On the Robustness of Langevin Dynamics to Score Function Error

Daniel Yiming Cao, August Y. Chen, Karthik Sridharan, Yuchen Wu
Proceedings of the 43rd International Conference on Machine Learning, PMLR 306:11099-11119, 2026.

Abstract

We consider the robustness of score-based generative modeling to errors in the estimate of the score function. In particular, we show that Langevin dynamics is not robust to the $L^2$ errors (more generally $L^p$ errors) in the estimate of the score function. It is well-established that with small $L^2$ errors in the estimate of the score function, diffusion models can sample faithfully from the target distribution under fairly mild regularity assumptions in a polynomial time horizon. In contrast, our work shows that even for simple distributions in high dimensions, Langevin dynamics run for any polynomial time horizon will produce a distribution far from the target distribution in Total Variation (TV) distance, even when the $L^2$ error (more generally $L^p$) of the estimate of the score function is arbitrarily small. Considering such an error in the estimate of the score function is unavoidable in practice when learning the score function from data, our results provide further justification for diffusion models over Langevin dynamics and serve to caution against the use of Langevin dynamics with estimated scores.

Cite this Paper

BibTeX
@InProceedings{pmlr-v306-cao26b, title = {On the Robustness of {L}angevin Dynamics to Score Function Error}, author = {Cao, Daniel Yiming and Chen, August Y. and Sridharan, Karthik and Wu, Yuchen}, booktitle = {Proceedings of the 43rd International Conference on Machine Learning}, pages = {11099--11119}, 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/cao26b/cao26b.pdf}, url = {https://proceedings.mlr.press/v306/cao26b.html}, abstract = {We consider the robustness of score-based generative modeling to errors in the estimate of the score function. In particular, we show that Langevin dynamics is not robust to the $L^2$ errors (more generally $L^p$ errors) in the estimate of the score function. It is well-established that with small $L^2$ errors in the estimate of the score function, diffusion models can sample faithfully from the target distribution under fairly mild regularity assumptions in a polynomial time horizon. In contrast, our work shows that even for simple distributions in high dimensions, Langevin dynamics run for any polynomial time horizon will produce a distribution far from the target distribution in Total Variation (TV) distance, even when the $L^2$ error (more generally $L^p$) of the estimate of the score function is arbitrarily small. Considering such an error in the estimate of the score function is unavoidable in practice when learning the score function from data, our results provide further justification for diffusion models over Langevin dynamics and serve to caution against the use of Langevin dynamics with estimated scores.} }
Endnote
%0 Conference Paper %T On the Robustness of Langevin Dynamics to Score Function Error %A Daniel Yiming Cao %A August Y. Chen %A Karthik Sridharan %A Yuchen Wu %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-cao26b %I PMLR %P 11099--11119 %U https://proceedings.mlr.press/v306/cao26b.html %V 306 %X We consider the robustness of score-based generative modeling to errors in the estimate of the score function. In particular, we show that Langevin dynamics is not robust to the $L^2$ errors (more generally $L^p$ errors) in the estimate of the score function. It is well-established that with small $L^2$ errors in the estimate of the score function, diffusion models can sample faithfully from the target distribution under fairly mild regularity assumptions in a polynomial time horizon. In contrast, our work shows that even for simple distributions in high dimensions, Langevin dynamics run for any polynomial time horizon will produce a distribution far from the target distribution in Total Variation (TV) distance, even when the $L^2$ error (more generally $L^p$) of the estimate of the score function is arbitrarily small. Considering such an error in the estimate of the score function is unavoidable in practice when learning the score function from data, our results provide further justification for diffusion models over Langevin dynamics and serve to caution against the use of Langevin dynamics with estimated scores.
APA
Cao, D.Y., Chen, A.Y., Sridharan, K. & Wu, Y.. (2026). On the Robustness of Langevin Dynamics to Score Function Error. Proceedings of the 43rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 306:11099-11119 Available from https://proceedings.mlr.press/v306/cao26b.html.

Related Material