Dimension-Free Multimodal Sampling via Preconditioned Annealed Langevin Dynamics

Designing sampling algorithms for multimodal targets that remain stable under refinement of the finite-dimensional approximation of an underlying function-space problem is a central challenge. Annealed Langevin dynamics (ALD) is a natural alternative to classical Langevin in this context, since it is often observed to improve exploration across modes. Yet a gap remains between its empirical success and existing theory: under which conditions can ALD be guaranteed to remain stable across dimensions? In this paper, we bridge this gap by providing a uniform-in-dimension analysis of continuous-time ALD for Gaussian-mixture targets. Along an explicit annealing path obtained by gradually removing Gaussian smoothing from the target, we identify spectral conditions linking the smoothing covariance to the component covariances under which ALD achieves a prescribed accuracy in Kullback-Leibler divergence within a dimension-uniform time horizon. We then establish stability in a perturbative regime with imperfect initialization and approximate scores. Under a misspecified-mixture score model, we show that preconditioning ALD with an operator whose spectrum decays sufficiently fast prevents error terms from accumulating across coordinates and thereby preserves dimension-uniform control.

Sampling from a complex distribution is like exploring a landscape with several deep valleys. As we represent a problem in finer detail, the number of quantities that must be described can grow dramatically, leading to what mathematicians call a high-dimensional distribution. In such settings, a standard exploration strategy may stay trapped in one valley and miss the others. We study a method that first blurs the landscape, making it easier to move between valleys, and then gradually sharpens it back to the original target. We prove that this strategy can remain stable as the level of detail increases, provided the smoothing and the way the sampler scales its motion in different directions are chosen carefully. Our results give mathematical guidance for designing samplers whose performance remains reliable as problems are described in finer detail.