Robust Regression via Hard Thresholding

We study the problem of Robust Least Squares Regression (RLSR) where several response variables can be adversarially corrupted. More specifically, for a data matrix X ∈ Rp×n and an underlying model w∗, the response vector is generated as y = XTw∗+b where b ∈ R is the corruption vector supported over at most C ·n coordinates. Existing exact recovery results for RLSR focus solely on L1-penalty based convex formulations and impose relatively strict model assumptions such as requiring the corruptions b to be selected independently of X. In this work, we study a simple hard-thresholding algorithm called Torrent which, under mild conditions on X, can recover w∗ exactly even if b corrupts the response variables in an adversarial manner, i.e. both the support and entries of b are selected adversarially after observing X and w∗. Our results hold under deterministic assumptions which are satisfied if X is sampled from any sub-Gaussian distribution. Finally unlike existing results that apply only to a fixed w∗, generated independently of X, our results are universal and hold for any w∗ ∈ R. Next, we propose gradient descent-based extensions of Torrent that can scale efficiently to large scale problems, such as high dimensional sparse recovery and prove similar recovery guarantees for these extensions. Empirically we find Torrent, and more so its extensions, offering significantly faster recovery than the state-of-the-art L1 solvers. For instance, even on moderate-sized datasets (with p = 50K) with around 40% corrupted responses, a variant of our proposed method called Torrent-HYB is more than 20× faster than the best L1 solver. “If among these errors are some which appear too large to be admissible, then those equations which produced these errors will be rejected, as coming from too faulty experiments, and the unknowns will be determined by means of the other equations, which will then give much smaller errors.” A. M. Legendre, On the Method of Least Squares. 1805.