Bellman operator convergence enhancements in reinforcement learning algorithms
David Krame Kadurha, Domini Jocema Leko Moutouo, Yae Ulrich Gaba
TL;DR
The paper investigates the topological underpinnings of reinforcement learning by situating state, action, and policy spaces within metric and Banach spaces and framing RL convergence through contraction mappings and Bellman operators. It formalizes the Banach fixed-point perspective on RL, then introduces and analyzes alternative Bellman operators, including the Consistent Bellman Operator and a Modified Robust Stochastic Operator that incorporates advantage learning. Theoretical results establish contraction and monotonicity for the classical and consistent operators, while the modified operator offers optimality-preservation and gap-increasing properties despite not being a contraction. Empirical evaluations on MountainCar, CartPole, and Acrobot demonstrate practical benefits of the refined operators, especially in sample efficiency and convergence speed, highlighting a path toward more robust, theory-grounded RL algorithms. The work thus bridges rigorous topological analysis with practical algorithm design, urging further exploration of operator families that respect monotonic contraction properties while improving convergence in complex RL settings.
Abstract
This paper reviews the topological groundwork for the study of reinforcement learning (RL) by focusing on the structure of state, action, and policy spaces. We begin by recalling key mathematical concepts such as complete metric spaces, which form the foundation for expressing RL problems. By leveraging the Banach contraction principle, we illustrate how the Banach fixed-point theorem explains the convergence of RL algorithms and how Bellman operators, expressed as operators on Banach spaces, ensure this convergence. The work serves as a bridge between theoretical mathematics and practical algorithm design, offering new approaches to enhance the efficiency of RL. In particular, we investigate alternative formulations of Bellman operators and demonstrate their impact on improving convergence rates and performance in standard RL environments such as MountainCar, CartPole, and Acrobot. Our findings highlight how a deeper mathematical understanding of RL can lead to more effective algorithms for decision-making problems.