A Policy-Gradient Approach to Solving Imperfect-Information Games with Iterate Convergence (2408.00751v1)

Abstract: Policy gradient methods have become a staple of any single-agent reinforcement learning toolbox, due to their combination of desirable properties: iterate convergence, efficient use of stochastic trajectory feedback, and theoretically-sound avoidance of importance sampling corrections. In multi-agent imperfect-information settings (extensive-form games), however, it is still unknown whether the same desiderata can be guaranteed while retaining theoretical guarantees. Instead, sound methods for extensive-form games rely on approximating counterfactual values (as opposed to Q values), which are incompatible with policy gradient methodologies. In this paper, we investigate whether policy gradient can be safely used in two-player zero-sum imperfect-information extensive-form games (EFGs). We establish positive results, showing for the first time that a policy gradient method leads to provable best-iterate convergence to a regularized Nash equilibrium in self-play.

A Policy-Gradient Approach to Solving Imperfect-Information Extensive-Form Games with Iterate Convergence

The paper "A Policy-Gradient Approach to Solving Imperfect-Information Extensive-Form Games with Iterate Convergence" by Mingyang Liu, Gabriele Farina, and Asuman Ozdaglar addresses a fundamental challenge in the domain of deep reinforcement learning (DRL) applied to multi-agent imperfect-information extensive-form games (EFGs), exemplified by games such as Texas Hold'em poker. Traditional DRL algorithms typically fail in these settings due to issues such as getting trapped in cycles without making significant progress.

Key Contributions

This work notably diverges from conventional counterfactual regret minimization (CFR) methods, which, while empirically effective, offer guarantees only for average-strategy convergence rather than iterate convergence. The primary contribution of the paper is the development of a policy gradient method termed Q-Function based Regret Minimization (QFR) that achieves best-iterate convergence in solving EFGs. This method represents a fundamental shift, leveraging trajectory Q-values for efficient, scalable value estimation without resorting to importance sampling.

Methodology

The QFR algorithm relies on several essential components:

• Trajectory Q-Values: A distinct notion of Q-values for EFGs that is compatible with trajectory-based estimation, eliminating the need for importance sampling and the associated high variance.
• Bidilated Regularizer: A novel regularization technique designed specifically for EFGs, which ensures stability across varying game depths.
• Iterate Convergence: Achieved through a learning rate schedule that dynamically adjusts with the game depth, ensuring the stability of ancestor strategies while updating descendant strategies.

Theoretical Foundations

The authors provide rigorous theoretical guarantees for QFR, establishing best-iterate convergence to a regularized Nash equilibrium. They detail the gradients' boundedness and stability, ensuring convergence through appropriate regularization and learning rate schedules. The paper includes a comprehensive analysis of convergence under both full-information feedback and stochastic feedback settings, demonstrating the robustness of QFR against variations in the feedback mechanism.

Experimental Validation

The empirical evaluation includes several benchmark games such as 4-Sided Liar's Dice, Leduc Poker, Kuhn Poker, and 2x2 Abrupt Dark Hex. The results indicate that QFR outperforms traditional CFR methods, particularly in terms of last-iterate performance. This highlights the algorithm’s practical viability beyond mere theoretical elegance, especially in large and complex game settings.

Implications and Future Directions

The proposed QFR algorithm bridges a significant gap between DRL and imperfect-information EFGs, bringing EFG technology closer to the state-of-the-art in DRL. This development has substantial implications for multi-agent systems, potentially enhancing strategies in competitive domains such as automated negotiations, strategic decision-making, and beyond.

Future developments could explore further optimization of learning rates uniformly across all game infosets, as suggested by the empirical evidence. Moreover, scaling QFR for much larger game spaces, possibly integrating neural network-based function approximation, represents an exciting avenue for extending this line of research.

In conclusion, this paper makes a substantial contribution to the field of multi-agent reinforcement learning, providing both a novel theoretical framework and practical algorithmic solutions for achieving best-iterate convergence in two-player zero-sum imperfect-information EFGs. These advancements are expected to significantly enhance the applicability of DRL methodologies in complex, strategic multi-agent environments.