Dimension-Free Multimodal Sampling via Preconditioned Annealed Langevin Dynamics

TL;DR

This paper provides a theoretical analysis of annealed Langevin dynamics (ALD) for sampling from high-dimensional multimodal distributions, establishing conditions for its stability and accuracy that are uniform across dimensions.

Contribution

It offers the first dimension-uniform analysis of ALD for Gaussian mixture models, identifying spectral conditions for stability and robustness to initialization and score approximation.

Findings

ALD achieves prescribed accuracy within a dimension-uniform time horizon.

Preconditioning ALD with a decaying spectrum is necessary for robustness.

Numerical experiments validate the theoretical conditions.

Abstract

Designing algorithms that can explore multimodal target distributions accurately across successive refinements of an underlying high-dimensional problem is a central challenge in sampling. Annealed Langevin dynamics (ALD) is a widely used alternative to classical Langevin since it often yields much faster mixing on multimodal targets, but there is still a gap between this empirical success and existing theory: when, and under which design choices, can ALD be guaranteed to remain stable as dimension increases? In this paper, we help bridge this gap by providing a uniform-in-dimension analysis of continuous-time ALD for multimodal targets that can be well-approximated by Gaussian mixture models. Along an explicit annealing path obtained by progressively removing Gaussian smoothing of the target, we identify sufficient spectral conditions - linking smoothing covariance and the covariances of the Gaussian components of the mixture - under which ALD achieves a prescribed accuracy within a single, dimension-uniform time horizon. We then establish dimension-robustness to imperfect initialization and score approximation: under a misspecified-mixture score model, we derive explicit conditions showing that preconditioning the ALD algorithm with a sufficiently decaying spectrum is necessary to prevent error terms from accumulating across coordinates and destroying dimension-uniform control. Finally, numerical experiments illustrate and validate the theory.