# Faster Game Solving via Asymmetry of Step Sizes

arXiv: https://arxiv.org/abs/2503.12770 · HTML: https://mlsanity.com/paper/2503.12770/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 32 references that are in the map.
- First posted on arXiv 17 Mar 2025.

## Builds on (most cited first, 13 of 13)

- [Introduction to Online Convex Optimization](https://mlsanity.com/paper/1909.05207/) (2019)
- [DeepStack: Expert-Level Artificial Intelligence in No-Limit Poker](https://mlsanity.com/paper/1701.01724/) (2017)
- [Optimization, Learning, and Games with Predictable Sequences](https://mlsanity.com/paper/1311.1869/) (2013)
- [OpenSpiel: A Framework for Reinforcement Learning in Games](https://mlsanity.com/paper/1908.09453/) (2019)
- [Solving Large Imperfect Information Games Using CFR+](https://mlsanity.com/paper/1407.5042/) (2014)
- [Solving Imperfect-Information Games via Discounted Regret Minimization](https://mlsanity.com/paper/1809.04040/) (2018)
- [From Poincar'e Recurrence to Convergence in Imperfect Information Games: Finding Equilibrium via Regularization](https://mlsanity.com/paper/2002.08456/) (2020)
- [Faster Game Solving via Predictive Blackwell Approachability: Connecting Regret Matching and Mirror Descent](https://mlsanity.com/paper/2007.14358/) (2020)
- [Online Convex Optimization for Sequential Decision Processes and Extensive-Form Games](https://mlsanity.com/paper/1809.03075/) (2018)
- [Regret Matching+: (In)Stability and Fast Convergence in Games](https://mlsanity.com/paper/2305.14709/) (2023)
- [Equivalence Analysis between Counterfactual Regret Minimization and Online Mirror Descent](https://mlsanity.com/paper/2110.04961/) (2021)
- [Faster Game Solving via Hyperparameter Schedules](https://mlsanity.com/paper/2404.09097/) (2024)
- [Minimizing Weighted Counterfactual Regret with Optimistic Online Mirror Descent](https://mlsanity.com/paper/2404.13891/) (2024)

## Built on by (most cited first, 1 of 1)

- [Agents That Certify Their Own Exploits: Confidence-Scheduled Restricted Responses for Safe Opponent Exploitation](https://mlsanity.com/paper/2607.28520/) (2026)

## Abstract (arXiv)

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).