Zhenxing Ge

Linjian Meng, Youzhi Zhang, Zhenxing Ge et al.

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.

Linjian Meng, Tianpei Yang, Youzhi Zhang et al.

Counterfactual Regret Minimization (CFR) algorithms are widely used to compute a Nash equilibrium (NE) in two-player zero-sum imperfect-information extensive-form games (IIGs). Among them, Predictive CFR$^+$ (PCFR$^+$) is particularly powerful, achieving an exceptionally fast empirical convergence rate via the prediction in many games.However, the empirical convergence rate of PCFR$^+$ would significantly degrade if the prediction is inaccurate, leading to unstable performance on certain IIGs. To enhance the robustness of PCFR$^+$, we propose Asymmetric PCFR$^+$ (APCFR$^+$), which employs an adaptive asymmetry of step sizes between the updates of implicit and explicit accumulated counterfactual regrets to mitigate the impact of the prediction inaccuracy on convergence. We present a theoretical analysis demonstrating why APCFR$^+$ can enhance the robustness. To the best of our knowledge, we are the first to propose the asymmetry of step sizes, a simple yet novel technique that effectively improves the robustness of PCFR$^+$. Then, to reduce the difficulty of implementing APCFR$^+$ caused by the adaptive asymmetry, we propose a simplified version of APCFR$^+$ called Simple APCFR$^+$ (SAPCFR$^+$), which uses a fixed asymmetry of step sizes to enable only a single-line modification compared to original PCFR$^+$.Experimental results on five standard IIG benchmarks and two heads-up no-limit Texas Hold' em (HUNL) Subagems show that (i) both APCFR$^+$ and SAPCFR$^+$ outperform PCFR$^+$ in most of the tested games, (ii) SAPCFR$^+$ achieves a comparable empirical convergence rate with APCFR$^+$,and (iii) our approach can be generalized to improve other CFR algorithms, e.g., Discount CFR (DCFR).