We present the first efficient and provably consistent estimator for the robust regression problem. The area of robust learning and optimization has generated a significant amount of interest in the learning and statistics communities in recent years owing to its applicability in scenarios with corrupted data, as well as in handling model mis-specifications. In particular, special interest has been devoted to the fundamental problem of robust linear regression where estimators that can tolerate corruption in up to a constant fraction of the response variables are widely studied. Surprisingly however, to this date, we are not aware of a polynomial time estimator that offers a consistent estimate in the presence of dense, unbounded corruptions. In this work we present such an estimator, called CRR. This solves an open problem put forward in the work of (Bhatia et al, 2015). Our consistency analysis requires a novel two-stage proof technique involving a careful analysis of the stability of ordered lists which may be of independent interest. We show that CRR not only offers consistent estimates, but is empirically far superior to several other recently proposed algorithms for the robust regression problem, including extended Lasso and the TORRENT algorithm. In comparison, CRR offers comparable or better model recovery but with runtimes that are faster by an order of magnitude.

Summary: The paper presents a provably consistent, polynomial-time algorithm (that they call consistent robust regression, or CRR) for linear regression with corrupted samples under the oblivious adversary model. The algorithm is a simple iterative hard thresholding (IHT) procedure, and the authors show that this algorithm exhibits linear convergence. The analysis is supported by some representative synthetic numerical results. Strengths: Quality: Somewhat surprisingly, this method appears to be the first consistent procedure for robust linear regression, i.e., the parameter estimation error vanishes as the number of samples tends to infinity. Other estimation procedures (such as LASSO-type methods) seem to only provide solutions with error comparable to the noise level, even in the large sample limit.

We present the first efficient and provably consistent estimator for the robust regression problem. The area of robust learning and optimization has generated a significant amount of interest in the learning and statistics communities in recent years owing to its applicability in scenarios with corrupted data, as well as in handling model mis-specifications. In particular, special interest has been devoted to the fundamental problem of robust linear regression where estimators that can tolerate corruption in up to a constant fraction of the response variables are widely studied. Surprisingly however, to this date, we are not aware of a polynomial time estimator that offers a consistent estimate in the presence of dense, unbounded corruptions. In this work we present such an estimator, called CRR.