On the Theory of Policy Gradient Methods: Optimality, Approximation, and Distribution Shift (1908.00261v5)

Abstract: Policy gradient methods are among the most effective methods in challenging reinforcement learning problems with large state and/or action spaces. However, little is known about even their most basic theoretical convergence properties, including: if and how fast they converge to a globally optimal solution or how they cope with approximation error due to using a restricted class of parametric policies. This work provides provable characterizations of the computational, approximation, and sample size properties of policy gradient methods in the context of discounted Markov Decision Processes (MDPs). We focus on both: "tabular" policy parameterizations, where the optimal policy is contained in the class and where we show global convergence to the optimal policy; and parametric policy classes (considering both log-linear and neural policy classes), which may not contain the optimal policy and where we provide agnostic learning results. One central contribution of this work is in providing approximation guarantees that are average case -- which avoid explicit worst-case dependencies on the size of state space -- by making a formal connection to supervised learning under distribution shift. This characterization shows an important interplay between estimation error, approximation error, and exploration (as characterized through a precisely defined condition number).

On the Theory of Policy Gradient Methods: Optimality, Approximation, and Distribution Shift

The paper "On the Theory of Policy Gradient Methods: Optimality, Approximation, and Distribution Shift" by Agarwal et al. provides a rigorous theoretical investigation into the underlying principles and behaviors of policy gradient methods within the field of reinforcement learning (RL). The paper centers on identifying and addressing various challenges such as convergence properties, dealing with approximation errors, and handling distribution shifts. The work primarily examines these aspects within the framework of discounted Markov Decision Processes (MDPs).

Main Contributions

1. Convergence in Tabular Settings: The authors delve into policy gradient methods under a tabular parameterization, where each parameter corresponds directly to a state-action pair, ensuring the policy class includes the optimal policy. They present global convergence guarantees using both conventional gradient ascent and natural policy gradient methods. Table 1 in the paper provides iteration complexity bounds, indicating the number of iterations necessary to achieve an ε-optimal policy, assuming access to exact gradients. Notably, the natural policy gradient method exhibits no dependence on state or action space size, highlighting its potential efficiency in this regard.
2. Handling Distribution Shift: One significant achievement of the paper lies in offering a deeper understanding of how policy gradient methods interact with distribution shifts. Particularly, they propose a distribution mismatch coefficient that captures the difficulty of exploration in optimization problems and show its influence on convergence rates. The paper outlines how various factors, including estimation error, approximation error, and exploration, interplay through an explicitly defined condition number.
3. Function Approximation Approaches: The authors further extend their analysis to policy gradient methods using generalized function approximation techniques such as log-linear and neural policies. They introduce a novel agnostic learning framework where the objective is to perform as well as the best policy within a predefined function class. This approach allows for performance bounds that are milder in their dependence on density ratios, which are often a limiting factor in practical scenarios.
4. Log Barrier Regularization: To combat issues of vanishing gradients in the softmax policy parameterization, the paper incorporates a log barrier regularization technique. This method is shown to preserve exploration potential by preventing probability weights from collapsing prematurely to certain actions, thereby allowing the optimization process to retain its efficacy throughout.

Implications and Future Directions

• Practical Implementations: The insights gained regarding convergence in both tabular and function approximation settings have direct implications on the design of more robust and sample-efficient RL algorithms. This is particularly relevant with the emphasis on robust policy optimization techniques such as those based on natural policies.
• Theoretical Insights: The proposed decomposition into estimation and transfer errors enriches the understanding of how policy gradient methods manage distributional changes, thereby providing theoretical insights that can guide the creation of new policy update rules that are less sensitive to such shifts.
• Improving Sample Efficiency: The results related to natural policy gradients are promising in indicating pathways towards achieving dimension-free convergence rates. This raises important questions about how established methods can be further developed to leverage function approximation without suffering from sample inefficiencies.

The paper charts a comprehensive course through the theoretical underpinnings of policy gradient methods, shedding light on aspects previously not understood sufficiently. While it presents formidable progress in the field, there remains substantial room for research into further mitigating the effects of distribution shifts and ensuring robust policy optimization in increasingly complex and high-dimensional spaces.