilon$ has bounded variance and (ii) $X$ is sub-Gaussian with identity second moment and $\epsilon$ is sub-Gaussian. In both settings, our estimators: (a) Achieve optimal sample complexities and recovery guarantees up to log factors and (b) Run in near linear time ($\tilde{O}(nd / \eta^6)$). Prior to our work, polynomial time algorithms achieving near optimal sample complexities were only known in the setting where $X$ is Gaussian with identity covariance and $\epsilon$ is Gaussian, and no linear time estimators were known for robust linear regression in any setting. Our estimators and their analysis leverage recent developments in the construction of faster algorithms for robust mean estimation to improve runtimes, and refined concentration of measure arguments alongside Gaussian rounding techniques to improve statistical sample complexities.

研究了高维稳健线性回归问题，在受到对抗性破坏的情况下提出了估计方法，包括样本复杂度，恢复保证，运行时间等关键指标，并利用近期算法发展的加速算法和高斯舍入技术等方法来优化估计器的运行时间和统计样本复杂性。

近线性时间内的最优鲁棒线性回归