# High-Order Langevin Diffusion Yields an Accelerated MCMC Algorithm

arXiv: https://arxiv.org/abs/1908.10859 · HTML: https://mlsanity.com/paper/1908.10859/

## Key facts

- Cited by 36 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Aug 2019.

## Built on by (most cited first, 30 of 36)

- [Score-Based Generative Modeling with Critically-Damped Langevin Diffusion](https://mlsanity.com/paper/2112.07068/) (2021)
- [Estimating High Order Gradients of the Data Distribution by Denoising](https://mlsanity.com/paper/2111.04726/) (2021)
- [Private Convex Optimization via Exponential Mechanism](https://mlsanity.com/paper/2203.00263/) (2022)
- [Fast Solvers for Discrete Diffusion Models: Theory and Applications of High-Order Algorithms](https://mlsanity.com/paper/2502.00234/) (2025)
- [Minimax Mixing Time of the Metropolis-Adjusted Langevin Algorithm for Log-Concave Sampling](https://mlsanity.com/paper/2109.13055/) (2021)
- [Reverse Diffusion Monte Carlo](https://mlsanity.com/paper/2307.02037/) (2023)
- [On the Posterior Distribution in Denoising: Application to Uncertainty Quantification](https://mlsanity.com/paper/2309.13598/) (2023)
- [Where to Diffuse, How to Diffuse, and How to Get Back: Automated Learning for Multivariate Diffusions](https://mlsanity.com/paper/2302.07261/) (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)
- [Underdamped Diffusion Bridges with Applications to Sampling](https://mlsanity.com/paper/2503.01006/) (2025)
- [Sqrt(d) Dimension Dependence of Langevin Monte Carlo](https://mlsanity.com/paper/2109.03839/) (2021)
- [Lower Bounds on Metropolized Sampling Methods for Well-Conditioned Distributions](https://mlsanity.com/paper/2106.05480/) (2021)
- [Heavy-tailed denoising score matching](https://mlsanity.com/paper/2112.09788/) (2021)
- [A connection between Tempering and Entropic Mirror Descent](https://mlsanity.com/paper/2310.11914/) (2023)
- [When does Metropolized Hamiltonian Monte Carlo provably outperform Metropolis-adjusted Langevin algorithm?](https://mlsanity.com/paper/2304.04724/) (2023)
- [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)
- [When is the Convergence Time of Langevin Algorithms Dimension Independent? A Composite Optimization Viewpoint](https://mlsanity.com/paper/2110.01827/) (2021)
- [Langevin Monte Carlo for strongly log-concave distributions: Randomized midpoint revisited](https://mlsanity.com/paper/2306.08494/) (2023)
- [Wasserstein distance estimates for the distributions of numerical approximations to ergodic stochastic differential equations](https://mlsanity.com/paper/2104.12384/) (2021)
- [Hamiltonian Monte Carlo for efficient Gaussian sampling: long and random steps](https://mlsanity.com/paper/2209.12771/) (2022)
- [Solving Linear Inverse Problems using Higher-Order Annealed Langevin Diffusion](https://mlsanity.com/paper/2305.05014/) (2023)
- [Parallel Simulation for Log-concave Sampling and Score-based Diffusion Models](https://mlsanity.com/paper/2412.07435/) (2024)
- [Generative Modelling with High-Order Langevin Dynamics](https://mlsanity.com/paper/2404.12814/) (2024)
- [Faster Sampling via Stochastic Gradient Proximal Sampler](https://mlsanity.com/paper/2405.16734/) (2024)
- [Sequential Monte Carlo approximations of Wasserstein--Fisher--Rao gradient flows](https://mlsanity.com/paper/2506.05905/) (2025)
- [Accelerated Bayesian imaging by relaxed proximal-point Langevin sampling](https://mlsanity.com/paper/2308.09460/) (2023)
- [Accelerating Langevin Monte Carlo Sampling: A Large Deviations Analysis](https://mlsanity.com/paper/2503.19066/) (2025)
- [De-randomizing MCMC dynamics with the diffusion Stein operator](https://mlsanity.com/paper/2110.03768/) (2021)
- [Learning and Inference in Sparse Coding Models with Langevin Dynamics](https://mlsanity.com/paper/2204.11150/) (2022)

## Abstract (arXiv)

We propose a Markov chain Monte Carlo (MCMC) algorithm based on third-order Langevin dynamics for sampling from distributions with log-concave and smooth densities. The higher-order dynamics allow for more flexible discretization schemes, and we develop a specific method that combines splitting with more accurate integration. For a broad class of d-dimensional distributions arising from generalized linear models, we prove that the resulting third-order algorithm produces samples from a distribution that is at most varepsilon > 0 in Wasserstein distance from the target distribution in Oleft(fracd1/4 varepsilon1/2 right) steps. This result requires only Lipschitz conditions on the gradient. For general strongly convex potentials with α-th order smoothness, we prove that the mixing time scales as O left(fracd1/4varepsilon1/2 + fracd1/2varepsilon1/(α - 1) right).