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arXiv · 2604.18972

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

Yaowei Zheng·Richong Zhang·Shenxi Wu·Shirui Bian·Haosong Zhang·Li Zeng·Xingjian Ma·Yichi Zhang
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Abstract

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.

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Yaowei Zheng, Richong Zhang, Shenxi Wu, Shirui Bian, Haosong Zhang, Li Zeng, Xingjian Ma, Yichi Zhang. 2026-05-07. Beyond Bellman: High-Order Generator Regression for Continuous-Time Policy Evaluation. https://arxiv.org/abs/2604.18972

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