We study the problem of robust linear regression with response variable corruptions. We consider the oblivious adversary model, where the adversary corrupts a fraction of the responses in complete ignorance of the data. We provide a nearly linear time estimator which consistently estimates the true regression vector, even with $1-o(1)$ fraction of corruptions. Existing results in this setting either don't guarantee consistent estimates or can only handle a small fraction of corruptions. We also extend our estimator to robust sparse linear regression and show that similar guarantees hold in this setting. Finally, we apply our estimator to the problem of linear regression with heavy-tailed noise and show that our estimator consistently estimates the regression vector even when the noise has unbounded variance (e.g., Cauchy distribution), for which most existing results don't even apply. Our estimator is based on a novel variant of outlier removal via hard thresholding in which the threshold is chosen adaptively and crucially relies on randomness to escape bad fixed points of the non-convex hard thresholding operation.

通过使用硬阈值化的新颖变体，本文提出了一种快速的鲁棒估计器，可以有效地解决使用响应变量损坏的鲁棒线性回归问题，并通过应用于不同的扰动模型，展示了其估计能力的稳健性。

自适应硬阈值近似最优一致鲁棒回归