- First-order algorithm with $O(ln(1/\epsilon))$ convergence for $\epsilon$-equilibrium in two-person zero-sum games Andrew Gilpin (gilpincs.cmu.edu) Javier Pena (jfpandrew.cmu.edu) Tuomas Sandholm (sandholmcs.cmu.edu) Abstract: 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). Keywords: games, equilibrium, smoothing Category 1: Convex and Nonsmooth Optimization (Convex Optimization ) Category 2: Other Topics (Game Theory ) Citation: 23rd National Conference on Artificial Intelligence (AAAI'08) Download: [PDF]Entry Submitted: 04/16/2008Entry Accepted: 05/02/2008Entry Last Modified: 05/03/2008Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Programming Society and by the Optimization Technology Center.