On the Global Convergence Rates of Softmax Policy Gradient Methods (2005.06392v3)

Abstract: We make three contributions toward better understanding policy gradient methods in the tabular setting. First, we show that with the true gradient, policy gradient with a softmax parametrization converges at a $O(1/t)$ rate, with constants depending on the problem and initialization. This result significantly expands the recent asymptotic convergence results. The analysis relies on two findings: that the softmax policy gradient satisfies a \L{}ojasiewicz inequality, and the minimum probability of an optimal action during optimization can be bounded in terms of its initial value. Second, we analyze entropy regularized policy gradient and show that it enjoys a significantly faster linear convergence rate $O(e^{-c \cdot t})$ toward softmax optimal policy $(c > 0)$. This result resolves an open question in the recent literature. Finally, combining the above two results and additional new $\Omega(1/t)$ lower bound results, we explain how entropy regularization improves policy optimization, even with the true gradient, from the perspective of convergence rate. The separation of rates is further explained using the notion of non-uniform \L{}ojasiewicz degree. These results provide a theoretical understanding of the impact of entropy and corroborate existing empirical studies.

On the Global Convergence Rates of Softmax Policy Gradient Methods

The paper in question presents a comprehensive paper on the global convergence properties of softmax policy gradient methods within the field of policy optimization for reinforcement learning (RL). The softmax policy gradient approach is known for its utility in finding optimal policies by incrementally updating policy parameters using gradient estimates. This paper meticulously investigates and quantifies the convergence rates of these gradient methods, offering a theoretical underpinning that aligns with and extends existing empirical findings in the field.

Contributions and Findings

The paper makes three significant contributions:

1. Convergence Rate of Vanilla Softmax Policy Gradient: The paper establishes that with the true gradient, the softmax policy gradient converges globally at an $O(1/t)$ rate, where $t$ is the number of iterations. This analysis is predicated on the utilization of a \L{}ojasiewicz inequality, which asserts a gradient dominance condition over the expected rewards, thus ensuring that the gradient method avoids getting stuck prematurely. This finding significantly extends previous asymptotic convergence results by quantifying the rate, which had hitherto remained unspecified.
2. Entropy Regularized Policy Gradient: By incorporating entropy regularization—a method that encourages exploration by discouraging deterministic policies—the convergence rate improves to linear, i.e., $O(e^{-c \cdot t})$ for some $c > 0$. This result answers an open question in the literature concerning how such regularization can expedite convergence. The entropy regularization is shown to effectively act as a form of strong convexity in the objective landscape, thereby accelerating the convergence rate substantially.
3. Theoretical Insights into Entropy Regularization: The paper addresses how entropy regularization contributes to better convergence properties both by providing a $O(1/t)$ lower bound without regularization, and by demonstrating that entropy induces a positive non-uniform \L{}ojasiewicz degree. This change explains the observed discrepancy in convergence speeds, illustrating that entropy influences the gradient landscape in a manner that allows for more efficient optimization.

Implications and Future Work

These findings have profound implications for both theoretical and practical reinforcement learning. Theoretically, they underscore the importance of examining the subtle interactions between policy parameterizations and convergence properties in non-convex optimization scenarios typical of RL. Practically, the insights shed light on how entropy regularization serves not just as an explorative mechanism but as a convergence accelerator. This work invites further research into adaptive mechanisms for tuning regularization strength and extending these results to more complex function approximation settings or noisy gradient scenarios.

Future research directions could include extending these convergence results to deep reinforcement learning contexts where function approximation introduces additional challenges, such as stability and scalability of policy gradient methods. Additionally, adaptive schedules for regularization parameters that dynamically balance exploration and exploitation could further enhance learning efficiency without compromising convergence guarantees.

Overall, this paper provides a rigorous and quantifiable analysis of softmax policy gradient methods, offering deep insights into their performance characteristics and shedding light on the role of entropy in improving policy optimization procedures.