Exponential integrator achieves dimension-uniform bounds for Langevin samplers

Diffusion-based samplers struggle in high-dimensional spaces because discretization errors accumulate across high-frequency coordinates, causing instability. A new study from Baldassari, Garnier, Solna, and de Hoop tackles this by analyzing preconditioned annealed Langevin dynamics (ALD) for Gaussian mixtures. They show that the standard Euler-Maruyama (EM) discretization treats the stiff linear part of the annealed score with a forward Euler step, forcing a stability constraint that couples the preconditioner with the annealed covariance scale. This constraint requires the initial smoothed law to be uniformly close to the target across dimensions, limiting practical application.

The researchers then present an exponential-integrator scheme that integrates the stiff linear part exactly. Under explicit spectral summability conditions linking the smoothing covariance, component covariance spectra, and preconditioner, they prove a dimension-uniform Kullback-Leibler (KL) bound. This bound can be made arbitrarily small by allowing enough annealing time and refining the time mesh, regardless of dimension. Importantly, the conditions allow regimes where the KL divergence between target and initial smoothed law diverges with dimension, demonstrating that EM's restrictions are scheme-dependent, not intrinsic to ALD. This work provides a theoretical foundation for stable high-dimensional sampling.