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Well-Conditioned Methods for Ill-Conditioned Systems: Linear Regression with Semi-Random Noise

4 August 2020
A. Jambulapati
Jingkai Li
Christopher Musco
Aaron Sidford
ArXiv (abs)PDFHTML
Abstract

Classical iterative algorithms for linear system solving and regression are brittle to the condition number of the data matrix. Even a semi-random adversary, constrained to only give additional consistent information, can arbitrarily hinder the resulting computational guarantees of existing solvers. We show how to overcome this barrier by developing a framework which takes state-of-the-art solvers and "robustifies" them to achieve comparable guarantees against a semi-random adversary. Given a matrix which contains an (unknown) well-conditioned submatrix, our methods obtain computational and statistical guarantees as if the entire matrix was well-conditioned. We complement our theoretical results with preliminary experimental evidence, showing that our methods are effective in practice.

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