Infinitesimal Operator World Models: Finite-Sample Guarantees for Offline Reinforcement Learning

arXiv:2511.10383 · stat.ML, cs.LG, cs.SY, eess.SY, math.OC · Submitted 2025-11-13 · Read on arXiv

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Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: I'm Tom, and with me are Jane, Lu, senior AI researcher at Tsinghua, Meng, lead engineer at a mysterious AI startup and Lalam, the in-house Large Language Model.

Jane: Today's paper: "Infinitesimal Operator World Models".

Tom: The gist: This paper proposes an operator-based algorithm, O-CTRL,

Jane: First, who's behind it and why it matters.

Title and authors: Tom: Let's talk about who wrote this. The paper, "Infinitesimal Operator World Models: Finite-Sample Guarantees for Offline Reinforcement Learning," was put out by Nicolas Hoischen, Petar Bevanda, Max Beier, Stefan Sosnowski, Boris Houska from TU Munich and Sandra Hirche from TU Munich.

Jane: Those are some heavy hitters in the operator theory space. It tells us right away this isn't just a simple RL tweak; they’re leaning on deep mathematical machinery to handle the continuous nature of the problem.

Lu: The authors are bringing in statistical learning theory for linear operators and convex optimization to make sure they get global convergence for the value function, which is pretty ambitious when you start with offline data.

Meng: That sounds like a lot of work just to prove that the solution actually exists and stays stable across different data samples. How do they handle all that complexity in practice?

Lalam: They are using this operator model to decompose the RL problem into learning the dynamics, which is then used for policy optimization, creating this kind of end-to-end pipeline.

The paper's summary: Tom: Now for what they actually did in "Infinitesimal Operator World Models: Finite-Sample Guarantees for Offline Reinforcement Learning." They propose an operator-based algorithm called O-CTRL that solves the RL problem by linking it to the Hamilton–Jacobi–Bellman equation through this learned world model.

Jane: So, instead of training a policy directly on the data, they first learn an operator representing the system dynamics in a Reproducing Kernel Hilbert Space, or RKHS.

Lu: They represent that infinitesimal generator of controlled diffusion processes using Markov semigroups, which is key because it directly captures how the probability distribution of states evolves under different actions.

Meng: This means they aren't just learning a snapshot; they are learning the rule for how the system moves from one state to another over time, which sounds much more robust than standard model learning.

Lalam: That representation lets them split the dynamics into an autonomous part and an action-dependent part, which is essential because it lets them handle both how things happen naturally and how we can steer them with control.

The paper's improvements: Tom: The authors suggest a few key improvements over existing offline RL work. They focus on using statistical learning theory for linear operator learning and convex optimization to establish global convergence for the value function.

Jane: They are providing finite-sample guarantees, which means they can actually give you a bound on how good their solution is based on how much data you have and the properties of the system itself, like smoothness and stability.

Lu: They achieve this by relating the approximation errors of the operators—specifically E(Abγ) and E(Bbγ)—to an error term E(Gbγ) which they bound by a constant delta greater than zero.

Meng: That structure is interesting; they are essentially showing that if you control the error in learning the operator G, you can control the final policy evaluation error. It’s a clean way to manage uncertainty.

Lalam: They also connect this entire structure to an optimal control problem where the value function has to satisfy a stationary discounted HJB equation, which gives it a structural link to methods like proximal policy optimization and mirror descent.

Conclusion: Tom: So, wrapping up "Infinitesimal Operator World Models: Finite-Sample Guarantees for Offline Reinforcement Learning," the authors show that O-CTRL provides a way to solve continuous-time offline RL by building an operator world model in an RKHS and using dynamic programming recursion guided by the HJB equation.

Jane: The main implication is that they’ve established a framework with global convergence guarantees, meaning we have a theoretical way to trust the value function approximation derived from historical data.

Lu: This work opens up a new direction for offline RL by treating it as an optimal control problem and using operator methods to handle the complex, continuous dynamics inherent in many real-world systems.

Meng: For practical AI, this suggests that separating the world model learning from the task optimization might make performance less sensitive to dataset quality alone, if we can get those theoretical guarantees working reliably.

Lalam: I think what's most powerful here is the end-to-end pipeline they construct, moving from dynamical system data all the way to a value function approximation with mathematical guarantees attached.

Tom: That’s exactly it—a solid analytical tool for continuous-time offline reinforcement learning. We’ve been looking at how these operator methods can help us understand and control systems that evolve smoothly.

