Subsampled Newton methods approximate Hessian matrices through subsampling techniques to alleviate the per-iteration cost. Previous results require Ω (d) samples to approximate Hessians, where d is the dimension of data points, making it less practical for high-dimensional data. The situation is deteriorated when d is comparably as large as the number of data points n, which requires to take the whole dataset into account, making subsampling not useful. This paper theoretically justifies the effectiveness of subsampled Newton methods on strongly convex empirical risk minimization with high dimensional data. Specifically, we provably require only Θ˜(deffγ) samples for approximating the Hessian matrices, where deffγ is the γ-ridge leverage and can be much smaller than d as long as nγ ≫ 1. Our theories work for three types of Newton methods: subsampled Netwon, distributed Newton, and proximal Newton.
Published Date: 2020-06-02
Registration: ISSN 2374-3468 (Online) ISSN 2159-5399 (Print) ISBN 978-1-57735-835-0 (10 issue set)
Copyright: Published by AAAI Press, Palo Alto, California USA Copyright © 2020, Association for the Advancement of Artificial Intelligence All Rights Reserved