Dimension-Free Multimodal Sampling via Preconditioned Annealed Langevin Dynamics

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.

💡 Research Summary

The paper addresses the long‑standing challenge of sampling from high‑dimensional multimodal distributions in a way that remains stable as the dimension grows. The authors focus on target distributions that can be well‑approximated by Gaussian mixture models (GMMs) and study a continuous‑time version of annealed Langevin dynamics (ALD), a method that progressively removes Gaussian smoothing from an initially highly regularized distribution.

The key contributions are threefold. First, the authors introduce a precise annealing schedule: for each truncation dimension d they define a smoothing covariance C_d that is diagonal in a fixed orthonormal basis with eigenvalues λ_j that form a summable sequence. The annealed target at time t∈