Optimizing Irreversible Perturbations of the Unadjusted Langevin Algorithm

2026-06-03

AI Summary

This work addresses the slow convergence and discretization bias of the Unadjusted Langevin Algorithm under non-Gaussian target distributions by proposing a joint optimization framework that simultaneously enhances mixing efficiency and controls discretization error while preserving the invariant measure. Performance is quantified via spectral gap–type metrics and weighted expected squared jumping distance, enabling the first explicit derivation of an optimal position-independent irreversible perturbation that unifies the trade-off between acceleration and accuracy. The theoretical analysis integrates constrained optimization, spectral theory, and irreversible Markov chain theory. Numerical experiments confirm that the designed perturbations significantly accelerate convergence, effectively suppress bias, and reduce mean squared estimation error.

Abstract

Irreversible perturbations accelerate the convergence of Langevin dynamics, breaking detailed balance while preserving the invariant measure. The design of optimal irreversible perturbations has been studied in the continuous-time Gaussian setting, but extensions to non-Gaussian target distributions, and the impact of time discretization on the design of optimal perturbations, have not been well understood. Numerical discretizations of Langevin dynamics introduce bias, which is typically exacerbated by irreversible perturbations; handling this interaction demands a joint treatment of acceleration and accuracy. This paper develops a systematic framework for optimizing position-independent irreversible perturbations of the unadjusted Langevin algorithm (ULA). We formulate a constrained optimization problem that simultaneously accounts for mixing efficiency and discretization bias, where the former is characterized by a spectral gap analogue and the latter is quantified via a weighted expected squared jump distance. Within this framework, we derive an explicit characterization of the optimal position-independent irreversible perturbation. Extensive numerical experiments demonstrate that our design yields faster convergence with controlled bias, and improves mean squared estimation errors compared to other choices of irreversible perturbation.

Problem

• irreversible perturbations
• unadjusted Langevin algorithm
• discretization bias
• non-Gaussian target distributions
• mixing efficiency

Innovation

• irreversible perturbation
• unadjusted Langevin algorithm
• discretization bias
• spectral gap
• constrained optimization