Beyond Exact Gradients: Convergence of Stochastic Soft-Max Policy Gradient Methods with Entropy Regularization
Yuhao Ding, Junzi Zhang, Hyunin Lee, Javad Lavaei
TL;DR
This work addresses the lack of global convergence theory for stochastic entropy-regularized policy gradient methods with soft-max parametrization. It introduces two nearly unbiased estimators—an unbiased visitation-measure estimator and a practical trajectory-based estimator—both with uniformly bounded variance despite unbounded regularized rewards. To overcome the non-coercive landscape, it proposes a two-phase algorithm that uses a large batch in phase I and a small batch in phase II, achieving a global convergence guarantee with sample complexity $\tilde{O}(1/\epsilon^2)$. The results provide the first global convergence and tight sample complexity bounds for stochastic, entropy-regularized vanilla PG, offering theoretical grounding and practical guidance for entropy-regularized RL in stochastic settings.
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 $\widetilde{\mathcal{O}}(\frac{1}{ε^2})$ for the proposed algorithm. Our result is the first global convergence and sample complexity results for the stochastic entropy-regularized vanilla PG method.