TU Munich

stat.ML, cs.LG, cs.SY, eess.SY, math.OC

Submitted: 2025-11-13

Updated: 2026-10-09

Journal ref: IEEE Open Journal of Control Systems, 01 October 2026

DOI: 10.1109/OJCSYS.2026.3739678

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 87/100

The gist: The gist: This paper proposes an operator-based algorithm, O-CTRL, to solve offline continuous-time reinforcement learning by linking it to Hamilton–Jacobi–Bellman theory through an operator

Key concepts

Infinitesimal Generator: This mathematical tool describes how the probability distribution of states evolves over time in a continuous-time system. In this paper, it is used to create a world model that captures both autonomous state changes and action-dependent transitions, allowing the system's dynamics to be represented as an operator.
Reproducing Kernel Hilbert Space (RKHS): An RKHS is a space where complex functions can be approximated using kernel methods. The paper uses this space to learn the infinitesimal generator of the world model from data. This allows for a statistical learning approach to build a precise, albeit approximate, mathematical representation of how the environment changes.
Hamilton-Jacobi-Bellman (HJB) Equation: The HJB equation is central to optimal control and reinforcement learning. It defines the optimal value function by balancing expected rewards against state transitions. The paper reformulates the policy optimization step into a structural analogue of this equation, enabling the use of operator theory for finding the best control strategy.
Operator-Based Algorithm (O-CTRL): O-CTRL is a proposed algorithm that solves offline RL by decomposing it into learning a world model and optimizing a policy. It uses operator theory to handle the continuous dynamics and statistical learning methods to ensure global convergence of the value function, offering theoretical guarantees for performance.

Terminology

Reinforcement Learning — 4× in this paper · explained in 4 episodes
World Model — 2× in this paper · explained in 7 episodes
Diffusion Processes — 1× in this paper · explained in 2 episodes
Hilbert Space — 1× in this paper · explained in 2 episodes

Summary

The gist: This paper proposes an operator-based algorithm, O-CTRL, to solve offline continuous-time reinforcement learning by linking it to Hamilton–Jacobi–Bellman theory through an operator world model learned in a reproducing kernel Hilbert space.

Operator Models for Continuous-Time Offline Reinforcement Learning

The core idea is to decompose the offline continuous-time RL problem into world model operator learning and policy optimization, which enables the construction of O-CTRL (Figure 1) to solve the RL problem for the optimal value function > Operator Models for Continuous-Time Offline Reinforcement Learning, Figure 1. This approach utilizes statistical learning theory for linear operator learning and convex optimization to establish global convergence of the value function and derive finite-sample guarantees tied to system properties such as smoothness and stability > Operator Models for Continuous-Time Offline Reinforcement Learning, Page 1.

World Model Construction via RKHS

The world model is represented in terms of the infinitesimal generator of controlled diffusion processes learned in a reproducing kernel Hilbert space > Operator Models for Continuous-Time Offline Reinforcement Learning, Page 1. This representation allows one to model the evolution of probability densities via Markov semigroups, which directly captures the distributional evolution of states central to policy evaluation and optimization/planning > Operator Models for Continuous-Time Offline Reinforcement Learning, Page 2. The infinitesimal generator Gt at time t is defined on a domain with sufficient regularity D(G) (Engel et al., 2000), splitting into autonomous and action-dependent dynamics: [Gtϕ](s, a) = Aϕ + [Btϕ](s, a), where Aϕ = ∇sϕ(s)⊤f(s) + ϵ ∆sϕ(s), and [Btϕ](S, a] = ∇S ϕ(S))TG(S, a) at > Operator Models for Continuous-Time Offline Reinforcement Learning, Page 3. To obtain data-based surrogates of the operators A and B, the paper constructs an empirical risk formulation (Equation 18) to estimate G in HS(HS, HZ), which is solved using Kernel Ridge Regression (KRR) approximation (Equation 19) > Operator Models for Continuous-Time Offline Reinforcement Learning, Page 5.

Policy Optimization via Optimal Control

The policy optimization step is formulated by connecting the value function to the optimal control problem where V⋆ must satisfy the stationary discounted HJB equation: max∥π∥L∞ Operator Models for Continuous-Time Offline Reinforcement Learning, Page 2. Under Assumption 2 (Reward), the maximization term is identified with the Fenchel conjugate of the action penalty, Da(λ):= max a ∈ A [⟨a,λ⟩ − ca(s, a)], which admits an unique maximizer a⋆(λ) = ∇Da(λ) > Operator Models for Continuous-Time Offline Reinforcement Learning, Page 14. This leads to the infinitesimal HJB formulation T (V) = −ρI − A V + rs + Da(B V), where B is the controlled part of G, and this structure provides a structural analogue to policy mirror descent (PMD) and proximal policy optimization (PPO) > Operator Models for Continuous-Time Offline Reinforcement Learning, Page 14.

