Time-Inhomogeneous Preconditioned Langevin Dynamics

• URL: http://arxiv.org/abs/2605.06091v1
• Date: Thu, 07 May 2026 12:12:22 GMT
• Title: Time-Inhomogeneous Preconditioned Langevin Dynamics
• Abstract summary: Langevin sampling from distributions of the form $p(x) propto exp(-(x))$ faces two major challenges.<n>The first challenge is relevant for multi-modal distributions with disjoint modes, whereas the second arises when the potential $$ exhibits diverse and ill-conditioned local mode geometry.<n>We propose the TIPreL, which introduces a time- and position-dependent preconditioner.
• Score: 45.12207971054227
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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