Rapid convergence of Langevin AIS for multimodal distributions on ℝd
Abstract
We study annealed importance sampling (AIS) driven by the overdamped Langevin diffusion, targeting a multimodal Gibbs measure πε ∝ e−U/ε on ℝd at low temperature ε. For a potential U with nondegenerate critical points, finitely many wells of comparable mass, and at least quadratic growth at infinity, we show that AIS with a harmonically spaced temperature schedule produces a weighted empirical measure that approximates πε to accuracy δ at a total cost of order ε−2δ−2, up to logarithmic factors, and the exponents do not depend on the dimension. This is an exponential improvement over plain Langevin Monte Carlo, whose relaxation time obeys the Arrhenius law. The assumptions cover, in particular, the potentials of Gaussian mixtures, as long as their critical points are nondegenerate and their wells carry comparable mass. The analysis extends that of arXiv:2604.17526 [math.PR] from the torus 𝕋d to the noncompact state space ℝd. The two new ingredients are a weighted L∞ bound on the eigenfunctions of the Langevin generator, obtained by combining Agmon estimates at infinity with a local maximum principle near the critical points, and a Weyl-type eigenvalue count derived from the Cwikel-Lieb-Rozenblum and Lieb-Thirring inequalities. For potentials with more than two wells the eigenfunction bound requires an additional argument, which we give by constructing a perturbation of U that renders every saddle an effective local maximum.
Related
- Reversible Stratonovich perturbations for accelerating convergence to equilibrium of Langevin dynamics
- Initialization Stability and Convergence of Projected Langevin Dynamics
- The Fenchel Game of Underdamped Langevin Dynamics: Insights into Accelerated Convergence
- Accelerating Langevin Dynamics with Weighted Laplacian Flow: Global Well-Posedness and Exponential Convergence
- Global Convergence of Third-Order Langevin Dynamics for Non-Convex Optimization via Simulated Annealing
BibTeX
@misc{son2026rapid,
title = {{Rapid convergence of Langevin AIS for multimodal distributions on $\mathbbR^d$}},
author = {Seungjae Son},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24567},
eprint = {2610.24567},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24567},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Rapid convergence of Langevin AIS for multimodal distributions on ℝ^d
AU - Seungjae Son
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.24567
UR - https://arxiv.org/abs/2610.24567
AB - We study annealed importance sampling (AIS) driven by the overdamped Langevin diffusion, targeting a multimodal Gibbs measure π_ε ∝ e^−U/ε on ℝ^d at low temperature ε. For a potential U with nondegenerate critical points, finitely many wells of comparable mass, and at least quadratic growth at infinity, we show that AIS with a harmonically spaced temperature schedule produces a weighted empirical measure that approximates π_ε to accuracy δ at a total cost of order ε^−2δ^−2, up to logarithmic factors, and the exponents do not depend on the dimension. This is an exponential improvement over plain Langevin Monte Carlo, whose relaxation time obeys the Arrhenius law. The assumptions cover, in particular, the potentials of Gaussian mixtures, as long as their critical points are nondegenerate and their wells carry comparable mass. The analysis extends that of arXiv:2604.17526 [math.PR] from the torus 𝕋^d to the noncompact state space ℝ^d. The two new ingredients are a weighted L^∞ bound on the eigenfunctions of the Langevin generator, obtained by combining Agmon estimates at infinity with a local maximum principle near the critical points, and a Weyl-type eigenvalue count derived from the Cwikel-Lieb-Rozenblum and Lieb-Thirring inequalities. For potentials with more than two wells the eigenfunction bound requires an additional argument, which we give by constructing a perturbation of U that renders every saddle an effective local maximum.
ER -