Second-Order Smooth Planning with Optimal-Transport Bellman Smoothing

Tuan Quang Dam
Proceedings of the 43rd International Conference on Machine Learning, PMLR 306:22689-22728, 2026.

Abstract

Planning with a generative model aims to estimate the value of a state using as few simulator calls as possible. SmoothCruiser achieves problem-independent complexity $\widetilde O(\varepsilon^{-4})$ by exploiting the smoothness of the entropy-regularized Bellman backup, but its estimator is only first-order. We show that the sample-complexity exponent of SmoothCruiser-type planners is governed by the order $\beta$ of the local Taylor remainder, giving oracle complexity $\widetilde O(\varepsilon^{-(2+2/(\beta-1))})$: the first-order case $\beta=2$ recovers SmoothCruiser, while a second-order/cubic remainder $\beta=3$ yields $\widetilde O(\varepsilon^{-3})$. We reach this regime with an optimal-transport-smoothed Bellman backup over action distributions, which has a closed form, a policy gradient, and a Lipschitz Hessian, and whose quadratic correction admits an unbiased cross-product estimator. The resulting SecondOrderSmoothCruiser achieves $\widetilde O(\varepsilon^{-3})$ oracle complexity for fixed OT parameters, and we relate the OT, entropy-regularized, and unregularized objectives through explicit regularization-bias bounds.

Cite this Paper

BibTeX
@InProceedings{pmlr-v306-dam26a, title = {Second-Order Smooth Planning with Optimal-Transport {B}ellman Smoothing}, author = {Dam, Tuan Quang}, booktitle = {Proceedings of the 43rd International Conference on Machine Learning}, pages = {22689--22728}, year = {2026}, editor = {Zhang, Tong and Dudik, Miroslav and Jaggi, Martin and Agarwal, Alekh and Li, Sharon and Schuurmans, Dale and Zhu, Jerry and Berkenkamp, Felix and Dong, Hanze and Bietti, Alberto}, volume = {306}, series = {Proceedings of Machine Learning Research}, month = {06--11 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v306/main/assets/dam26a/dam26a.pdf}, url = {https://proceedings.mlr.press/v306/dam26a.html}, abstract = {Planning with a generative model aims to estimate the value of a state using as few simulator calls as possible. SmoothCruiser achieves problem-independent complexity $\widetilde O(\varepsilon^{-4})$ by exploiting the smoothness of the entropy-regularized Bellman backup, but its estimator is only first-order. We show that the sample-complexity exponent of SmoothCruiser-type planners is governed by the order $\beta$ of the local Taylor remainder, giving oracle complexity $\widetilde O(\varepsilon^{-(2+2/(\beta-1))})$: the first-order case $\beta=2$ recovers SmoothCruiser, while a second-order/cubic remainder $\beta=3$ yields $\widetilde O(\varepsilon^{-3})$. We reach this regime with an optimal-transport-smoothed Bellman backup over action distributions, which has a closed form, a policy gradient, and a Lipschitz Hessian, and whose quadratic correction admits an unbiased cross-product estimator. The resulting SecondOrderSmoothCruiser achieves $\widetilde O(\varepsilon^{-3})$ oracle complexity for fixed OT parameters, and we relate the OT, entropy-regularized, and unregularized objectives through explicit regularization-bias bounds.} }
Endnote
%0 Conference Paper %T Second-Order Smooth Planning with Optimal-Transport Bellman Smoothing %A Tuan Quang Dam %B Proceedings of the 43rd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2026 %E Tong Zhang %E Miroslav Dudik %E Martin Jaggi %E Alekh Agarwal %E Sharon Li %E Dale Schuurmans %E Jerry Zhu %E Felix Berkenkamp %E Hanze Dong %E Alberto Bietti %F pmlr-v306-dam26a %I PMLR %P 22689--22728 %U https://proceedings.mlr.press/v306/dam26a.html %V 306 %X Planning with a generative model aims to estimate the value of a state using as few simulator calls as possible. SmoothCruiser achieves problem-independent complexity $\widetilde O(\varepsilon^{-4})$ by exploiting the smoothness of the entropy-regularized Bellman backup, but its estimator is only first-order. We show that the sample-complexity exponent of SmoothCruiser-type planners is governed by the order $\beta$ of the local Taylor remainder, giving oracle complexity $\widetilde O(\varepsilon^{-(2+2/(\beta-1))})$: the first-order case $\beta=2$ recovers SmoothCruiser, while a second-order/cubic remainder $\beta=3$ yields $\widetilde O(\varepsilon^{-3})$. We reach this regime with an optimal-transport-smoothed Bellman backup over action distributions, which has a closed form, a policy gradient, and a Lipschitz Hessian, and whose quadratic correction admits an unbiased cross-product estimator. The resulting SecondOrderSmoothCruiser achieves $\widetilde O(\varepsilon^{-3})$ oracle complexity for fixed OT parameters, and we relate the OT, entropy-regularized, and unregularized objectives through explicit regularization-bias bounds.
APA
Dam, T.Q.. (2026). Second-Order Smooth Planning with Optimal-Transport Bellman Smoothing. Proceedings of the 43rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 306:22689-22728 Available from https://proceedings.mlr.press/v306/dam26a.html.

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