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...
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.