Convex Regularization and Convergence of Policy Gradient Flows under Safety Constraints

This paper studies reinforcement learning (RL) in infinite-horizon dynamic decision processes with almost-sure safety constraints. Such safety-constrained decision processes are central to applications in autonomous systems, finance, and resource management, where policies must satisfy strict, state-dependent constraints. We consider a doubly-regularized RL framework that combines reward and parameter regularization to address these constraints within continuous state-action spaces. Specifically, we formulate the problem as a convex regularized objective with parametrized policies in the mean-field regime. Our approach leverages recent developments in mean-field theory and Wasserstein gradient flows to model policies as elements of an infinite-dimensional statistical manifold, with policy updates evolving via gradient flows on the space of parameter distributions. Our main contributions include establishing solvability conditions for safety-constrained problems, defining smooth and bounded approximations that facilitate gradient flows, and demonstrating exponential convergence towards global solutions under sufficient regularization. We provide general conditions on regularization functions, encompassing standard entropy regularization as a special case. The results also enable a particle method implementation for practical RL applications. The theoretical insights and convergence guarantees presented here offer a robust framework for safe RL in complex, high-dimensional decision-making problems.