Non-asymptotic Error Bounds in $\mathcalW_2$-Distance with Sqrt(d) Dimension Dependence and First Order Convergence for Langevin Monte Carlo beyond Log-Concavity

Bin Yang, Xiaojie Wang
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:71358-71382, 2025.

Abstract

Generating samples from a high dimensional probability distribution is a fundamental task with wide-ranging applications in the area of scientific computing, statistics and machine learning. This article revisits the popular Langevin Monte Carlo (LMC) sampling algorithms and provides a non-asymptotic error analysis in $\mathcal{W}_2$-distance in a non-convex setting. In particular, we prove an error bound $O(\sqrt{d} h)$, which guarantees a mixing time $ \tilde{O} (\sqrt{d} \epsilon^{-1})$ to achieve the accuracy tolerance $\epsilon$, under certain log-smooth conditions and the assumption that the target distribution satisfies a log-Sobolev inequality, as opposed to the strongly log-concave condition used in (Li et al., 2019; 2022). This bound matches the best one in the strongly log-concave case and improves upon the best-known convergence rates in non-convex settings. To prove it, we establish a new framework of uniform-in-time convergence for discretizations of SDEs. Distinct from (Li et al., 2019; 2022), we start from the finite-time mean-square fundamental convergence theorem, which combined with uniform-in-time moment bounds of LMC and the exponential ergodicity of SDEs in the non-convex setting gives the desired uniform-in-time convergence. Our framework also applies to the case when the gradient of the potential $U$ is non-globally Lipschitz with superlinear growth, for which modified LMC samplers are proposed and analyzed, with a non-asymptotic error bound in $\mathcal{W}_2$-distance obtained. Numerical experiments corroborate the theoretical analysis.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-yang25al, title = {Non-asymptotic Error Bounds in $\mathcal{W}_2$-Distance with Sqrt(d) Dimension Dependence and First Order Convergence for {L}angevin {M}onte {C}arlo beyond Log-Concavity}, author = {Yang, Bin and Wang, Xiaojie}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {71358--71382}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/yang25al/yang25al.pdf}, url = {https://proceedings.mlr.press/v267/yang25al.html}, abstract = {Generating samples from a high dimensional probability distribution is a fundamental task with wide-ranging applications in the area of scientific computing, statistics and machine learning. This article revisits the popular Langevin Monte Carlo (LMC) sampling algorithms and provides a non-asymptotic error analysis in $\mathcal{W}_2$-distance in a non-convex setting. In particular, we prove an error bound $O(\sqrt{d} h)$, which guarantees a mixing time $ \tilde{O} (\sqrt{d} \epsilon^{-1})$ to achieve the accuracy tolerance $\epsilon$, under certain log-smooth conditions and the assumption that the target distribution satisfies a log-Sobolev inequality, as opposed to the strongly log-concave condition used in (Li et al., 2019; 2022). This bound matches the best one in the strongly log-concave case and improves upon the best-known convergence rates in non-convex settings. To prove it, we establish a new framework of uniform-in-time convergence for discretizations of SDEs. Distinct from (Li et al., 2019; 2022), we start from the finite-time mean-square fundamental convergence theorem, which combined with uniform-in-time moment bounds of LMC and the exponential ergodicity of SDEs in the non-convex setting gives the desired uniform-in-time convergence. Our framework also applies to the case when the gradient of the potential $U$ is non-globally Lipschitz with superlinear growth, for which modified LMC samplers are proposed and analyzed, with a non-asymptotic error bound in $\mathcal{W}_2$-distance obtained. Numerical experiments corroborate the theoretical analysis.} }
Endnote
%0 Conference Paper %T Non-asymptotic Error Bounds in $\mathcalW_2$-Distance with Sqrt(d) Dimension Dependence and First Order Convergence for Langevin Monte Carlo beyond Log-Concavity %A Bin Yang %A Xiaojie Wang %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-yang25al %I PMLR %P 71358--71382 %U https://proceedings.mlr.press/v267/yang25al.html %V 267 %X Generating samples from a high dimensional probability distribution is a fundamental task with wide-ranging applications in the area of scientific computing, statistics and machine learning. This article revisits the popular Langevin Monte Carlo (LMC) sampling algorithms and provides a non-asymptotic error analysis in $\mathcal{W}_2$-distance in a non-convex setting. In particular, we prove an error bound $O(\sqrt{d} h)$, which guarantees a mixing time $ \tilde{O} (\sqrt{d} \epsilon^{-1})$ to achieve the accuracy tolerance $\epsilon$, under certain log-smooth conditions and the assumption that the target distribution satisfies a log-Sobolev inequality, as opposed to the strongly log-concave condition used in (Li et al., 2019; 2022). This bound matches the best one in the strongly log-concave case and improves upon the best-known convergence rates in non-convex settings. To prove it, we establish a new framework of uniform-in-time convergence for discretizations of SDEs. Distinct from (Li et al., 2019; 2022), we start from the finite-time mean-square fundamental convergence theorem, which combined with uniform-in-time moment bounds of LMC and the exponential ergodicity of SDEs in the non-convex setting gives the desired uniform-in-time convergence. Our framework also applies to the case when the gradient of the potential $U$ is non-globally Lipschitz with superlinear growth, for which modified LMC samplers are proposed and analyzed, with a non-asymptotic error bound in $\mathcal{W}_2$-distance obtained. Numerical experiments corroborate the theoretical analysis.
APA
Yang, B. & Wang, X.. (2025). Non-asymptotic Error Bounds in $\mathcalW_2$-Distance with Sqrt(d) Dimension Dependence and First Order Convergence for Langevin Monte Carlo beyond Log-Concavity. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:71358-71382 Available from https://proceedings.mlr.press/v267/yang25al.html.

Related Material