Sublinear classical and quantum algorithms for general matrix games
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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∈ℝ^n× d, sublinear algorithms for the matrix game min_x∈𝒳max_y∈𝒴 y^⊤ Ax were previously known only for two special cases: (1) 𝒴 being the ℓ_1-norm unit ball, and (2) 𝒳 being either the ℓ_1- or the ℓ_2-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 ℓ_q-norm unit ball within additive error ϵ in time Õ((n+d)/ϵ^2). We also provide a corresponding sublinear quantum algorithm that solves the same task in time Õ((√(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 ℓ_q-margin support vector machines as applications.
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