Fast Global Convergence of Natural Policy Gradient Methods with Entropy Regularization (2007.06558v5)

Abstract: Natural policy gradient (NPG) methods are among the most widely used policy optimization algorithms in contemporary reinforcement learning. This class of methods is often applied in conjunction with entropy regularization -- an algorithmic scheme that encourages exploration -- and is closely related to soft policy iteration and trust region policy optimization. Despite the empirical success, the theoretical underpinnings for NPG methods remain limited even for the tabular setting. This paper develops $\textit{non-asymptotic}$ convergence guarantees for entropy-regularized NPG methods under softmax parameterization, focusing on discounted Markov decision processes (MDPs). Assuming access to exact policy evaluation, we demonstrate that the algorithm converges linearly -- or even quadratically once it enters a local region around the optimal policy -- when computing optimal value functions of the regularized MDP. Moreover, the algorithm is provably stable vis-`a-vis inexactness of policy evaluation. Our convergence results accommodate a wide range of learning rates, and shed light upon the role of entropy regularization in enabling fast convergence.

Fast Global Convergence of Natural Policy Gradient Methods with Entropy Regularization

The paper under discussion offers a rigorous theoretical framework for natural policy gradient (NPG) methods within the context of reinforcement learning (RL), with a focus on entropy regularization. This approach is examined within discounted Markov Decision Processes (MDPs) using softmax parameterization. The paper targets the development of non-asymptotic convergence guarantees and aims to provide a comprehensive understanding of the stability and efficiency associated with these methods.

Methodology and Key Results

Natural policy gradient methods serve as an extension of policy gradient techniques, leveraging the Fisher information matrix to precondition gradient computations for enhanced convergence rates. The inclusion of entropy regularization in NPG methods aims to mitigate the challenges posed by non-concave optimization landscapes, facilitating exploration and preventing convergence to suboptimal policies by smoothing the optimization landscape.

The authors present the following significant findings:

1. Linear Convergence: Entropy-regularized NPG methods demonstrate linear convergence for deriving optimal policies of regularized MDPs, even with approximate policy evaluation. Linear convergence is confirmed for both soft Q-functions and the corresponding log policies.
2. Impact of Learning Rates and Regularization: The analysis reveals that the convergence rate is near-dimension free and robust across a range of learning rates. Entropy regularization generally accelerates convergence, substantiating its role in improving the efficiency of NPG methods.
3. Quadratic Convergence: In scenarios requiring high accuracy, the method shows quadratic convergence once approaching a local region near the optimal policy, provided the problem is small-$\epsilon$.
4. Stability under Approximation: The paper extends the theoretical guarantees to scenarios involving inexact policy evaluation, demonstrating that NPG methods maintain a stable convergence pattern in such cases.
5. Computational Implications: The investigation concludes that a higher degree of regularization parameter $\tau$ promotes faster convergence, irrespective of its role in approximating the original problem solution. For the special case of soft policy iteration (SPI), convergence rates align with those of standard policy iteration methods.

Implications and Future Research

The insights into NPG methods with entropy regularization have several practical and theoretical implications. Practically, the findings suggest that RL algorithms implementing NPG could benefit from including entropy regularization as a tool for enhancing convergence efficiency. Theoretically, the paper advances our understanding of the role of entropy in regularized policy optimization, offering a framework that could be adapted to more complex environments and constraints, including function approximation scenarios and partially observable MDPs.

Future research directions may include exploring the implications of these theoretical advancements in sample-based RL settings, examining their efficiency in scenarios beyond tabular MDPs where function approximation is required. Moreover, the investigation of other parameterization schemes and the potential optimization of sample complexity remain open areas of inquiry.

By formalizing the convergence properties of entropy-regularized natural policy gradient methods, this paper not only offers a significant contribution to theoretical RL research but also lays a solid foundation for the development of more robust and efficient reinforcement learning algorithms.