On the Global Convergence Rates of Softmax Policy Gradient Methods

TL;DR

This paper provides a theoretical analysis of the convergence rates of softmax policy gradient methods, showing that entropy regularization accelerates convergence from sublinear to linear, with implications for policy optimization.

Contribution

It establishes the first $O(1/t)$ convergence rate with true gradient and demonstrates that entropy regularization achieves a faster linear rate, explaining its empirical benefits.

Findings

Policy gradient with true gradient converges at $O(1/t)$ rate.

Entropy regularization leads to linear convergence rate.

Entropy improves policy optimization efficiency.

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.