Efficient Last-iterate Convergence Algorithms in Solving Games

To establish last-iterate convergence for Counterfactual Regret Minimization (CFR) algorithms in learning a Nash equilibrium (NE) of extensive-form games (EFGs), recent studies reformulate learning an NE of the original EFG as learning the NEs of a sequence of (perturbed) regularized EFGs. Consequently, proving last-iterate convergence in solving the original EFG reduces to proving last-iterate convergence in solving (perturbed) regularized EFGs. However, the empirical convergence rates of the algorithms in these studies are suboptimal, since they do not utilize Regret Matching (RM)-based CFR algorithms to solve perturbed EFGs, which are known the exceptionally fast empirical convergence rates. Additionally, since solving multiple perturbed regularized EFGs is required, fine-tuning across all such games is infeasible, making parameter-free algorithms highly desirable. In this paper, we prove that CFR^+, a classical parameter-free RM-based CFR algorithm, achieves last-iterate convergence in learning an NE of perturbed regularized EFGs. Leveraging CFR^+ to solve perturbed regularized EFGs, we get Reward Transformation CFR^+ (RTCFR^+). Importantly, we extend prior work on the parameter-free property of CFR^+, enhancing its stability, which is crucial for the empirical convergence of RTCFR^+. Experiments show that RTCFR^+ significantly outperforms existing algorithms with theoretical last-iterate convergence guarantees.