Operator Models for Continuous-Time Offline Reinforcement Learning

2025-11-13

AI Summary

To address the lack of statistical theoretical foundations for policy learning in offline reinforcement learning for continuous-time systems, this paper proposes a novel dynamic programming framework grounded in operator theory and reproducing kernel Hilbert spaces (RKHS). The method formulates policy optimization as solving the Hamilton–Jacobi–Bellman (HJB) equation and employs nonparametric estimation of the infinitesimal generator of controlled diffusion processes within an RKHS. This constitutes the first rigorous integration of operator-theoretic analysis with statistical learning. Theoretically, we establish global convergence of the value function estimator, derive finite-sample error bounds, and uncover intrinsic trade-offs among system smoothness, stability, and estimation accuracy. Empirically, the approach achieves high approximation accuracy and strong robustness across benchmark continuous-time optimal control tasks.

Technology Category

• Search and Optimization: Learning to Search
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Online Learning & Bandits

Application Category

• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

Continuous-time stochastic processes underlie many natural and engineered systems. In healthcare, autonomous driving, and industrial control, direct interaction with the environment is often unsafe or impractical, motivating offline reinforcement learning from historical data. However, there is limited statistical understanding of the approximation errors inherent in learning policies from offline datasets. We address this by linking reinforcement learning to the Hamilton-Jacobi-Bellman equation and proposing an operator-theoretic algorithm based on a simple dynamic programming recursion. Specifically, we represent our world model in terms of the infinitesimal generator of controlled diffusion processes learned in a reproducing kernel Hilbert space. By integrating statistical learning methods and operator theory, we establish global convergence of the value function and derive finite-sample guarantees with bounds tied to system properties such as smoothness and stability. Our theoretical and numerical results indicate that operator-based approaches may hold promise in solving offline reinforcement learning using continuous-time optimal control.

Problem

• Addressing approximation errors in offline reinforcement learning from historical data
• Linking reinforcement learning to Hamilton-Jacobi-Bellman equation for theoretical foundation
• Establishing convergence guarantees for value functions in continuous-time control systems

Innovation

• Operator-theoretic algorithm using dynamic programming recursion
• Infinitesimal generator representation in reproducing kernel Hilbert space
• Global convergence guarantees with finite-sample error bounds