Theoretical Guarantees and Convergence Rates

The paper establishes global convergence of the value function through operator-norm error analysis, showing that the approximation errors E(Abγ) and E(Bbγ) are bounded by E(Gbγ) = δ > 0 in Section D.3 > Operator Models for Continuous-Time Offline Reinforcement Learning, Page 17. Theorem 1 provides a finite-sample bound for the learning error:∥V⋆ − Vbk∥L2µ ≤ (λbgap+ρ)−1δz>learning + O((∆t)p)z>discretization + κ e−(λbgap+ρ)k∆tVb⋆L2µconvergence > Operator Models for Continuous-Time Offline Reinforcement Learning, Page 7. Corollary 2 further provides a finite-sample bound on the learning error:∥V⋆ − Vb⋆∥L2µ ≤ c (λbgap+ρ)−1N−α2(α+β)ln ξ−1 > Operator Models for Continuous-Time Offline Reinforcement Learning, Page 23.

Numerical Validation and Comparison

The proposed algorithm is evaluated on linear and nonlinear process dynamics, including an Ornstein-Uhlenbeck process and a nonlinear SDE with affine action > Operator Models for Continuous-Time Offline Reinforcement Learning, Page 5. The results show that O-CTRL is competitive on Random data and surpasses the expert on Replay data, suggesting the decoupling of world model learning from task optimization makes performance less sensitive to dataset quality > Operator Models for Continuous-Time Offline Reinforcement Learning, Page 8. The computational complexity of Algorithm 1 is dominated by O(N 3) terms arising from the factorization of Kγ and subsequent solves required to obtain A, with the dynamic programming recursion also being dominated by an O(N 3) factorization of M > Operator Models for Continuous-Time Offline Reinforcement Learning, Page 24. The analysis confirms that operator methods provide a promising direction for offline RL using continuous-time optimal control > Operator Models for Continuous-Time Offline Reinforcement Learning, Page 6.

Limitations and Future Work

A key limitation is the requirement of independent data for the generator learning scheme, which might be costly to obtain > Operator Models for Continuous-Time Offline Reinforcement Learning, Page 9. Furthermore, while universal RKHS guarantees arbitrarily accurate value function approximations, convergence to the true value function can be slow without additional assumptions > Operator Models for Continuous-Time Offline Reinforcement Learning, Page 12. Open questions include extending the analysis to dependent (trajectory) data and determining the algorithmic requirement for ensuring that ρ + λgap > LDE(Gb) – sufficient stability > Operator Models for Continuous-Time Offline Reinforcement Learning, Page 9. The work presents the first end-to-end learning theory for continuous-time offline reinforcement learning, highlighting the power of operator methods as an analytical tool > Operator Models for Continuous-Time Offline Reinforcement Learning, Page 6.

Improvements for AI systems

Continuous-Time Offline RL via O-CTRL Algorithm: This system can learn optimal policies in continuous-time environments by first learning a world model operator using Reproducing Kernel Hilbert Spaces (RKHS) and then solving the Hamilton–Jacobi–Bellman (HJB) equation through a dynamic programming recursion, as described in Equation 1.
Global Convergence Guarantees: The system provides global convergence of the value function and derives finite-sample guarantees with bounds tied to system properties such as smoothness and stability, which are established by relating operator learning errors to value-function errors.
Operator Model Risk Minimization: The system employs a regularized risk minimization approach, specifically Equation 39, where the empirical risk is approximated using Kernel Ridge Regression (KRR) via Gram matrices like Kγ, to obtain an estimator Gγ that minimizes the squared operator norm error E(Gbγ).
Nonparametric Policy Optimization: By utilizing RKHSs as a universal hypothesis class, the system enables learning arbitrarily good finite-rank approximations of the infinitesimal generator, allowing for policy optimization without relying on fixed parametric models.
End-to-End Learning Pipeline: The O-CTRL algorithm constructs an end-to-end pipeline that starts from dynamical system data to construct a value function approximation with guarantees, which decouples the learning of system dynamics from the downstream task defined by the reward.

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 under sufficiently small generator-approximation error and derive explicit finite-sample guarantees with bounds tied directly to system properties such as smoothness and stability. Our theoretical and proof-of-concept results indicate that operator-based approaches may hold promise for solving offline reinforcement learning using continuous-time optimal control.