First-order algorithm with O(ln(1/ε )) convergence for ε -equilibrium in two-person zero-sum games

Abstract

We propose an iterated version of Nesterov's first-order smoothing method for the two-person zero-sum game equilibrium problem minx∈Q1 maxy∈Q2 xT Ay = maxy∈Q2 minx∈Q1 xTAy. This formulation applies to matrix games as well as sequential games. Our new algorithmic scheme computes an E-equilibrium to this min-max problem in O(κ(A) In(1/E)) first-order iterations, where κ(A) is a certain condition measure of the matrix A. This improves upon the previous first-order methods which required O(1/E) iterations, and it matches the iteration complexity bound of interior-point methods in terms of the algorithm's dependence on E. Unlike the interior-point methods that are inapplicable to large games due to their memory requirements, our algorithm retains the small memory requirements of prior first-order methods. Our scheme supplements Nesterov's algorithm with an outer loop that lowers the target E between iterations (this target affects the amount of smoothing in the inner loop). We find it surprising that such a simple modification yields an exponential speed improvement. Finally, computational experiments both in matrix games and sequential games show that a significant speed improvement is obtained in practice as well, and the relative speed improvement increases with the desired accuracy (as suggested by the complexity bounds).

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