Optimizing Irreversible Perturbations of the Unadjusted Langevin Algorithm

Abstract

A central task in Bayesian inference and scientific computing is to compute expectations with respect to probability distributions that are only known up to a normalizing constant. Markov chain Monte Carlo (MCMC) methods, and in particular Langevin dynamics, provide a powerful framework for this task by constructing stochastic processes that converge to the target distribution. However, practical implementations face two challenges: slow mixing when the target distribution is anisotropic or multimodal, and persistent discretization bias introduced by numerical schemes. This thesis investigates irreversible perturbations of overdamped Langevin dynamics, aiming to accelerate mixing while controlling discretization error. Irreversible perturbations introduce skew-symmetric drift terms that preserve the target distribution while inducing rotational flow, thereby enhancing exploration. Although prior work has established their benefits in continuous-time settings, the impact of discretization and the design of optimal perturbations for discrete-time algorithms remain open problems. We develop a framework for optimizing constant (position-independent) irreversible perturbations in the Unadjusted Langevin Algorithm (ULA). Our approach balances two competing objectives: maximizing the spectral gap of the continuous dynamics to accelerate convergence, and minimizing discretization error that drives estimation bias. Motivated by this, we introduce new criteria that jointly evaluate bias and efficiency, and we show how these criteria identify perturbations that improve performance beyond existing constructions. Theoretical analysis is complemented by numerical experiments on Gaussian and nonGaussian targets. These experiments demonstrate that appropriately designed irreversible perturbations can reduce mean-squared error without sacrificing stability, while poorly chosen perturbations can degrade performance. The results highlight the importance of geometry-aware design and motivate systematic optimization strategies for irreversible perturbations. Overall, this work extends the theoretical and practical understanding of irreversible Langevin dynamics, bridging the gap between continuous-time spectral analysis and discrete-time numerical performance. It provides principled tools for constructing efficient MCMC samplers, with potential applications in high-dimensional Bayesian inference and modern machine learning.