Block-Coordinate Methods and Restarting for Solving Extensive-Form Games

Abstract

Coordinate descent methods are popular in machine learning and optimization for their simple sparse updates and excellent practical performance. In the context of large-scale sequential game solving, these same properties would be attractive, but until now no such methods were known, because the strategy spaces do not satisfy the typical separable block structure exploited by such methods. We present the first cyclic coordinate-descent-like method for the polytope of sequence-form strategies, which form the strategy spaces for the players in an extensive-form game (EFG). Our method exploits the recursive structure of the proximal update induced by what are known as dilated regularizers, in order to allow for a pseudo block-wise update. We show that our method enjoys a $O(1/T)$ convergence rate to a two-player zero-sum Nash equilibrium, while avoiding the worst-case polynomial scaling with the number of blocks common to cyclic methods. We empirically show that our algorithm usually performs better than other state-of-the-art first-order methods (i.e., mirror prox), and occasionally can even beat CFR$^+$, a state-of-the-art algorithm for numerical equilibrium computation in zero-sum EFGs. We then introduce a restarting heuristic for EFG solving. We show empirically that restarting can lead to speedups, sometimes huge, both for our cyclic method, as well as for existing methods such as mirror prox and predictive CFR$^+$.