[Paper Review] Optimality and Approximation with Policy Gradient Methods in Markov Decision Processes

Alekh Agarwal, Sham M. Kakade+2 more|arXiv (Cornell University)|Aug 1, 2019

Reinforcement Learning in Robotics33 citations

TL;DR

This paper establishes theoretical foundations for policy gradient methods in discounted Markov Decision Processes, proving global convergence to the optimal policy under tabular parameterizations and providing agnostic learning guarantees under restricted policy classes. It formalizes the role of favorable initial state distributions in overcoming exploration challenges, offering convergence rates and approximation error bounds that place policy gradients on par with value-based methods in theory.

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 (say with a sufficiently rich policy class); how they cope with approximation error due to using a restricted class of parametric policies; or their finite sample behavior. Such characterizations are important not only to compare these methods to their approximate value function counterparts (where such issues are relatively well understood, at least in the worst case), but also to help with more principled approaches to algorithm design. This work provides provable characterizations of computational, approximation, and sample size issues with regards to policy gradient methods in the context of discounted Markov Decision Processes (MDPs). We focus on both: 1) tabular policy parameterizations, where the optimal policy is contained in the class and where we show global convergence to the optimal policy, and 2) restricted policy classes, which may not contain the optimal policy and where we provide agnostic learning results. One insight of this work is in formalizing the importance how a favorable initial state distribution provides a means to circumvent worst-case exploration issues. Overall, these results place policy gradient methods under a solid theoretical footing, analogous to the global convergence guarantees of iterative value function based algorithms.

Motivation & Objective

• To establish provable convergence properties of policy gradient methods in discounted Markov Decision Processes (MDPs), particularly in terms of computational, approximation, and sample size behavior.
• To analyze how policy gradient methods perform when the optimal policy is not contained in the parametric policy class, providing agnostic learning guarantees.
• To investigate the impact of initial state distribution on exploration efficiency and convergence, formalizing its role in circumventing worst-case exploration issues.
• To compare policy gradient methods to value-based methods by providing theoretical guarantees analogous to those of iterative value function algorithms.
• To bridge the theoretical gap in understanding policy gradient methods, especially regarding convergence speed and approximation error in practical settings.

Proposed method

• The authors analyze policy gradient methods in the context of discounted MDPs using both tabular policy parameterizations and restricted parametric policy classes.
• For tabular policies, they prove global convergence to the optimal policy using gradient ascent on the expected cumulative reward, leveraging smoothness and strong concavity properties.
• For restricted policy classes, they derive agnostic learning bounds that quantify the approximation error relative to the best policy in the class.
• They introduce a formal analysis of how the initial state distribution influences convergence, showing that favorable distributions can eliminate worst-case exploration bottlenecks.
• Theoretical results are derived using tools from stochastic approximation, Markov chain theory, and optimization, including bounds on gradient noise and convergence rates.
• Key components include the use of the policy gradient theorem and analysis of the Hessian of the performance objective to establish local and global convergence behavior.

Experimental results

Research questions

• RQ1 Under what conditions do policy gradient methods converge globally to the optimal policy in tabular MDPs?
• RQ2 How do policy gradient methods behave when the optimal policy lies outside the parametric policy class, and what performance guarantees can be provided?
• RQ3 What is the impact of the initial state distribution on the convergence and exploration efficiency of policy gradient methods?
• RQ4 How do approximation errors in the policy class affect the performance of policy gradient methods, and can these be bounded?
• RQ5 What are the finite-sample and computational convergence rates of policy gradient methods in the presence of function approximation?

Key findings

• Policy gradient methods achieve global convergence to the optimal policy when using tabular parameterizations in discounted MDPs, under standard regularity conditions.
• For restricted policy classes that do not contain the optimal policy, the method provides agnostic learning guarantees, bounding the suboptimality gap in terms of approximation error.
• A favorable initial state distribution significantly improves convergence by mitigating worst-case exploration issues, effectively reducing the need for extensive exploration.
• The paper establishes finite-sample convergence rates for policy gradient methods, showing that convergence speed depends on the curvature of the performance landscape and the quality of the policy initialization.
• Approximation error due to restricted policy classes is formally quantified, with bounds that depend on the distance between the best policy in the class and the true optimal policy.
• The theoretical framework provides a foundation for policy gradient methods that is comparable in rigor to the convergence guarantees of value-based iterative algorithms.