Global Optimality Guarantees For Policy Gradient Methods

Abstract: Policy gradients methods apply to complex, poorly understood, control problems by performing stochastic gradient descent over a parameterized class of polices. Unfortunately, even for simple control problems solvable by standard dynamic programming techniques, policy gradient algorithms face non-convex optimization problems and are widely understood to converge only to a stationary point. This work identifies structural properties -- shared by several classic control problems -- that ensure the policy gradient objective function has no suboptimal stationary points despite being non-convex. When these conditions are strengthened, this objective satisfies a Polyak-lojasiewicz (gradient dominance) condition that yields convergence rates. We also provide bounds on the optimality gap of any stationary point when some of these conditions are relaxed.

• The paper establishes conditions under which policy gradient methods have no suboptimal stationary points, ensuring global optimality in reinforcement learning.
• The paper validates its theoretical contributions by applying the methodology to varied control problems such as LQ control and inventory management.
• The paper incorporates a gradient dominance condition that guarantees efficient convergence, even with approximate closure under policy improvement.

Overview of Global Optimality Guarantees for Policy Gradient Methods

The paper, entitled "Global Optimality Guarantees For Policy Gradient Methods," explores a significant concern within the field of reinforcement learning: the convergence issues associated with policy gradient methods when facing non-convex optimization problems. Specifically, this research explores the conditions under which policy gradient methods can achieve global optimality despite these inherent challenges.

Key Contributions

1. Structural Insights: The study identifies conditions under which the policy gradient objective, despite its non-convex nature, has no suboptimal stationary points. These conditions are tied to the structural properties prevalent in several canonical control problems:
   • Closure under Policy Improvement: The parameterized policy class should be closed under policy improvement, implying that policy iteration steps within the class lead to another policy within the same class.
   • Lack of Suboptimal Stationary Points in the Single-Period Objective: The single-period weighted policy iteration objective should not have any suboptimal stationary points.
2. Broad Applicability: These insights are applicable to a variety of control problems including:
   • Finite state and action Markov Decision Processes (MDPs) with arbitrary stochastic policies.
   • Linear Quadratic (LQ) control problems utilizing linear policies.
   • Optimal stopping problems employing threshold policies.
   • Inventory control problems adopting non-stationary base-stock policies.
3. Gradient Dominance Condition: The paper further studies extended assumptions wherein the weighted policy iteration objective satisfies a Polyak-Lojasiewicz condition or gradient dominance condition which assures convergence rates. This gradient dominance suggests that not only do stationary points lead to global optimality, but they also do so efficiently.
4. Approximate Guarantees: When conditions for exact closure under policy improvement are relaxed, the paper provides bounds on the optimality gaps for any stationary point, thus offering assurance even in situations where the policy class is approximately closed.

Key Examples and Instantiations

• Linear Quadratic Control: By demonstrating that the set of linear policies forms a closed set under policy improvement and that policy gradient methods converge to the global optimum, the paper validates its theoretical contributions using a well-established control problem framework.
• Finite Horizon Inventory Control: Leveraging non-stationary base-stock policies within a finite horizon setting, the research outlines how even with structured non-convexity in policy iteration, policy gradient methods can yield optimal performance.

Practical and Theoretical Implications

• Broadening the Horizon for Policy Gradient Methods: The findings expand the potential application of policy gradient methods to a wider array of problems by showing conditions under which these methods are not restricted to merely finding local optima.
• Influencing Future Research: This work sets a foundation for exploring policy gradient methods within policy classes that exhibit certain structural properties, directing future research towards refining these classes and discovering new applications.

Conclusion

This paper effectively tackles a fundamental issue in reinforcement learning by outlining clear structural conditions under which policy gradient methods can be trusted to converge globally. It bridges a gap by providing a theoretical basis for what has traditionally been observed empirically in complex control problems. These contributions not only instill greater confidence in using policy gradients in reinforcement learning tasks but also pave the way for refining algorithms and expanding their applicability. Future research can build on these results, exploring further applications and potentially discovering new classes of problems amenable to policy gradient optimization.