High Dimensional Robust Sparse Regression

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Liu Liu, Yanyao Shen, Tianyang Li, Constantine Caramanis ;
Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, PMLR 108:411-421, 2020.

Abstract

We provide a novel – and to the best of our knowledge, the first – algorithm for high dimensional sparse regression with constant fraction of corruptions in explanatory and/or response variables. Our algorithm recovers the true sparse parameters with sub-linear sample complexity,in the presence of a constant fraction of arbitrary corruptions. Our main contribution is a robust variant of Iterative Hard Thresholding. Using this, we provide accurate estimators:when the covariance matrix in sparse regression is identity, our error guarantee is near information-theoretically optimal. We then deal with robust sparse regression with unknown structured covariance matrix. We propose a filtering algorithm whichconsists of a novel randomized outlier removal technique for robust sparse mean estimation that may be of interest in its own right: the filtering algorithm is flexible enough to deal with unknown covariance.Also, it is orderwise more efficient computationally than the ellipsoid algorithm.Using sub-linear sample complexity, our algorithm achieves the best known (and first) error guarantee. We demonstrate the effectiveness on large-scale sparse regression problems with arbitrary corruptions.

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