Optimizing Irreversible Perturbations of the Unadjusted Langevin Algorithm
For practitioners of Markov chain Monte Carlo, this work provides a principled way to design irreversible perturbations that accelerate convergence while controlling bias, addressing a gap in non-Gaussian and discretized settings.
This paper develops a framework for optimizing position-independent irreversible perturbations of the unadjusted Langevin algorithm (ULA), balancing mixing efficiency and discretization bias. The optimal perturbation is derived explicitly and shown to improve mean squared estimation errors in numerical experiments.
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.