We propose an iterated version of Nesterov's first-order smoothing method for the two-person zero-sum game equilibrium problem $$\min_{x\in Q_1} \max_{y\in Q_2} \ip{x}{Ay} = \max_{y\in Q_2} \min_{x\in Q_1} \ip{x}{Ay}.$$ This formulation applies to matrix games as well as sequential games. Our new algorithmic scheme computes an $\epsilon$-equilibrium to this min-max problem in $\Oh(\kappa(A) \ln(1/\epsilon))$ first-order iterations, where $\kappa(A)$ is a certain condition measure of the matrix $A$. This improves upon the previous first-order methods which required $\Oh(1/\epsilon)$ iterations, and it matches the iteration complexity bound of interior-point methods in terms of the algorithm's dependence on $\epsilon$. 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 $\epsilon$ 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).
Citation
23rd National Conference on Artificial Intelligence (AAAI'08)