In this work we revisit two classic high-dimensional online learning problems, namely regression and linear 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, where the sequence of tasks we are asked to solve is adaptively and adversarially chosen? We answer this question in the affirmative for both regression and linear contextual bandits. In fact our algorithms succeed where convex surrogates failin the sense that we show strong lower bounds for the existing approaches . Our approach is based on a novel alternating minimization scheme that interleaves ordinary least-squares with a semidefinite program for finding appropriate reweightings of the distribution . Moreover we give extensions of our main results to infinite dimensional settings where the feature vectors are represented implicitly via a kernel map.
In this work we revisit two classic high-dimensional online learning problems, namely 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, where 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. In particular 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. Our results obtain essentially optimal dependence on the contamination level \eta, reach the optimal breakdown point, and naturally apply to infinite dimensional settings where the feature vectors are represented implicitly via a kernel map.
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regression and linear
linear regression and
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regression and linear
linear regression and
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convex surrogates failin the sense that
conventional methods fail. In particular
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for the existing approaches
against Huber regression and more generally any convex M-estimator
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semidefinite program for finding appropriate reweightings
simple convex program that finds the optimal reweighting
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. Moreover we give extensions of our main results
under a spectral constraint. Our results obtain essentially optimal dependence on the contamination level \eta, reach the optimal breakdown point, and naturally apply