Landscape of Policy Optimization for Finite Horizon MDPs with General State and Action (2409.17138v1)

Abstract: Policy gradient methods are widely used in reinforcement learning. Yet, the nonconvexity of policy optimization imposes significant challenges in understanding the global convergence of policy gradient methods. For a class of finite-horizon Markov Decision Processes (MDPs) with general state and action spaces, we develop a framework that provides a set of easily verifiable assumptions to ensure the Kurdyka-Lojasiewicz (KL) condition of the policy optimization. Leveraging the KL condition, policy gradient methods converge to the globally optimal policy with a non-asymptomatic rate despite nonconvexity. Our results find applications in various control and operations models, including entropy-regularized tabular MDPs, Linear Quadratic Regulator (LQR) problems, stochastic inventory models, and stochastic cash balance problems, for which we show an $\epsilon$-optimal policy can be obtained using a sample size in $\tilde{\mathcal{O}}(\epsilon^{-1})$ and polynomial in terms of the planning horizon by stochastic policy gradient methods. Our result establishes the first sample complexity for multi-period inventory systems with Markov-modulated demands and stochastic cash balance problems in the literature.

Overview of "Landscape of Policy Optimization for Finite Horizon MDPs with General State and Action" by Chen, Hu, and Zhao

The paper "Landscape of Policy Optimization for Finite Horizon MDPs with General State and Action" by Xin Chen, Yifan Hu, and Minda Zhao addresses the nonconvexity challenges in policy gradient methods for finite horizon Markov Decision Processes (MDPs). The authors present a framework with easily verifiable assumptions that guarantee the Kurdyka-Łojasiewicz (KŁ) condition for policy optimization, ensuring global convergence of policy gradient methods and providing non-asymptotic convergence rates.

Key Contributions

1. Kurdyka-Łojasiewicz (KŁ) Condition Framework: The authors propose a general framework to establish the KŁ condition for policy gradient optimization in finite-horizon MDPs. The KŁ condition ensures that any point satisfying the first-order necessary optimality condition will be globally optimal.
2. Non-Asymptotic Convergence Rates: Utilizing the KŁ condition, the paper demonstrates that policy gradient methods achieve global convergence with a linear rate for exact methods and an $\tilde{\mathcal{O}}(\epsilon^{-1})$ sample complexity for stochastic methods.
3. Application to Various Models: The framework is applied to several control and operations models, including:
   • Entropy-regularized tabular MDPs
   • Linear Quadratic Regulator (LQR) problems
   • Multi-period inventory systems with Markov-modulated demands
   • Stochastic cash balance problems

Numerical Results

The paper provides the first sample complexity results for multi-period inventory systems with Markov-modulated demands and stochastic cash balance problems. Specifically, an $\epsilon$-optimal policy can be obtained using a sample size polynomial in terms of the planning horizon by stochastic policy gradient methods. This is significant as it offers a polynomial dependence on the time horizon, contrasting with prior results that had an exponential dependence for similar problems.

Theoretical Implications

The paper intersects with several areas of research:

1. Nonconvex Landscape Conditions: It contributes to the literature on nonconvex optimization by providing conditions under which the policy gradient objective function satisfies the KŁ condition.
2. Global Optimality of Policy Gradient Methods: By guaranteeing the KŁ condition, the results offer theoretical support for the global convergence of policy gradient methods in nonconvex settings, extending the understanding of convergence beyond asymptotic guarantees.
3. Data-Driven Operations Management: The findings have practical implications for real-world applications such as inventory management and cash balance decisions, where the framework can be directly applied to design efficient algorithms with guaranteed convergence.

Practical Implications and Future Developments

The practical implications of this research are substantial for fields employing finite-horizon MDPs. By ensuring that policy gradient methods converge globally and efficiently, the framework can be utilized in various industries for more reliable and effective decision-making.

Future Directions:

• Exploring Specific Parameter Dependencies: Further research may refine the polynomial dependence on the planning horizon, potentially improving the computational efficiency of the proposed methods.
• Generalizing to Broader Cost Classes: Extending the framework to include general convex per-period costs could broaden its applicability and utility.
• Regularization Techniques: Investigating the impact of regularization on improving convergence rates without significantly increasing complexity.

In conclusion, this paper makes significant strides in elucidating the landscape of policy optimization for finite-horizon MDPs. The authors' contributions provide robust theoretical underpinnings for policy gradient methods, ensuring their global convergence and offering new avenues for practical application and future research in reinforcement learning and operations management.