Sublinear classical and quantum algorithms for general matrix games

TL;DR

This paper introduces new sublinear classical and quantum algorithms for solving general matrix games, extending previous work to a broader class of norm constraints with provable efficiency and optimality.

Contribution

It develops a unified sublinear algorithm for matrix games with $ ext{l}_q$-norm constraints, and provides quantum algorithms with quadratic speedups, also applying to related optimization problems.

Findings

Classical algorithm solves matrix games with $ ext{l}_q$-norm constraints in near-linear time.

Quantum algorithm achieves quadratic speedup over classical in dimension parameters.

Both algorithms are proven to be optimal up to poly-logarithmic factors.

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 $A\in\mathbb{R}^{n\times d}$, sublinear algorithms for the matrix game $\min_{x\in\mathcal{X}}\max_{y\in\mathcal{Y}} y^{\top} Ax$ were previously known only for two special cases: (1) $\mathcal{Y}$ being the $\ell_{1}$-norm unit ball, and (2) $\mathcal{X}$ being either the $\ell_{1}$- or the $\ell_{2}$-norm unit ball. We give a sublinear classical algorithm that can interpolate smoothly between these two cases: for any fixed $q\in (1,2]$, we solve the matrix game where $\mathcal{X}$ is a $\ell_{q}$-norm unit ball within additive error $\epsilon$ in time $\tilde{O}((n+d)/{\epsilon^{2}})$. We also provide a corresponding sublinear quantum algorithm that solves the same task in time $\tilde{O}((\sqrt{n}+\sqrt{d})\textrm{poly}(1/\epsilon))$ 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 Carath\'eodory problem and the $\ell_{q}$-margin support vector machines as applications.