Beyond Bellman: High-Order Generator Regression for Continuous-Time Policy Evaluation
For researchers in continuous-time reinforcement learning, this work provides an interpretable method with a clear operating region that improves policy evaluation accuracy, though the gains are incremental over existing approaches.
This paper introduces a high-order generator regression method for continuous-time policy evaluation that improves upon the first-order Bellman baseline by using multi-step transitions and moment-matching coefficients. The second-order estimator consistently outperforms the Bellman baseline across calibration studies and benchmarks, with stable performance in the predicted regime.
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.