Global Convergence of Policy Gradient Methods to (Almost) Locally Optimal Policies (1906.08383v3)

Abstract: Policy gradient (PG) methods are a widely used reinforcement learning methodology in many applications such as video games, autonomous driving, and robotics. In spite of its empirical success, a rigorous understanding of the global convergence of PG methods is lacking in the literature. In this work, we close the gap by viewing PG methods from a nonconvex optimization perspective. In particular, we propose a new variant of PG methods for infinite-horizon problems that uses a random rollout horizon for the Monte-Carlo estimation of the policy gradient. This method then yields an unbiased estimate of the policy gradient with bounded variance, which enables the tools from nonconvex optimization to be applied to establish global convergence. Employing this perspective, we first recover the convergence results with rates to the stationary-point policies in the literature. More interestingly, motivated by advances in nonconvex optimization, we modify the proposed PG method by introducing periodically enlarged stepsizes. The modified algorithm is shown to escape saddle points under mild assumptions on the reward and the policy parameterization. Under a further strict saddle points assumption, this result establishes convergence to essentially locally-optimal policies of the underlying problem, and thus bridges the gap in existing literature on the convergence of PG methods. Results from experiments on the inverted pendulum are then provided to corroborate our theory, namely, by slightly reshaping the reward function to satisfy our assumption, unfavorable saddle points can be avoided and better limit points can be attained. Intriguingly, this empirical finding justifies the benefit of reward-reshaping from a nonconvex optimization perspective.

Analysis of Global Convergence in Policy Gradient Methods

The paper "Global Convergence of Policy Gradient Methods to (Almost) Locally Optimal Policies" by Kaiqing Zhang, Alec Koppel, Hao Zhu, and Tamer Başar, deals with addressing a prominent gap in the field of reinforcement learning (RL) in relation to the theoretical underpinnings of policy gradient (PG) methods. These methods are a subset of RL algorithms widely applied in complex continuous space problems but have historically lacked rigorous proofs of global convergence in nonconvex optimization landscapes often encountered in real-world applications.

Problem Statement and Methodology

The core focus of the paper is to substantiate the global convergence of PG methods by re-framing them within the context of nonconvex optimization. The authors propose a modified variant for infinite-horizon tasks which employs a stochastic rollout methodology. This alteration permits an unbiased estimation of the PG with bounded variance. The theoretical angle pivots on constructing convergence results adhering to stationary-point criteria, progressing further by using enlarged stepsizes as an escape mechanism from saddle points under specific assumptions on reward functions and policy parameterizations.

The inclusion of an unbiased policy gradient estimate was realized through incorporating random rollout horizons in Monte Carlo simulations to ensure unbiased expectations of the Q-value, which previously introduced bias when approximated conventionally in PG methods. This approach aligns the PG estimates more naturally with the framework of classical stochastic programming algorithms.

Main Contributions and Results

1. Unbiased PG Estimation: The paper introduces the concept of random-horizon PG methods, which use stochastic rollouts. This method produces a PG estimate that is theoretically unbiased over infinite horizons, significantly enhancing reliability and robustness of the estimated gradients.
2. Convergence Analysis: They rigorously establish the convergence to stationary points with formal proofs that underscore the algorithm's reliability over time with non-diminishing stepsizes. They manage to draw parallels with nonconvex optimization theories suggesting guaranteed convergence to stationary-point policies.
3. Escape From Saddle Points: A key innovation is in mitigating the limitation of stagnation at saddle points using periodically increased stepsizes. This technique ensures that the PG methods can, with high probability, avoid saddle points and converge towards policies that exhibit local optimality.
4. Practical Implications through Reward Reshaping: Empirical validation illustrates how altering the reward structure to enforce positive-definiteness accelerates the escape from undesirable equilibria, substantiating the claims with simulation outcomes from a robotics testbed on an inverted pendulum task.

Theoretical and Practical Implications

The research bridges an essential gap by tying together contemporary non-convex optimization methods with PG in dynamic programming, thereby broadening the application scope of RL algorithms in environments that are analytically or computationally expensive to model otherwise. The escape mechanism from saddle points presents a robust pathway for broadening the applicability of PG methods in fields where reaching the absolute or local optimal swiftly is mission-critical, such as autonomous navigation and complex robotic control systems.

Future Directions

Building upon this work, there lies an opportunity to explore more sophisticated schemes involving adaptive learning rates and dynamic policy parameters that adjust based on the gradient’s behavior and the structure of the state space. Future research may expand upon these foundations by incorporating multi-agent environments and considering asynchronous updates, which are prevalent in distributed AI systems, thus propelling the extensive utilization of PG methods in intricate decision-making applications.