Sublinear classical and quantum algorithms for general matrix games

TitleSublinear classical and quantum algorithms for general matrix games
Publication TypeJournal Article
Year of Publication2020
AuthorsLi, T, Wang, C, Chakrabarti, S, Wu, X
JournalTo appear in the Thirty-Fifth AAAI Conference on Artificial Intelligence (AAAI 2021)
Date Published12/11/2020
Abstract

We investigate sublinear classical and quantum algorithms for matrix games, a fundamental problem in optimization and machine learning, with provable guarantees. Given a matrix A∈Rn×d, sublinear algorithms for the matrix game minx∈Xmaxy∈Yy⊤Ax were previously known only for two special cases: (1) Y being the ℓ1-norm unit ball, and (2) X being either the ℓ1- or the ℓ2-norm unit ball. We give a sublinear classical algorithm that can interpolate smoothly between these two cases: for any fixed q∈(1,2], we solve the matrix game where X is a ℓq-norm unit ball within additive error ε in time O~((n+d)/ε2). We also provide a corresponding sublinear quantum algorithm that solves the same task in time O~((n−−√+d−−√)poly(1/ε)) with a quadratic improvement in both n and d. Both our classical and quantum algorithms are optimal in the dimension parameters n and d up to poly-logarithmic factors. Finally, we propose sublinear classical and quantum algorithms for the approximate Carathéodory problem and the ℓq-margin support vector machines as applications.

URLhttps://arxiv.org/abs/2012.06519