A Theory of Regularized Markov Decision Processes (1901.11275v2)

Abstract: Many recent successful (deep) reinforcement learning algorithms make use of regularization, generally based on entropy or Kullback-Leibler divergence. We propose a general theory of regularized Markov Decision Processes that generalizes these approaches in two directions: we consider a larger class of regularizers, and we consider the general modified policy iteration approach, encompassing both policy iteration and value iteration. The core building blocks of this theory are a notion of regularized BeLLMan operator and the Legendre-Fenchel transform, a classical tool of convex optimization. This approach allows for error propagation analyses of general algorithmic schemes of which (possibly variants of) classical algorithms such as Trust Region Policy Optimization, Soft Q-learning, Stochastic Actor Critic or Dynamic Policy Programming are special cases. This also draws connections to proximal convex optimization, especially to Mirror Descent.

• The paper presents a novel framework that extends regularized MDPs by incorporating a broad range of regularizers and a generalized policy iteration method.
• It introduces the regularized Bellman operator via the Legendre-Fenchel transformation to ensure unique optimality and contraction properties.
• The study connects reinforcement learning methods to convex optimization techniques, highlighting implications for error propagation, convergence, and algorithmic stability.

A Theory of Regularized Markov Decision Processes

The paper, "A Theory of Regularized Markov Decision Processes" by Matthieu Geist, Bruno Scherrer, and Olivier Pietquin, presents a comprehensive theoretical framework that extends the Regularized Markov Decision Processes (MDPs). By employing a wider class of regularizers and adopting a generalized modified policy iteration approach, the authors offer numerous advancements over existing heuristic regularization methods such as entropy and Kullback-Leibler (KL) divergence.

Summary of Contributions

The paper extends the framework of regularization in MDPs by integrating a broader set of regularizers beyond conventional entropy-based methods. Key elements of this formalism include the regularized BeLLMan operator, which encapsulates the BeLLMan evaluation operator within a Legendre-Fenchel transformation, a cornerstone in convex optimization. This theoretical apparatus allows for an analysis framework applicable to various reinforcement learning algorithms, including Trust Region Policy Optimization (TRPO), Soft Q-learning, and others, laying the groundwork for more systematic investigations into convergence and error propagation.

Theoretical Analyses

1. Regularized BeLLMan Operators: Defined within the context of the Legendre-Fenchel transformation, these operators provide the basis for regularized value functions and optimality guarantees. The contracting nature of these operators ensures the unique fixed-point corresponding to the optimal regularized value function.
2. Error Propagation and Convergence: Utilizing the regularized BeLLMan operators, the paper develops a rigorous error propagation analysis in regularized MDPs, establishing conditions for convergence that mirror traditional MDP methods under a regularization perspective. The paper demonstrates that robust theoretical frameworks can ensure performance consistency across varied reinforcement learning methodologies.
3. Connections to Optimization: The paper astutely links dynamics of regularized policy iteration to proximal methods in convex optimization, including Mirror Descent, thus positioning regularization as an optimization strategy rather than just a heuristic tactic.

Practical Implications and Future Directions

• Refined Reinforcement Learning Algorithms: By accommodating a variety of regularizers, this framework permits refined adjustments to trade-off exploration versus exploitation, mitigate overfitting, and induce algorithmic stability across different environments—factors crucial to deploying reinforcement learning in complex domains.
• Beyond Theoretical Regularization: Though primarily theoretical, implications extend to potential empirical exploration of how different regularizers impact policy convergence, robustness, or even generalization capabilities across various benchmarks or real-world applications.
• Broader Applications: Further studies could explore extending this regularized framework to domains such as zero-sum games, inverse reinforcement learning, and other areas where policy uniqueness and optimality play pivotal roles.

Conclusion

Overall, the paper underscores the efficacy of a generalized regularization theory in MDPs, furnishing rigorous mathematical validation and potentially revolutionizing the design of reinforcement learning algorithms. By bridging conventional reinforcement learning and convex optimization principles, this research not only dispels the heuristic nature of regularization but also aligns with systematic, theoretically grounded optimization techniques. Future advancements could expand upon these foundations, illuminating new paths in both applied and theoretical research in artificial intelligence.