Sublinear classical and quantum algorithms for general matrix games

• URL: http://arxiv.org/abs/2012.06519v1
• Date: Fri, 11 Dec 2020 17:36:33 GMT
• Title: Sublinear classical and quantum algorithms for general matrix games
• Authors: Tongyang Li, Chunhao Wang, Shouvanik Chakrabarti, and Xiaodi Wu
• Abstract summary: We investigate sublinear classical and quantum algorithms for matrix games. For any fixed $qin (1,2), we solve the matrix game where $mathcalX$ is a $ell_q$-norm unit ball within additive error. We also provide a corresponding sublinear quantum algorithm that solves the same task in time.
• Score: 11.339580074756189
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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