Spectral Sparsification and Regret Minimization Beyond Matrix Multiplicative Updates
Abstract
In this paper, we provide a novel construction of the linear-sized spectral sparsifiers of Batson, Spielman and Srivastava [BSS14]. While previous constructions required running time [BSS14, Zou12], our sparsification routine can be implemented in almost-quadratic running time . The fundamental conceptual novelty of our work is the leveraging of a strong connection between sparsification and a regret minimization problem over density matrices. This connection was known to provide an interpretation of the randomized sparsifiers of Spielman and Srivastava [SS11] via the application of matrix multiplicative weight updates (MWU) [CHS11, Vis14]. In this paper, we explain how matrix MWU naturally arises as an instance of the Follow-the-Regularized-Leader framework and generalize this approach to yield a larger class of updates. This new class allows us to accelerate the construction of linear-sized spectral sparsifiers, and give novel insights on the motivation behind Batson, Spielman and Srivastava [BSS14].
Cite
@article{arxiv.1506.04838,
title = {Spectral Sparsification and Regret Minimization Beyond Matrix Multiplicative Updates},
author = {Zeyuan Allen-Zhu and Zhenyu Liao and Lorenzo Orecchia},
journal= {arXiv preprint arXiv:1506.04838},
year = {2015}
}