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

2026-05-19 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences · Depth: Expert, extended

Summary

This research analyzes preconditioned annealed Langevin dynamics (ALD) for multimodal Gaussian mixtures in high- and infinite-dimensional settings, addressing challenges of error accumulation during finite-dimensional approximations. The study compares two discretization schemes: Euler-Maruyama (EM) and an exact-linear-part (ELP) exponential integrator. It demonstrates that EM discretization, by using a forward Euler step for the stiff linear part of the annealed score, imposes a stability constraint that forces the initial smoothed law to remain uniformly close to the target across dimensions. In contrast, the ELP scheme, which integrates the stiff linear part exactly, achieves dimension-uniform Kullback-Leibler (KL) bounds under explicit spectral summability conditions. These conditions are less restrictive, allowing the initial smoothed law to diverge from the target as dimension increases. Numerical experiments confirm EM's instability in high-frequency coordinates and ELP's robustness.

Key takeaway

Research Scientists working on high-dimensional sampling problems with annealed Langevin dynamics should prioritize discretization schemes like the Exact-Linear-Part (ELP) exponential integrator over Euler-Maruyama. The ELP scheme offers provable dimension-uniform Kullback-Leibler bounds, even when the initial smoothed distribution is far from the target, by effectively managing high-frequency stiffness. This approach is critical for maintaining stability and accuracy as finite-dimensional approximations are refined, preventing error accumulation that destabilizes EM.

Key insights

Dimension-uniform control in annealed Langevin dynamics requires specific discretization and preconditioning for stability.

Principles

• Discretization errors accumulate in high-frequency coordinates.
• Preconditioners balance acceleration and error damping.
• Scheme-dependent restrictions impact dimension-uniform guarantees.

Method

The Exact-Linear-Part (ELP) scheme integrates the stiff linear part of the annealed score exactly, freezing the nonlinear term, to achieve dimension-uniform KL bounds.

In practice

• Avoid Euler-Maruyama for high-dimensional ALD with stiff linear parts.
• Use ELP schemes for dimension-uniform KL control.
• Balance preconditioner damping with annealing efficiency.

Topics

• Preconditioned Annealed Langevin Dynamics
• Gaussian Mixtures
• Dimension-Uniform Control
• Euler-Maruyama Discretization
• Exact-Linear-Part Scheme

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.