Regression And Contextual Bandits With Huber Contamination

We revisit two classical (online) high-dimensional learning problems, linear regression and contextual bandits, from the perspective of adversarial robustness. Existing works in algorithmic robust statistics make strong distributional assumptions that ensure that the input data is evenly spread out or comes from a nice generative model. Is it possible to achieve strong robustness guarantees even without distributional assumptions altogether, and even when the sequence of tasks we are asked to solve is adaptively and adversarially chosen? We answer this question in the affirmative for both linear regression and contextual bandits. In fact our algorithms succeed where conventional methods fail --- we show strong lower bounds against Huber regression and more generally any convex M-estimator. Our approach is based on a novel alternating minimization scheme that interleaves ordinary least-squares with a simple convex program that finds the optimal reweighting of the distribution under a spectral constraint. This method obtains essentially optimal dependence on the contamination level η, reach the optimal breakdown point, and naturally apply to infinite dimensional settings where the feature vectors are represented implicitly via a kernel map. Based on joint work with Sitan Chen, Ankur Moitra, and Morris Yau.