# On sampling from a log-concave density using kinetic Langevin diffusions

arXiv: https://arxiv.org/abs/1807.09382 · HTML: https://mlsanity.com/paper/1807.09382/

## Key facts

- Cited by 62 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Jul 2018.

## Built on by (most cited first, 30 of 62)

- [Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions](https://mlsanity.com/paper/2209.11215/) (2022)
- [Variational inference via Wasserstein gradient flows](https://mlsanity.com/paper/2205.15902/) (2022)
- [Faster high-accuracy log-concave sampling via algorithmic warm starts](https://mlsanity.com/paper/2302.10249/) (2023)
- [Minimax Mixing Time of the Metropolis-Adjusted Langevin Algorithm for Log-Concave Sampling](https://mlsanity.com/paper/2109.13055/) (2021)
- [Towards a Complete Analysis of Langevin Monte Carlo: Beyond Poincar'e Inequality](https://mlsanity.com/paper/2303.03589/) (2023)
- [Improved dimension dependence of a proximal algorithm for sampling](https://mlsanity.com/paper/2302.10081/) (2023)
- [Improved Discretization Analysis for Underdamped Langevin Monte Carlo](https://mlsanity.com/paper/2302.08049/) (2023)
- [Sqrt(d) Dimension Dependence of Langevin Monte Carlo](https://mlsanity.com/paper/2109.03839/) (2021)
- [A Proximal Algorithm for Sampling](https://mlsanity.com/paper/2202.13975/) (2022)
- [Sampling via Gradient Flows in the Space of Probability Measures](https://mlsanity.com/paper/2310.03597/) (2023)
- [Lower Bounds on Metropolized Sampling Methods for Well-Conditioned Distributions](https://mlsanity.com/paper/2106.05480/) (2021)
- [Zeroth-Order Sampling Methods for Non-Log-Concave Distributions: Alleviating Metastability by Denoising Diffusion](https://mlsanity.com/paper/2402.17886/) (2024)
- [Exploring the Optimal Choice for Generative Processes in Diffusion Models: Ordinary vs Stochastic Differential Equations](https://mlsanity.com/paper/2306.02063/) (2023)
- [Sampling from Bayesian Neural Network Posteriors with Symmetric Minibatch Splitting Langevin Dynamics](https://mlsanity.com/paper/2410.19780/) (2024)
- [When is the Convergence Time of Langevin Algorithms Dimension Independent? A Composite Optimization Viewpoint](https://mlsanity.com/paper/2110.01827/) (2021)
- [Kinetic Langevin MCMC Sampling Without Gradient Lipschitz Continuity -- the Strongly Convex Case](https://mlsanity.com/paper/2301.08039/) (2023)
- [Query lower bounds for log-concave sampling](https://mlsanity.com/paper/2304.02599/) (2023)
- [Fast parallel sampling under isoperimetry](https://mlsanity.com/paper/2401.09016/) (2024)
- [Langevin Monte Carlo for strongly log-concave distributions: Randomized midpoint revisited](https://mlsanity.com/paper/2306.08494/) (2023)
- [Parallelized Midpoint Randomization for Langevin Monte Carlo](https://mlsanity.com/paper/2402.14434/) (2024)
- [Wasserstein distance estimates for the distributions of numerical approximations to ergodic stochastic differential equations](https://mlsanity.com/paper/2104.12384/) (2021)
- [Sampling from Log-Concave Distributions with Infinity-Distance Guarantees](https://mlsanity.com/paper/2111.04089/) (2021)
- [Unadjusted Hamiltonian MCMC with Stratified Monte Carlo Time Integration](https://mlsanity.com/paper/2211.11003/) (2022)
- [Accelerating Hamiltonian Monte Carlo via Chebyshev Integration Time](https://mlsanity.com/paper/2207.02189/) (2022)
- [Kinetic Interacting Particle Langevin Monte Carlo](https://mlsanity.com/paper/2407.05790/) (2024)
- [Solving Linear Inverse Problems using Higher-Order Annealed Langevin Diffusion](https://mlsanity.com/paper/2305.05014/) (2023)
- [A Separation in Heavy-Tailed Sampling: Gaussian vs. Stable Oracles for Proximal Samplers](https://mlsanity.com/paper/2405.16736/) (2024)
- [The Poisson Midpoint Method for Langevin Dynamics: Provably Efficient Discretization for Diffusion Models](https://mlsanity.com/paper/2405.17068/) (2024)
- [Wasserstein Convergence of Critically Damped Langevin Diffusions](https://mlsanity.com/paper/2511.02419/) (2025)
- [Mixing of the No-U-Turn Sampler and the Geometry of Gaussian Concentration](https://mlsanity.com/paper/2410.06978/) (2024)

## Abstract (arXiv)

Langevin diffusion processes and their discretizations are often used for sampling from a target density. The most convenient framework for assessing the quality of such a sampling scheme corresponds to smooth and strongly log-concave densities defined on mathbb Rp. The present work focuses on this framework and studies the behavior of Monte Carlo algorithms based on discretizations of the kinetic Langevin diffusion. We first prove the geometric mixing property of the kinetic Langevin diffusion with a mixing rate that is, in the overdamped regime, optimal in terms of its dependence on the condition number. We then use this result for obtaining improved guarantees of sampling using the kinetic Langevin Monte Carlo method, when the quality of sampling is measured by the Wasserstein distance. We also consider the situation where the Hessian of the log-density of the target distribution is Lipschitz-continuous. In this case, we introduce a new discretization of the kinetic Langevin diffusion and prove that this leads to a substantial improvement of the upper bound on the sampling error measured in Wasserstein distance.