# Beyond Bellman: High-Order Generator Regression for Continuous-Time Policy Evaluation

arXiv: https://arxiv.org/abs/2604.18972 · HTML: https://mlsanity.com/paper/2604.18972/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 44 references that are in the map.
- First posted on arXiv 21 Apr 2026.

## Builds on (most cited first, 18 of 18)

- [Offline Reinforcement Learning: Tutorial, Review, and Perspectives on Open Problems](https://mlsanity.com/paper/2005.01643/) (2020)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Doubly Robust Off-policy Value Evaluation for Reinforcement Learning](https://mlsanity.com/paper/1511.03722/) (2015)
- [Data-Efficient Off-Policy Policy Evaluation for Reinforcement Learning](https://mlsanity.com/paper/1604.00923/) (2016)
- [Safe and Efficient Off-Policy Reinforcement Learning](https://mlsanity.com/paper/1606.02647/) (2016)
- [More Robust Doubly Robust Off-policy Evaluation](https://mlsanity.com/paper/1802.03493/) (2018)
- [Empirical Study of Off-Policy Policy Evaluation for Reinforcement Learning](https://mlsanity.com/paper/1911.06854/) (2019)
- [Towards Optimal Off-Policy Evaluation for Reinforcement Learning with Marginalized Importance Sampling](https://mlsanity.com/paper/1906.03393/) (2019)
- [Policy Gradient and Actor-Critic Learning in Continuous Time and Space: Theory and Algorithms](https://mlsanity.com/paper/2111.11232/) (2021)
- [q-Learning in Continuous Time](https://mlsanity.com/paper/2207.00713/) (2022)
- [Data-driven approximation of the Koopman generator: Model reduction, system identification, and control](https://mlsanity.com/paper/1909.10638/) (2019)
- [Efficiently Breaking the Curse of Horizon in Off-Policy Evaluation with Double Reinforcement Learning](https://mlsanity.com/paper/1909.05850/) (2019)
- [Consistent On-Line Off-Policy Evaluation](https://mlsanity.com/paper/1702.07121/) (2017)
- [Bootstrapping Fitted Q-Evaluation for Off-Policy Inference](https://mlsanity.com/paper/2102.03607/) (2021)
- [Solving high-dimensional eigenvalue problems using deep neural networks: A diffusion Monte Carlo like approach](https://mlsanity.com/paper/2002.02600/) (2020)
- [Hamilton-Jacobi Deep Q-Learning for Deterministic Continuous-Time Systems with Lipschitz Continuous Controls](https://mlsanity.com/paper/2010.14087/) (2020)
- [Off-Policy Fitted Q-Evaluation with Differentiable Function Approximators: Z-Estimation and Inference Theory](https://mlsanity.com/paper/2202.04970/) (2022)
- [Reliable Off-policy Evaluation for Reinforcement Learning](https://mlsanity.com/paper/2011.04102/) (2020)

## Abstract (arXiv)

We study finite-horizon continuous-time policy evaluation from discrete closed-loop trajectories under time-inhomogeneous dynamics. The target value surface solves a backward parabolic equation, but the Bellman baseline obtained from one-step recursion is only first-order in the grid width. We estimate the time-dependent generator from multi-step transitions using moment-matching coefficients that cancel lower-order truncation terms, and combine the resulting surrogate with backward regression. The main theory gives an end-to-end decomposition into generator misspecification, projection error, pooling bias, finite-sample error, and start-up error, together with a decision-frequency regime map explaining when higher-order gains should be visible. Across calibration studies, four-scale benchmarks, feature and start-up ablations, and gain-mismatch stress tests, the second-order estimator consistently improves on the Bellman baseline and remains stable in the regime where the theory predicts visible gains. These results position high-order generator regression as an interpretable continuous-time policy-evaluation method with a clear operating region.