On Coresets for Regularized Regression

Jul 12, 2020

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We study the effect of norm based regularization on the size of coresets for the regularized regression problems. Specifically, given a matrix π€βˆˆπ‘^n xd with n≫ d and a vector π›βˆˆπ‘ ^ n and Ξ» > 0, we analyze the size of coresets for regularized versions of regression of the form ||Ax-b||_p^r + Ξ»||x||_q^s . It has been shown for the case of ridge regression (p,q,r,s=2) that we can obtain a coreset smaller than the coreset for its unregularized counterpart i.e. least squares regression<cit.>. However we show that when r β‰  s, no coreset for some regularized regression can have size smaller than the optimal coreset of the unregularized version. The well known LASSO problem falls under this category and hence does not allow a coreset smaller than the one for least squares regression. We propose a modified version of the LASSO problem and obtain for it a coreset of size smaller than the least square regression. We empirically show that the modified version of LASSO also induces sparsity in solution like the LASSO. We also obtain smaller coresets for β„“_p-regression with β„“_p-regularization. We extend our methods to multi response regularized regression. Finally, we empirically demonstrate the coreset performance for the modified LASSO and the β„“_1-regression with β„“_1- regularization.

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The International Conference on Machine Learning (ICML) is the premier gathering of professionals dedicated to the advancement of the branch of artificial intelligence known as machine learning. ICML is globally renowned for presenting and publishing cutting-edge research on all aspects of machine learning used in closely related areas like artificial intelligence, statistics and data science, as well as important application areas such as machine vision, computational biology, speech recognition, and robotics. ICML is one of the fastest growing artificial intelligence conferences in the world. Participants at ICML span a wide range of backgrounds, from academic and industrial researchers, to entrepreneurs and engineers, to graduate students and postdocs.

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