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

Adaptive Hard Thresholding for Near-optimal Consistent Robust Regression