Variance Reduction for Matrix Games

07/03/2019

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by   Yair Carmon, et al.

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We present a randomized primal-dual algorithm that solves the problem _x_y y^ A x to additive error ϵ in time nnz(A) + √(nnz(A)n)/ϵ, for matrix A with larger dimension n and nnz(A) nonzero entries. This improves on Nemirovski's mirror-prox method by a factor of √(nnz(A)/n) and is faster than stochastic gradient methods in the accurate and/or sparse regime ϵ<√(n/nnz(A)). Our results hold for x,y in the simplex (matrix games, linear programming) and for x in an ℓ_2 ball and y in the simplex (perceptron / SVM, minimum enclosing ball). Our algorithm combines the mirror-prox method and a novel variance-reduced gradient estimator based on "sampling from the difference" between the current iterate and a reference point.