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

arXiv: https://arxiv.org/abs/2510.03949 · HTML: https://mlsanity.com/paper/2510.03949/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 24 references that are in the map.
- First posted on arXiv 4 Oct 2025.

## Builds on (most cited first, 11 of 11)

- [High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1605.01559/) (2016)
- [Score-Based Diffusion meets Annealed Importance Sampling](https://mlsanity.com/paper/2208.07698/) (2022)
- [Underdamped Diffusion Bridges with Applications to Sampling](https://mlsanity.com/paper/2503.01006/) (2025)
- [Improved Discretization Analysis for Underdamped Langevin Monte Carlo](https://mlsanity.com/paper/2302.08049/) (2023)
- [Bounding the error of discretized Langevin algorithms for non-strongly log-concave targets](https://mlsanity.com/paper/1906.08530/) (2019)
- [Schr"odinger Bridge Samplers](https://mlsanity.com/paper/1912.13170/) (2019)
- [Kinetic Langevin MCMC Sampling Without Gradient Lipschitz Continuity -- the Strongly Convex Case](https://mlsanity.com/paper/2301.08039/) (2023)
- [Unbiased Kinetic Langevin Monte Carlo with Inexact Gradients](https://mlsanity.com/paper/2311.05025/) (2023)
- [Accelerating Approximate Thompson Sampling with Underdamped Langevin Monte Carlo](https://mlsanity.com/paper/2401.11665/) (2024)
- [Multimeasurement Generative Models](https://mlsanity.com/paper/2112.09822/) (2021)
- [Reflection coupling for unadjusted generalized Hamiltonian Monte Carlo in the nonconvex stochastic gradient case](https://mlsanity.com/paper/2310.18774/) (2023)

## Abstract (arXiv)

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.