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We study the application of iterative first-order methods to the problem ofcomputing equilibria of large-scale two-player extensive-form games.First-order methods must typically be instantiated with a regularizer thatserves as a distance-generating function for the decision sets of the players.For the case of two-player zero-sum games, the state-of-the-art theoreticalconvergence rate for Nash equilibrium is achieved by using the dilated entropyfunction. In this paper, we introduce a new entropy-based distance-generatingfunction for two-player zero-sum games, and show that this function achievessignificantly better strong convexity properties than the dilated entropy,while maintaining the same easily-implemented closed-form proximal mapping.Extensive numerical simulations show that these superior theoretical propertiestranslate into better numerical performance as well. We then generalize our newentropy distance function, as well as general dilated distance functions, tothe scaled extension operator. The scaled extension operator is a way torecursively construct convex sets, which generalizes the decision polytope ofextensive-form games, as well as the convex polytopes corresponding tocorrelated and team equilibria. By instantiating first-order methods with ourregularizers, we develop the first accelerated first-order methods forcomputing correlated equilibra and ex-ante coordinated team equilibria. Ourmethods have a guaranteed $1/T$ rate of convergence, along with linear-timeproximal updates.

Better Regularization for Sequential Decision Spaces: Fast Convergence Rates for Nash, Correlated, and Team Equilibria