Contraction and Convergence Rates for Discretized Kinetic Langevin Dynamics

TL;DR

This paper analyzes the convergence rates of discretized kinetic Langevin dynamics for convex potentials, providing explicit stepsize conditions and introducing the concept of gamma-limit convergence to characterize underdamped schemes.

Contribution

It introduces a framework for convergence analysis with explicit rates, applies it to popular schemes, and defines the gamma-limit convergent property for Langevin methods.

Findings

Convergence rates of O(m/M) with explicit stepsize restrictions.

Identification of gamma-limit convergent schemes that are robust in high-friction limits.

Asymptotic bias estimates for the BAOAB scheme that remain accurate at high friction.

Abstract

We provide a framework to analyze the convergence of discretized kinetic Langevin dynamics for $M$-$\nabla$Lipschitz, $m$-convex potentials. Our approach gives convergence rates of $\mathcal{O}(m/M)$, with explicit stepsize restrictions, which are of the same order as the stability threshold for Gaussian targets and are valid for a large interval of the friction parameter. We apply this methodology to various integration schemes which are popular in the molecular dynamics and machine learning communities. Further, we introduce the property ``$\gamma$-limit convergent" (GLC) to characterize underdamped Langevin schemes that converge to overdamped dynamics in the high-friction limit and which have stepsize restrictions that are independent of the friction parameter; we show that this property is not generic by exhibiting methods from both the class and its complement. Finally, we provide asymptotic bias estimates for the BAOAB scheme, which remain accurate in the high-friction limit by comparison to a modified stochastic dynamics which preserves the invariant measure.