Kinetic Langevin Splitting Schemes for Constrained Sampling

TL;DR

This paper introduces new kinetic Langevin splitting algorithms for constrained sampling, offering improved efficiency and theoretical convergence guarantees, validated through numerical experiments.

Contribution

The paper develops novel splitting schemes for kinetic Langevin dynamics that enhance constrained sampling efficiency and provide theoretical convergence analysis.

Findings

Favorable strong order (bias) rates of the proposed schemes

Theoretical Wasserstein contraction and convergence results

Improved complexity bounds over existing methods

Abstract

Constrained sampling is an important and challenging task in computational statistics, concerned with generating samples from a distribution under certain constraints. There are numerous types of algorithm aimed at this task, ranging from general Markov chain Monte Carlo, to unadjusted Langevin methods. In this article we propose a series of new sampling algorithms based on the latter of these, specifically the kinetic Langevin dynamics. Our series of algorithms are motivated on advanced numerical methods which are splitting order schemes, which include the BU and BAO families of splitting schemes.Their advantage lies in the fact that they have favorable strong order (bias) rates and computationally efficiency. In particular we provide a number of theoretical insights which include a Wasserstein contraction and convergence results. We are able to demonstrate favorable results, such as improved complexity bounds over existing non-splitting methodologies. Our results are verified through numerical experiments on a range of models with constraints, which include a toy example and Bayesian linear regression.