Faster First-Order Methods for Extensive-Form Game Solving

Christian Kroer, Kevin Waugh, Fatma Kilinç-Karzan, Tuomas Sandholm

Abstract: We study the problem of computing a Nash equilibrium in large-scale two-player zero-sum extensive-form games. While this problem can be solved in polynomial time, first-order or regret-based methods are usually preferred for large games. Regret-based methods have largely been favored in practice, in spite of their theoretically inferior convergence rates. In this paper we investigate the acceleration of first-order methods both theoretically and experimentally. An important component of many first-order methods is a distance-generating function. Motivated by this, we investigate a specific distance-generating function, namely the dilated entropy function, over treeplexes, which are convex polytopes that encompass the strategy spaces of perfect-recall extensive-form games. We develop significantly stronger bounds on the associated strong convexity parameter. In terms of extensive-form game solving, this improves the convergence rate of several first-order methods by a factor of O((#information sets ⋅ depth ⋅ M)/(2depth)) where M is the maximum value of the l1 norm over the treeplex encoding the strategy spaces. Experimentally, we investigate the performance of three first-order methods (the excessive gap technique, mirror prox, and stochastic mirror prox) and compare their performance to the regret-based algorithms. In order to instantiate stochastic mirror prox, we develop a class of gradient sampling schemes for game trees. Equipped with our distance-generating function and sampling scheme, we find that mirror prox and the excessive gap technique outperform the prior regret-based methods for finding medium accuracy solutions