# Underdamped Langevin MCMC: A non-asymptotic analysis

arXiv: https://arxiv.org/abs/1707.03663 · HTML: https://mlsanity.com/paper/1707.03663/

## Key facts

- Cited by 133 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Jul 2017.

## Built on by (most cited first, 30 of 133)

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- [Faster high-accuracy log-concave sampling via algorithmic warm starts](https://mlsanity.com/paper/2302.10249/) (2023)
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- [Model-Based Diffusion for Trajectory Optimization](https://mlsanity.com/paper/2407.01573/) (2024)
- [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)
- [Beyond Log-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distance](https://mlsanity.com/paper/2501.02298/) (2025)
- [Sampling with Riemannian Hamiltonian Monte Carlo in a Constrained Space](https://mlsanity.com/paper/2202.01908/) (2022)
- [Improved Discretization Analysis for Underdamped Langevin Monte Carlo](https://mlsanity.com/paper/2302.08049/) (2023)
- [Particle algorithms for maximum likelihood training of latent variable models](https://mlsanity.com/paper/2204.12965/) (2022)
- [Langevin Diffusion Variational Inference](https://mlsanity.com/paper/2208.07743/) (2022)
- [Sqrt(d) Dimension Dependence of Langevin Monte Carlo](https://mlsanity.com/paper/2109.03839/) (2021)
- [A Proximal Algorithm for Sampling from Non-smooth Potentials](https://mlsanity.com/paper/2110.04597/) (2021)
- [A Proximal Algorithm for Sampling](https://mlsanity.com/paper/2202.13975/) (2022)
- [Resolving the Mixing Time of the Langevin Algorithm to its Stationary Distribution for Log-Concave Sampling](https://mlsanity.com/paper/2210.08448/) (2022)
- [A Convergence Theory for SVGD in the Population Limit under Talagrand's Inequality T1](https://mlsanity.com/paper/2106.03076/) (2021)
- [Lower Bounds on Metropolized Sampling Methods for Well-Conditioned Distributions](https://mlsanity.com/paper/2106.05480/) (2021)
- [Renormalizing Diffusion Models](https://mlsanity.com/paper/2308.12355/) (2023)
- [Lazy OCO: Online Convex Optimization on a Switching Budget](https://mlsanity.com/paper/2102.03803/) (2021)
- [On Convergence of Federated Averaging Langevin Dynamics](https://mlsanity.com/paper/2112.05120/) (2021)
- [Chain of Log-Concave Markov Chains](https://mlsanity.com/paper/2305.19473/) (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)
- [Fisher information lower bounds for sampling](https://mlsanity.com/paper/2210.02482/) (2022)

## Abstract (arXiv)

We study the underdamped Langevin diffusion when the log of the target distribution is smooth and strongly concave. We present a MCMC algorithm based on its discretization and show that it achieves varepsilon error (in 2-Wasserstein distance) in O(√d/varepsilon) steps. This is a significant improvement over the best known rate for overdamped Langevin MCMC, which is O(d/varepsilon2) steps under the same smoothness/concavity assumptions. The underdamped Langevin MCMC scheme can be viewed as a version of Hamiltonian Monte Carlo (HMC) which has been observed to outperform overdamped Langevin MCMC methods in a number of application areas. We provide quantitative rates that support this empirical wisdom.