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).

Urban Network Security Games (UNSGs), which model the strategic allocation of limited security resources on city road networks, are critical for urban safety. However, finding a Nash Equilibrium (NE) in large-scale UNSGs is challenging due to their massive and combinatorial action spaces. One common approach to addressing these games is the Policy-Space Response Oracle (PSRO) framework, which requires computing best responses (BR) at each iteration. However, precisely computing exact BRs is impractical in large-scale games, and employing reinforcement learning to approximate BRs inevitably introduces errors that limit the overall effectiveness of the PSRO methods. Recent advancements in leveraging non-convex stochastic optimization to approximate an NE offer a promising alternative to the burdensome BR computation. However, utilizing existing stochastic optimization techniques with an unbiased loss function for UNSGs remains challenging because the action spaces are too vast to be effectively represented by neural networks. To address these issues, we introduce Tree-based Stochastic Optimization (TSO), a framework that bridges the gap between the stochastic optimization paradigm for NE-finding and the demands of UNSGs. Specifically, we employ the tree-based action representation that maps the whole action space onto a tree structure, addressing the challenge faced by neural networks in representing actions when the action space cannot be enumerated. We then incorporate this representation into the loss function and theoretically demonstrate its equivalence to the unbiased loss function. To further enhance the quality of the converged solution, we introduce a sample-and-prune mechanism that reduces the risk of being trapped in suboptimal local optima. Extensive experimental results indicate the superiority of TSO over other baseline algorithms in addressing the UNSGs.

Siyuan Gan, Jiaheng Liu, Boyan Wang, Tianpei Yang, Runqing Miao, Yuyao Zhang, Fanyu Meng, Junlan Feng, Linjian Meng, Jing Huo, Yang Gao. Proceedings of the 64th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers). 2026.

Self-play methods have demonstrated remarkable success in enhancing model capabilities across various domains. In the context of Reinforcement Learning from Human Feedback (RLHF), self-play not only boosts Large Language Model (LLM) performance but also overcomes the limitations of traditional Bradley-Terry (BT) model assumptions by finding the Nash equilibrium (NE) of a preference-based, two-player constant-sum game. However, existing methods either guarantee only average-iterate convergence, incurring high storage and inference costs, or converge to the NE of a regularized game, failing to accurately reflect true human preferences. In this paper, we introduce Magnetic Preference Optimization (MPO), a novel approach capable of achieving last-iterate convergence to the NE of the original game, effectively overcoming the limitations of existing methods. Building upon Magnetic Mirror Descent (MMD), MPO attains a linear convergence rate, making it particularly suitable for fine-tuning LLMs. To ensure our algorithm is both theoretically sound and practically viable, we present a simple yet effective implementation that adapts the theoretical insights to the RLHF setting. Empirical results demonstrate that MPO can significantly enhance the performance of LLMs, highlighting the potential of self-play methods in alignment.

Nash equilibrium (NE) plays an important role in game theory. How to efficiently compute an NE in NFGs is challenging due to its complexity and non-convex optimization property. Machine Learning (ML), the cornerstone of modern artificial intelligence, has demonstrated remarkable empirical performance across various applications including non-convex optimization. To leverage non-convex stochastic optimization techniques from ML for approximating an NE, various loss functions have been proposed. Among these, only one loss function is unbiased, allowing for unbiased estimation under the sampled play. Unfortunately, this loss function suffers from high variance, which degrades the convergence rate. To improve the convergence rate by mitigating the high variance associated with the existing unbiased loss function, we propose a novel surrogate loss function named Nash Advantage Loss (NAL). NAL is theoretically proved unbiased and exhibits significantly lower variance than the existing unbiased loss function. Experimental results demonstrate that the algorithm minimizing NAL achieves a significantly faster empirical convergence rates compared to other algorithms, while also reducing the variance of estimated loss value by several orders of magnitude.

