Continuous-time reinforcement learning: ellipticity enables model-free value function approximation

2026-04-17 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences · Depth: Expert, medium

Summary

A new study addresses the challenge of off-policy reinforcement learning (RL) for continuous-time Markov diffusion processes, particularly when using discrete-time observations and actions with function approximation. The research introduces a model-free approach that learns value and advantage functions directly from data, bypassing the need for unrealistic structural assumptions on dynamics. By leveraging the ellipticity of diffusions, the authors establish novel Hilbert-space positive definiteness and boundedness properties for Bellman operators. These properties underpin the proposed Sobolev-prox fitted q-learning algorithm, which iteratively solves least-squares regression problems. The algorithm's estimation error is characterized by oracle inequalities, accounting for approximation error, localized complexity, optimization error, and numerical discretization error, suggesting that ellipticity simplifies RL with function approximation to a complexity comparable to supervised learning.

Key takeaway

For research scientists developing continuous-time reinforcement learning systems, understanding the role of ellipticity is crucial. This property enables model-free value function approximation with theoretical guarantees comparable to supervised learning, mitigating instability issues often seen with function approximation. You should consider incorporating the Sobolev-prox fitted q-learning algorithm when working with Markov diffusion processes to achieve robust and efficient learning from offline data.

Key insights

Ellipticity in continuous-time diffusions enables stable, model-free RL with function approximation, akin to supervised learning.

Principles

• Bellman operators exhibit Hilbert-space positive definiteness under ellipticity.
• Approximation error is controllable with appropriate function classes.

Method

The Sobolev-prox fitted q-learning algorithm updates advantage functions via least-squares regression, followed by a Sobolev-norm proximal step for value function updates.

In practice

• Apply to finance, robotics, queuing systems.
• Use for reward-guided fine-tuning of diffusion generative models.

Topics

• Continuous-time Reinforcement Learning
• Markov Diffusion Processes
• Model-Free Reinforcement Learning
• Function Approximation
• Ellipticity

Best for: Research Scientist, AI Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.