Bellman operator convergence enhancements in reinforcement learning algorithms

2025/05/20 by David Krame Kadurha, Kadurha, David Krame, Domini Jocema Leko Moutouo +3 · 1 citation
Computer Science · Engineering · #Adaptive Dynamic Programming Control #Artificial Intelligence (cs.AI) #Control and Stability of Dynamical Systems #FOS: Computer and information sciences #Machine Learning (cs.LG) #Reinforcement Learning in Robotics

paper · pdf · doi:10.48550/arxiv.2505.14564

openalex publication_date 2025/05/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Topological Foundations of Reinforcement Learning
• Metrics and continuity in reinforcement learning
• Efficient Model-free Reinforcement Learning in Metric Spaces
• The Value Function Polytope in Reinforcement Learning
• A General Family of Robust Stochastic Operators for Reinforcement Learning
• Lipschitz Continuity in Model-based Reinforcement Learning
• Neuronlike adaptive elements that can solve difficult learning control problems
• Reinforcement Learning: An Introduction

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