Adaptive Hard Thresholding for Near-optimal Consistent Robust Regression
This addresses robust regression for data with corruptions or heavy-tailed noise, offering consistent estimates where prior methods fail, though it is incremental in improving thresholding techniques.
The paper tackles robust linear regression with response corruptions, proposing a nearly linear-time estimator that consistently estimates the true regression vector even with up to 1-o(1) fraction of corruptions, and extends this to sparse regression and heavy-tailed noise cases like Cauchy distributions.
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.