Non-asymptotic error bounds for scaled underdamped Langevin MCMC

TL;DR

This paper investigates how scaling the underdamped Langevin MCMC algorithm can improve convergence error bounds, especially in relation to the condition number of the target density, by revisiting and refining existing non-asymptotic bounds.

Contribution

It introduces specific scaling conditions in the underdamped Langevin equation that enhance error bounds based on the density's condition number, advancing theoretical understanding.

Findings

Scaling improves error bounds in Langevin MCMC

Conditions identified for optimal scaling based on the condition number

Enhanced theoretical guarantees for convergence rates

Abstract

Recent works have derived non-asymptotic upper bounds for convergence of underdamped Langevin MCMC. We revisit these bound and consider introducing scaling terms in the underlying underdamped Langevin equation. In particular, we provide conditions under which an appropriate scaling allows to improve the error bounds in terms of the condition number of the underlying density of interest.