Dimension-Uniform Discretization Analysis of Preconditioned Annealed Langevin Dynamics for Multimodal Gaussian Mixtures

2026-05-15

AI Summary

This work addresses the challenge of discretization-induced instability in high- and infinite-dimensional diffusion sampling, where errors accumulate in high-frequency coordinates and disrupt dynamics. Focusing on preconditioned annealed Langevin dynamics for Gaussian mixture targets, the study demonstrates that the stability limitations of the Euler–Maruyama scheme stem from its numerical formulation rather than the underlying dynamics. The authors propose an exponential integrator that achieves dimension-independent KL divergence bounds under mild spectral summability conditions. Rigorous analysis shows that, given explicit spectral assumptions, this integrator guarantees arbitrarily small KL error upper bounds that remain uniform across dimensions—even when the KL divergence between the initial and target distributions diverges as dimensionality increases.

Abstract

Obtaining stable diffusion-based samplers in high- and infinite-dimensional settings is challenging because errors can accumulate across high-frequency coordinates and make the dynamics unstable under refinement of the finite-dimensional approximation of the underlying function-space problem. Discretization is a typical source of such errors, and preconditioning with a suitable spectral decay is one way to control their accumulation. In this paper, we study this problem for preconditioned annealed Langevin dynamics (ALD) applied to Gaussian mixtures. We first show that Euler-Maruyama (EM) discretization, by treating the stiff linear part of the annealed score with a forward Euler step, imposes a stability constraint coupling the preconditioner with the annealed covariance scale. Together with the conditions ensuring dimension-uniform control of the annealed dynamics, this constraint forces the initial smoothed law to remain uniformly close to the target across dimensions. We then consider an exponential-integrator scheme that integrates the stiff linear part of the annealed score exactly. Under explicit spectral summability conditions coupling the smoothing covariance, the component covariance spectra, and the preconditioner, we prove a dimension-uniform Kullback-Leibler (KL) bound for this scheme. This bound can be made arbitrarily small, uniformly in dimension, by allowing enough time for annealing and then refining the time mesh accordingly. Importantly, these conditions allow regimes in which the KL divergence between the target and the initial smoothed law diverges with dimension, showing that the restrictions imposed by EM are scheme-dependent rather than intrinsic to ALD.

Problem

• dimension-uniform
• discretization
• preconditioned annealed Langevin dynamics
• Gaussian mixtures
• stability

Innovation

• preconditioned annealed Langevin dynamics
• dimension-uniform discretization
• exponential integrator
• Gaussian mixtures
• KL divergence