# Bellman operator convergence enhancements in reinforcement learning algorithms

arXiv: https://arxiv.org/abs/2505.14564 · HTML: https://mlsanity.com/paper/2505.14564/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in RL theory and sample complexity, one of 15 directions in Deep reinforcement learning methods.
- Builds on 7 of its 21 references that are in the map.
- First posted on arXiv 20 May 2025.

## Placed in

- Deep reinforcement learning methods / [RL theory and sample complexity](https://mlsanity.com/deeprl/rl-theory-and-sample-complexity/)

## Builds on (most cited first, 7 of 7)

- [Lipschitz Continuity in Model-based Reinforcement Learning](https://mlsanity.com/paper/1804.07193/) (2018)
- [Increasing the Action Gap: New Operators for Reinforcement Learning](https://mlsanity.com/paper/1512.04860/) (2015)
- [The Value Function Polytope in Reinforcement Learning](https://mlsanity.com/paper/1901.11524/) (2019)
- [Metrics and continuity in reinforcement learning](https://mlsanity.com/paper/2102.01514/) (2021)
- [Efficient Model-free Reinforcement Learning in Metric Spaces](https://mlsanity.com/paper/1905.00475/) (2019)
- [A General Family of Robust Stochastic Operators for Reinforcement Learning](https://mlsanity.com/paper/1805.08122/) (2018)
- [Topological Foundations of Reinforcement Learning](https://mlsanity.com/paper/2410.03706/) (2024)

## Built on by (most cited first, 1 of 1)

- [TabQL: In-Context Q-Learning with Tabular Foundation Models](https://mlsanity.com/paper/2605.18979/) (2026)

## Abstract (arXiv)

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.