Analysis of kinetic Langevin Monte Carlo under the stochastic exponential Euler discretization from underdamped all the way to overdamped

TL;DR

This work analyzes the kinetic Langevin Monte Carlo (KLMC) method with the stochastic exponential Euler discretization across underdamped and overdamped regimes. By refining a synchronous Wasserstein coupling and exploiting a weighted norm, the authors establish a general contraction that holds under weaker parameter restrictions, and show that the exponential integrator remains stable in the overdamped limit when time is accelerated with $h \,\propto\,\gamma$. They derive explicit asymptotic-bias bounds that reveal distinct phase transitions around $h\gamma\approx1.69$ and provide nonasymptotic sampling complexity guarantees of $O(\kappa^{3/2} d^{1/2} \epsilon^{-1} \log(1/\epsilon))$ iterations to reach a prescribed Wasserstein accuracy. The results bridge the understanding of KLMC across regimes, align the overdamped behavior with Euler–Maruyama discretizations, and offer practical guidance for choosing discretization parameters. Altogether, the paper substantiates the overdamped viability of the exponential integrator and clarifies the bias-contraction trade-offs essential for efficient sampling in high dimensions.

Abstract

Simulating the kinetic Langevin dynamics is a popular approach for sampling from distributions, where only their unnormalized densities are available. Various discretizations of the kinetic Langevin dynamics have been considered, where the resulting algorithm is collectively referred to as the kinetic Langevin Monte Carlo (KLMC) or underdamped Langevin Monte Carlo. Specifically, the stochastic exponential Euler discretization, or exponential integrator for short, has previously been studied under strongly log-concave and log-Lipschitz smooth potentials via the synchronous Wasserstein coupling strategy. Existing analyses, however, impose restrictions on the parameters that do not explain the behavior of KLMC under various choices of parameters. In particular, all known results fail to hold in the overdamped regime, suggesting that the exponential integrator degenerates in the overdamped limit. In this work, we revisit the synchronous Wasserstein coupling analysis of KLMC with the exponential integrator. Our refined analysis results in Wasserstein contractions and bounds on the asymptotic bias that hold under weaker restrictions on the parameters, which assert that the exponential integrator is capable of stably simulating the kinetic Langevin dynamics in the overdamped regime, as long as proper time acceleration is applied.

Figures (4)

• Figure 1: \ref{['thm:general_contraction']}
• Figure 2: Effect of $h \gamma$ on \ref{['thm:general_contraction']}
• Figure 3: \ref{['thm:special_case_convergence_bound']}
• Figure 5: Scaling of the asymptotic error bound with respect to the step size $h$. The vertical dashed lines mark the point where $\zeta = h\gamma = 1.69$.

Theorems & Definitions (15)

• Corollary 3.1
• proof
• Corollary 3.2: Overdamped Limit
• Corollary 3.3: Overdamped Limit
• Corollary 3.4
• Lemma 5.1
• proof
• Lemma 5.2
• proof
• Lemma 5.3