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. Hence, proving last-iterate convergence in solving the original EFG reduces to proving last-iterate convergence in solving (perturbed) regularized EFGs. However, these studies only establish last-iterate convergence for Online Mirror Descent (OMD)-based CFR algorithms instead of Regret Matching (RM)-based CFR algorithms in solving perturbed regularized EFGs, resulting in a poor empirical convergence rate, as RM-based CFR algorithms typically outperform OMD-based CFR algorithms. In addition, as solving multiple perturbed regularized EFGs is required, fine-tuning across multiple perturbed regularized EFGs is infeasible, making parameter-free algorithms highly desirable. This paper show that CFR$^+$, a classical parameter-free RM-based CFR algorithm, achieves last-iterate convergence in learning an NE of perturbed regularized EFGs. This is the first parameter-free last-iterate convergence for RM-based CFR algorithms in 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 vital for the empirical convergence of RTCFR$^+$. Experiments show that RTCFR$^+$ exhibits a significantly faster empirical convergence rate than existing algorithms that achieve theoretical last-iterate convergence. Interestingly, RTCFR$^+$ show performance no worse than average-iterate convergence CFR algorithms. It is the first last-iterate convergence algorithm to achieve such performance. Our code is available at https://github.com/menglinjian/NeurIPS-2025-RTCFR.

Regret Matching$^+$ (RM$^+$) variants are widely used to build superhuman Poker AIs, yet few studies investigate their last-iterate convergence in learning a Nash equilibrium (NE). Although their last-iterate convergence is established for games satisfying the Minty Variational Inequality (MVI), no studies have demonstrated that these algorithms achieve such convergence in the broader class of games satisfying the weak MVI. A key challenge in proving last-iterate convergence for RM$^+$ variants in games satisfying the weak MVI is that even if the game's loss gradient satisfies the weak MVI, RM$^+$ variants operate on a transformed loss feedback which does not satisfy the weak MVI. To provide last-iterate convergence for RM$^+$ variants, we introduce a concise yet novel proof paradigm that involves: (i) transforming an RM$^+$ variant into an Online Mirror Descent (OMD) instance that updates within the original strategy space of the game to recover the weak MVI, and (ii) showing last-iterate convergence by proving the distance between accumulated regrets converges to zero via the recovered weak MVI of the feedback. Inspired by our proof paradigm, we propose Smooth Optimistic Gradient Based RM$^+$ (SOGRM$^+$) and show that it achieves last-iterate and finite-time best-iterate convergence in learning an NE of games satisfying the weak MVI, the weakest condition among all known RM$^+$ variants. Experiments show that SOGRM$^+$ significantly outperforms other algorithms. Our code is available at https://github.com/menglinjian/NeurIPS-2025-SOGRM.

Opponent exploitation is an important task for players to exploit the weaknesses of others in games. Existing approaches mainly focus on balancing between exploitation and exploitability but are often vulnerable to modeling errors and deceptive adversaries. To address this problem, our paper offers a novel perspective on the safety of opponent exploitation, named Adaptation Safety. This concept leverages the insight that strategies, even those not explicitly aimed at opponent exploitation, may inherently be exploitable due to computational complexities, rendering traditional safety overly rigorous. In contrast, adaptation safety requires that the strategy should not be more exploitable than it would be in scenarios where opponent exploitation is not considered. Building on such adaptation safety, we further propose an Opponent eXploitation Search (OX-Search) framework by incorporating real-time search techniques for efficient online opponent exploitation. Moreover, we provide theoretical analyses to show the adaptation safety and robust exploitation of OX-Search, even with inaccurate opponent models. Empirical evaluations in popular poker games demonstrate OX-Search’s superiority in both exploitability and exploitation compared to previous methods.

One of the most popular methods for learning Nash equilibrium (NE) in large-scale imperfect information extensive-form games (IIEFGs) is the neural variants of counterfactual regret minimization (CFR). CFR is a special case of Follow-The-Regularized-Leader (FTRL). At each iteration, the neural variants of CFR update the agent's strategy via the estimated counterfactual regrets. Then, they use neural networks to approximate the new strategy, which incurs an approximation error. These approximation errors will accumulate since the counterfactual regrets at iteration t are estimated using the agent's past approximated strategies. Such accumulated approximation error causes poor performance. To address this accumulated approximation error, we propose a novel FTRL algorithm called FTRL-ORW, which does not utilize the agent's past strategies to pick the next iteration strategy. More importantly, FTRL-ORW can update its strategy via the trajectories sampled from the game, which is suitable to solve large-scale IIEFGs since sampling multiple actions for each information set is too expensive in such games. However, it remains unclear which algorithm to use to compute the next iteration strategy for FTRL-ORW when only such sampled trajectories are revealed at iteration t. To address this problem and scale FTRL-ORW to large-scale games, we provide a model-free method called Deep FTRL-ORW, which computes the next iteration strategy using model-free Maximum Entropy Deep Reinforcement Learning. Experimental results on two-player zero-sum IIEFGs show that Deep FTRL-ORW significantly outperforms existing model-free neural methods and OS-MCCFR.