Neural Policy Gradient Methods: Global Optimality and Rates of Convergence (1909.01150v3)

Abstract: Policy gradient methods with actor-critic schemes demonstrate tremendous empirical successes, especially when the actors and critics are parameterized by neural networks. However, it remains less clear whether such "neural" policy gradient methods converge to globally optimal policies and whether they even converge at all. We answer both the questions affirmatively in the overparameterized regime. In detail, we prove that neural natural policy gradient converges to a globally optimal policy at a sublinear rate. Also, we show that neural vanilla policy gradient converges sublinearly to a stationary point. Meanwhile, by relating the suboptimality of the stationary points to the representation power of neural actor and critic classes, we prove the global optimality of all stationary points under mild regularity conditions. Particularly, we show that a key to the global optimality and convergence is the "compatibility" between the actor and critic, which is ensured by sharing neural architectures and random initializations across the actor and critic. To the best of our knowledge, our analysis establishes the first global optimality and convergence guarantees for neural policy gradient methods.

Global Optimality and Convergence of Neural Policy Gradient Methods

The paper "Neural Policy Gradient Methods: Global Optimality and Rates of Convergence" explores the theoretical underpinnings of policy gradient methods in the context of reinforcement learning, specifically addressing the global optimality and convergence concerns when using neural network parameterizations. Despite the empirical success of neural policy gradient methods, a comprehensive theoretical framework ensuring their global convergence remains elusive. This paper aims to address this gap by focusing on overparameterized two-layer neural networks.

Overview

The authors tackle two primary questions: whether neural policy gradient methods converge to globally optimal policies and whether they can guarantee convergence at all. The analysis centers around two variants of policy gradient methods—vanilla policy gradient and natural policy gradient—both of which employ actor-critic schemes for iterative updates. The exploration is undertaken under the assumption of an overparameterized regime, where the width of the neural networks exceeds certain thresholds to ensure sufficient approximation power.

Key Contributions

1. Convergence to Stationary Points: The paper establishes that neural vanilla policy gradient methods converge to a stationary point of the expected total reward with a sublinear convergence rate, specifically at a rate of $1/\sqrt{T}$, where $T$ denotes the number of policy improvement steps.
2. Global Optimality and Compatibility: It is demonstrated that under mild regularity conditions, all stationary points achieved by these methods are globally optimal. A critical condition for this global optimality is the compatibility between the actor and critic, which is fostered through shared neural architectures and random initializations.
3. Natural Policy Gradient: The paper provides evidence that neural natural policy gradient achieves convergence to a globally optimal policy at a sublinear rate similar to the vanilla method, but it is proven to attain global optimality directly without the additional regularity conditions required for the vanilla gradient method.

Theoretical Insights and Assumptions

To support these findings, various technical assumptions and conditions are imposed:

• Action-Value Function Class: The analysis assumes that action-value functions belong to a broad class of functions well-represented by overparameterized neural networks, enabling effective approximation.
• Regularity Conditions: Certain regularity conditions on state visitation measures and the smoothness of the expected total reward are posited to ensure the tractability and convergence of the methods analyzed.
• Error Bounds: The authors consider both approximation and algorithmic errors within neural Temporal-Difference learning, providing bounds for these errors under the large width assumption for neural networks.

Implications and Future Work

The results signify a theoretical breakthrough in understanding the convergence properties of neural policy gradient methods, suggesting that overparameterization in neural networks can furnish strong performance guarantees even in non-linear settings. These insights could impact the deployment of policy gradient methods in high-stakes applications, such as robotics and autonomous systems, where reliability and optimality are paramount.

Future work may explore diminishing the dependency on overparameterization by enhancing neural architectures or investigating alternative optimization landscapes. Extensions to more complex and larger-scale environments could further validate the robustness of these findings.

Overall, this paper not only reconciles theoretical analysis with empirical success but also provides a blueprint for future explorations into the intersections of deep learning and reinforcement learning.