Mathematical methods of reinforcement learning

2026-07-07

AI Summary

This work addresses the lack of a unified theoretical framework for reinforcement learning, which has hindered systematic analysis of its convergence, sample complexity, and generalization. Building upon Markov decision processes and Bellman operators, the paper introduces a cohesive analytical framework that integrates tools from operator theory, stochastic approximation, convex duality, and function approximation. This framework encompasses a broad range of algorithms, including value iteration, policy iteration, temporal difference methods, off-policy learning, and constrained MDPs. By leveraging contraction mappings, monotone operators, martingale techniques, mirror/proximal optimization, concentration inequalities, and mixing process theory, the study establishes finite-sample performance bounds and asymptotic convergence guarantees for diverse reinforcement learning algorithms, thereby forging a rigorous theoretical bridge between probability theory, optimization, and statistics.

Abstract

Reinforcement learning (RL) is increasingly grounded in tools from probability, optimization, and operator theory. This survey organizes the mathematical structures that underpin the design and analysis of modern algorithms in RL. We begin from Markov decision processes (MDPs) and the Bellman operators, emphasizing contraction mappings, monotonicity, and fixed-point theory that yield convergence guarantees and rates for value and policy iteration, and temporal-difference schemes. We then develop the optimization perspective: stochastic approximation and martingale methods, convex duality and the role of regularization linking mirror/proximal methods. Function approximation is treated through linear and non-linear settings, covering stabilization, error decomposition, and sample-complexity via concentration inequalities for dependent data and mixing processes. We further cover off-policy evaluation/learning, constrained RL and constrained MDPs (CMDPs). Throughout we unify algorithmic templates under common operator and variational lenses, highlighting both finite-sample bounds and asymptotic results. Our presentation is intended to provide a unified mathematical entry point for researchers in probability, optimization, and statistics interested in reinforcement learning.

Problem

• reinforcement learning
• mathematical foundations
• Markov decision processes
• optimization
• function approximation

Innovation

• Bellman operators
• stochastic approximation
• function approximation
• constrained MDPs
• concentration inequalities