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Contraction and Convergence Rates for Discretized Kinetic Langevin Dynamics

Benedict Leimkuhler, Daniel Paulin, Peter A. Whalley

TL;DR

This work develops a general, contraction-based framework for the nonasymptotic convergence of discretized kinetic Langevin dynamics in Wasserstein distance, under mild $M$-gradient-Lipschitz and $m$-strong convexity assumptions on the potential. It provides explicit step-size restrictions and contraction rates ${\cal O}(m/M)$ for a range of integrators, including EM, first- and higher-order splittings, and SES, with a detailed treatment of the high-friction limit. A novel $\gamma$-limit-convergent (GLC) property is introduced to distinguish integrators that converge to overdamped dynamics as $\gamma\to\infty$ from those that do not, with BAOAB and OBABO identified as GLC. The paper also gives asymptotic bias estimates for BAOAB that remain accurate in the high-friction regime by comparing to an invariant-measure-preserving HOH scheme, reinforcing BAOAB’s practicality for robust sampling. Overall, the results inform principled choices of discretization schemes for kinetic Langevin sampling in molecular dynamics and machine learning contexts, balancing stability, accuracy, and computational cost.

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 ``$γ$-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.

Contraction and Convergence Rates for Discretized Kinetic Langevin Dynamics

TL;DR

This work develops a general, contraction-based framework for the nonasymptotic convergence of discretized kinetic Langevin dynamics in Wasserstein distance, under mild -gradient-Lipschitz and -strong convexity assumptions on the potential. It provides explicit step-size restrictions and contraction rates for a range of integrators, including EM, first- and higher-order splittings, and SES, with a detailed treatment of the high-friction limit. A novel -limit-convergent (GLC) property is introduced to distinguish integrators that converge to overdamped dynamics as from those that do not, with BAOAB and OBABO identified as GLC. The paper also gives asymptotic bias estimates for BAOAB that remain accurate in the high-friction regime by comparing to an invariant-measure-preserving HOH scheme, reinforcing BAOAB’s practicality for robust sampling. Overall, the results inform principled choices of discretization schemes for kinetic Langevin sampling in molecular dynamics and machine learning contexts, balancing stability, accuracy, and computational cost.

Abstract

We provide a framework to analyze the convergence of discretized kinetic Langevin dynamics for -Lipschitz, -convex potentials. Our approach gives convergence rates of , 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 ``-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.
Paper Structure (28 sections, 12 theorems, 123 equations, 1 figure, 1 table)

This paper contains 28 sections, 12 theorems, 123 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Assumptions and definitions
  3. Assumptions on $U$
  4. Modified Euclidean Norms
  5. Wasserstein Distance
  6. Overdamped Langevin discretizations and contraction
  7. Convergence guarantees
  8. Kinetic Langevin Dynamics
  9. Discretization schemes
  10. The Euler-Maruyama method
  11. Splitting Methods
  12. The stochastic exponential Euler method
  13. Proof Strategy
  14. Convergence results
  15. Euler-Maruyama discretization
  16. ...and 13 more sections

Key Result

Proposition 2.4

\newlabelprop:Wasserstein0 Assume a numerical scheme for kinetic Langevin dynamics with a $m$-strongly convex $M$-$\nabla$Lipschitz potential $U$ and transition kernel $P_{h}$. Let $\left(x_{n},v_{n}\right)$ and $\left(\Tilde{x}_{n},\Tilde{v}_{n}\right)$ be two synchronously coupled chains of the for $\gamma^{2} \geq C_{\gamma}M$ and $h \leq C_{h}\left(\gamma,\sqrt{M}\right)$ for some $a,b >0$ s

Figures (1)

  • Figure 1: Contraction of two kinetic Langevin trajectories $x_{1}$ and $x_{2}$ with initial conditions $[-1,-1]$ and $[1,1]$ for a $2$-dimensional standard Gaussian with stepsize $h = 0.25 = 1/4\sqrt{M}$.

Theorems & Definitions (36)

  • Proposition 2.4
  • Proof 1
  • Example 4.1
  • Proposition 4.2
  • Proof 2
  • Remark 4.3
  • Theorem 4.4
  • Example 4.5
  • Theorem 4.6: BAO
  • Remark 4.7
  • ...and 26 more