Instance-Dependent Continuous-Time Reinforcement Learning via Maximum Likelihood Estimation
Runze Zhao, Yue Yu, Ruhan Wang, Chunfeng Huang, Dongruo Zhou
TL;DR
This paper introduces a model-based continuous-time reinforcement learning algorithm that adapts to environment difficulty by estimating state densities and employing randomized measurement schedules, with theoretical guarantees on performance.
Contribution
It presents a novel MLE-based CTRL method that estimates state densities instead of dynamics, providing instance-dependent regret bounds and adaptive measurement strategies.
Findings
Regret scales with total reward variance and measurement resolution.
Adaptive measurement frequency reduces dependence on measurement strategy.
Algorithm improves sample efficiency without extra measurement costs.
Abstract
Continuous-time reinforcement learning (CTRL) provides a natural framework for sequential decision-making in dynamic environments where interactions evolve continuously over time. While CTRL has shown growing empirical success, its ability to adapt to varying levels of problem difficulty remains poorly understood. In this work, we investigate the instance-dependent behavior of CTRL and introduce a simple, model-based algorithm built on maximum likelihood estimation (MLE) with a general function approximator. Unlike existing approaches that estimate system dynamics directly, our method estimates the state marginal density to guide learning. We establish instance-dependent performance guarantees by deriving a regret bound that scales with the total reward variance and measurement resolution. Notably, the regret becomes independent of the specific measurement strategy when the observation…
Peer Reviews
Decision·Submitted to ICLR 2026
1. Interesting and novel idea: While instance-dependent analysis has been studied in standard discrete-time RL, it is interesting to see how it adapts to continuous-time RL, where irregular measurement times introduce an extra layer of complexity. This paper tackles that setting. 2. New insights for CTRL: By analyzing instance-dependent regret, the paper offers guidance for designing measurement schedules. In particular, as long as the total measurement budget is upper bounded by the total cumu
1. Limited intuitive exposition in the main text: The theoretical results are dense. I suggest adding a brief proof sketch in the main paper. It would also help to discuss the key challenges and obstacles specific to the continuous-time setting, and how they differ from standard discrete-time RL (e.g., [1]). 2. My biggest concern is the quadratic density assumption (Proposition 5.9). Although Appendix A.3 provides some discussion, a more general treatment would strengthen the paper. If possible
- The continuous time RL problem is both interesting and important for the RL community as it can tackle more realistic settings than standard MDPs - The paper provides a theoretical analysis of the proposed algorithm, proving that its regret grows only logarithmically.
- The algorithm cannot be implemented in practice for meaningful and general classes of dynamics F,G and policies Pi. The authors address this in Appendix C with a relaxation of the considered optimization problems. However, this makes the algorithm loose its theoretical properties. - The analysis assumes Pi,F,G to be finite. Although the authors mention in Remark 5.2 that the analysis can be extended to more general sets using the same arguments as in a previous work (Wang et al. 2024b), they s
The shift from dynamic estimation to marginal density estimation via MLE is elegant and conceptually simple, offering a new angle on continuous-time RL problems. The authors derive a variance-dependent, nearly horizon-free regret bound, marking a notable advance over existing CTRL works that rely on worst-case analysis or have exponential dependence on the time horizon. The introduction of reward variance as a measure of instance difficulty is insightful, connecting continuous-time control wit
The empirical component appears secondary; the paper would benefit from more thorough experiments demonstrating practical performance, robustness, and scalability of CT-MLE. The assumptions of finite policy, drift, and diffusion classes simplify analysis but reduce applicability to large-scale or continuous policy spaces.
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Taxonomy
TopicsReinforcement Learning in Robotics · Advanced Bandit Algorithms Research · Gaussian Processes and Bayesian Inference