NDLI: First-order algorithm with $${\mathcal{O}({\rm ln}(1{/}\epsilon))}$$ convergence for $${\epsilon}$$ -equilibrium in two-person zero-sum games

Content Provider	Springer Nature Link
Author	Sandholm, Tuomas Gilpin, Andrew Peña, Javier
Copyright Year	2010
Abstract	We propose an iterated version of Nesterov’s first-order smoothing method for the two-person zero-sum game equilibrium problem $$\min_{x \in Q_1}\max_{y \in Q_2} {x}^{\rm T}{Ay} = \max_{y \in Q_2} \min_{x \in Q_1} {x}^{\rm T}{Ay}.$$ 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 $${\mathcal {O}\left(\frac{\\|A\\|}{\delta(A)} \, {\rm ln}(1{/}\epsilon)\right)}$$ first-order iterations, where δ(A) is a certain condition measure of the matrix A. This improves upon the previous first-order methods which required $${\mathcal {O}(1{/}\epsilon)}$$ iterations, and it matches the iteration complexity bound of interior-point methods in terms of the algorithm’s dependence on $${\epsilon}$$ . Unlike 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 method with an outer loop that lowers the target $${\epsilon}$$ between iterations (this target affects the amount of smoothing in the inner loop). 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).
Ending Page	298
Page Count	20
Starting Page	279
File Format	PDF
ISSN	00255610
e-ISSN	14364646
Journal	Mathematical Programming
Issue Number	1-2
Volume Number	133
Language	English
Publisher	Springer-Verlag
Publisher Date	2010-11-27
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Mathematical Methods in Physics 2-person games Mathematics of Computing Calculus of Variations and Optimal Control; Optimization Numerical Analysis Theoretical, Mathematical and Computational Physics Methods of reduced gradient type Combinatorics Minimax problems Complementarity and equilibrium problems and variational inequalities (finite dimensions)
Content Type	Text
Resource Type	Article
Subject	Mathematics Software

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