Single Deep Counterfactual Regret Minimization (1901.07621v4)

Abstract: Counterfactual Regret Minimization (CFR) is the most successful algorithm for finding approximate Nash equilibria in imperfect information games. However, CFR's reliance on full game-tree traversals limits its scalability. For this reason, the game's state- and action-space is often abstracted (i.e. simplified) for CFR, and the resulting strategy is then translated back to the full game, which requires extensive expert-knowledge and often converges to highly exploitable policies. A recently proposed method, Deep CFR, applies deep learning directly to CFR, allowing the agent to intrinsically abstract and generalize over the state-space from samples, without requiring expert knowledge. In this paper, we introduce Single Deep CFR (SD-CFR), a simplified variant of Deep CFR that has a lower overall approximation error by avoiding the training of an average strategy network. We show that SD-CFR is more attractive from a theoretical perspective and empirically outperforms Deep CFR with respect to exploitability and one-on-one play in poker.

• The paper introduces Single Deep CFR which eliminates an extra average strategy network, reducing both sampling and approximation errors in CFR.
• It applies trajectory-sampling to compute average strategies directly from stored value networks, maintaining theoretical accuracy.
• Empirical results in poker games demonstrate that Single Deep CFR achieves lower exploitability and enhanced performance over Deep CFR.

Single Deep Counterfactual Regret Minimization

In the domain of imperfect information games, Counterfactual Regret Minimization (CFR) stands as the predominant algorithm for identifying approximate Nash equilibria. Its iterative traversal of game trees has not only achieved notable successes, such as solving two-player Limit Texas Hold'em Poker, but also highlighted a significant limitation: scalability. Traditional CFR methods necessitate full game-tree traversals, rendering them impractical for large state spaces. Deep CFR introduced an innovative approach by incorporating deep learning to facilitate state-space generalization from samples, but this comes with its limitations, such as the need for training an extra average strategy network, which introduces additional approximation errors.

Introduction to Single Deep CFR

The paper at hand proposes Single Deep CFR (SD-CFR), a variation of Deep CFR designed to lower approximation errors by eliminating the necessity for an additional average strategy network training phase. SD-CFR aims to achieve a theoretically cleaner and empirically more robust performance by leveraging deep learning solely for value networks and eliminating redundant approximation steps.

Methodology

SD-CFR is distinct from Deep CFR in that it computes the average strategy directly from value networks stored from previous iterations. This approach addresses two primary errors in Deep CFR: sampling error and approximation error from the additional network. By maintaining a comprehensive buffer of value networks, SD-CFR can dynamically compute the average strategy on demand, ensuring alignment with theoretical principles of CFR.

Trajectory-Sampling

One specific method introduced in SD-CFR is trajectory-sampling. For freely playable trajectories, SD-CFR selects a value network at the trajectory's start and uses the corresponding policy throughout, ensuring that sampling aligns with the desired average strategy distribution. This mechanism maintains computational efficiency and correctness.

Theoretical Properties

Theoretical analysis demonstrates that SD-CFR precisely mimics the average strategy produced by traditional linear CFR, contingent on perfect function approximation of value networks. This theoretical validation posits SD-CFR as a reliable and sound variant of CFR, promising reduced practical exploitability.

Empirical Validation

Empirical results substantiate the theoretical claims, showcasing the superior performance of SD-CFR over Deep CFR in multiple poker variations, such as Leduc Poker and a modified version of Five-Flop Hold'em Poker (5-FHP). Key findings include:

• Leduc Poker: SD-CFR consistently demonstrates lower exploitability compared to Deep CFR, particularly in long training iterations, where Deep CFR's approximation errors become more pronounced.
• 5-FHP One-on-One Matches: Extensive matches between SD-CFR and Deep CFR reveal a significant advantage held by SD-CFR, particularly when both algorithms approach equilibrium strategies.

Practical Implications and Future Directions

The advancements cited in the paper imply several practical and theoretical benefits:

1. Scalability: SD-CFR's methodology can be extended to larger games and other domains of imperfect information games beyond poker.
2. Reduced Resource Requirements: By eliminating the need for average strategy networks, SD-CFR reduces computational and memory overhead, enhancing its applicability in resource-constrained environments.
3. Focused Sampling Techniques: Future research could explore more sophisticated sampling techniques, such as continuous approximations for large action spaces, enhancing generalization, and efficiency further.

Conclusion

Single Deep CFR represents a significant stride in applying deep learning to Counterfactual Regret Minimization by streamlining the approximation process and aligning theoretical principles with empirically validated performance improvements. This variant not only showcases more robust performance against exploitability but also provides a scalable, resource-efficient pathway for future advancements in imperfect information game strategies.

Acknowledgments and Code Availability

The author thanks collaborators for their contributions and provides access to the implementation and scripts used in the paper through a public repository, facilitating further research and replication of results.

References

The paper references foundational works and recent advancements within CFR and deep reinforcement learning, highlighting contributions from influential researchers and significant milestones in the field. These references provide context and validation for the proposed method's efficacy and relevance.