Variance Reduction for Matrix Games

Yair Carmon, Yujia Jin, Aaron Sidford, Kevin Tian

Abstract: We present a randomized primal-dual algorithm that solves the problem min_x max_y y^T A x to additive error epsilon in time nnz(A) + sqrt{nnz(A)n}/epsilon, for matrix A with larger dimension n and nnz(A) nonzero entries. This improves on Nemirovski's mirror-prox method by a factor of sqrt{nnz(A)/n} and is faster than stochastic gradient methods in the accurate and/or sparse regime epsilon < sqrt{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 is conceptually simple, leveraging a novel variance-reduced gradient estimator based on ``sampling from the difference'' between current iterates and a reference point.

CMT Num: 6073