Beyond Exact Gradients: Convergence of Stochastic Soft-Max Policy Gradient Methods with Entropy Regularization

TL;DR

This paper develops the first global convergence and sample complexity analysis for stochastic entropy-regularized policy gradient methods in reinforcement learning, introducing nearly unbiased estimators and a two-phase algorithm.

Contribution

It introduces nearly unbiased stochastic policy gradient estimators with entropy regularization and proves their convergence with a novel two-phase algorithm.

Findings

First global convergence result for stochastic entropy-regularized policy gradient.

Sample complexity of (rac{1}{\u03b5^2}) for the proposed method.

Bounded variance of the proposed stochastic estimators despite unboundedness in general.

Abstract

Entropy regularization is an efficient technique for encouraging exploration and preventing a premature convergence of (vanilla) policy gradient methods in reinforcement learning (RL). However, the theoretical understanding of entropy-regularized RL algorithms has been limited. In this paper, we revisit the classical entropy regularized policy gradient methods with the soft-max policy parametrization, whose convergence has so far only been established assuming access to exact gradient oracles. To go beyond this scenario, we propose the first set of (nearly) unbiased stochastic policy gradient estimators with trajectory-level entropy regularization, with one being an unbiased visitation measure-based estimator and the other one being a nearly unbiased yet more practical trajectory-based estimator. We prove that although the estimators themselves are unbounded in general due to the additional logarithmic policy rewards introduced by the entropy term, the variances are uniformly bounded. We then propose a two-phase stochastic policy gradient (PG) algorithm that uses a large batch size in the first phase to overcome the challenge of the stochastic approximation due to the non-coercive landscape, and uses a small batch size in the second phase by leveraging the curvature information around the optimal policy. We establish a global optimality convergence result and a sample complexity of $\widetilde{\mathcal{O}}(\frac{1}{\epsilon^2})$ for the proposed algorithm. Our result is the first global convergence and sample complexity results for the stochastic entropy-regularized vanilla PG method.