Time-Inhomogeneous Preconditioned Langevin Dynamics

2026-05-07

AI Summary

This work addresses the challenge in Langevin sampling of simultaneously achieving comprehensive coverage of global modes and fine-grained exploration of local modes in multimodal distributions. The authors propose a time- and space-dependent preconditioned Langevin dynamics framework that dynamically adapts the preconditioning matrix within a unified mechanism to enhance both global exploration and local refinement. For the first time, a time-varying, position-dependent preconditioner is introduced, overcoming the traditional trade-off between global and local sampling performance. The method establishes convergence guarantees in Wasserstein-2 distance under more general conditions. Experimental results demonstrate that the proposed algorithm significantly outperforms existing preconditioned approaches on both two-dimensional ill-conditioned examples and high-dimensional Bayesian logistic regression tasks, exhibiting superior efficiency and robustness.

Abstract

Langevin sampling from distributions of the form $p(x) \propto \exp(-Ψ(x))$ faces two major challenges: (global) mode coverage and (local) mode exploration. The first challenge is particularly relevant for multi-modal distributions with disjoint modes, whereas the second arises when the potential $Ψ$ exhibits diverse and ill-conditioned local mode geometry. To address these challenges, a common approach is to precondition Langevin dynamics with problem-specific information, such as the sample covariance or the local curvature of $Ψ$. However, existing preconditioner choices inherently involve a trade-off between global mode coverage and local mode exploration, and no prior method resolves both simultaneously. To overcome this limitation, we propose the TIPreL, which introduces a time- and position-dependent preconditioner. This design effectively addresses both challenges mentioned above within a single framework. We establish convergence of the resulting dynamics in the Wasserstein-2 distance both in continuous time and for a tamed Euler discretization. In particular, our analysis extends the existing state of the art by proving convergence under time- and space-dependent diffusion coefficients, and only locally Lipschitz drifts, which has not been covered by prior work. Finally, we experimentally compare TIPreL with competing preconditioning schemes on a two-dimensional, severely ill-posed example and on a Bayesian logistic regression task in higher dimensions, confirming the efficiency of the proposed method.

Problem

• Langevin dynamics
• multi-modal distributions
• mode exploration
• mode coverage
• preconditioning

Innovation

• time-inhomogeneous preconditioning
• Langevin dynamics
• multi-modal sampling
• Wasserstein convergence
• adaptive diffusion