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Underdamped Langevin MCMC: A non-asymptotic analysis

arXiv 1707.03663 ↗First posted 12 Jul 2017stat.ML cs.LG stat.COcited by 133

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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.

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

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Citing papers per year: 2021: 20; 2022: 18; 2023: 31; 2024: 24; 2025: 21; 2026: 19.

133 papers.

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  1. 2022Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptionscited by 445
  2. 2022Convergence for score-based generative modeling with polynomial complexitycited by 177
  3. 2021Bayesian imaging using Plug & Play priors: when Langevin meets Tweediecited by 97
  4. 2023Faster high-accuracy log-concave sampling via algorithmic warm startscited by 45
  5. 2021Minimax Mixing Time of the Metropolis-Adjusted Langevin Algorithm for Log-Concave Samplingcited by 44
  6. 2023Reverse Diffusion Monte Carlocited by 41
  7. 20233D molecule generation by denoising voxel gridscited by 40
  8. 2021MCMC Variational Inference via Uncorrected Hamiltonian Annealingcited by 39
  9. 2021Differentiable Annealed Importance Sampling and the Perils of Gradient Noisecited by 36
  10. 2024Model-Based Diffusion for Trajectory Optimizationcited by 32
  11. 2023Towards a Complete Analysis of Langevin Monte Carlo: Beyond Poincar'e Inequalitycited by 31
  12. 2023Improved dimension dependence of a proximal algorithm for samplingcited by 30
38 more
  1. 2025Beyond Log-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distancecited by 29
  2. 2022Sampling with Riemannian Hamiltonian Monte Carlo in a Constrained Spacecited by 28
  3. 2023Improved Discretization Analysis for Underdamped Langevin Monte Carlocited by 28
  4. 2022Particle algorithms for maximum likelihood training of latent variable modelscited by 26
  5. 2022Langevin Diffusion Variational Inferencecited by 26
  6. 2021Sqrt(d) Dimension Dependence of Langevin Monte Carlocited by 25
  7. 2021A Proximal Algorithm for Sampling from Non-smooth Potentialscited by 24
  8. 2022A Proximal Algorithm for Samplingcited by 22
  9. 2022Resolving the Mixing Time of the Langevin Algorithm to its Stationary Distribution for Log-Concave Samplingcited by 22
  10. 2021A Convergence Theory for SVGD in the Population Limit under Talagrand's Inequality T1cited by 21
  11. 2021Lower Bounds on Metropolized Sampling Methods for Well-Conditioned Distributionscited by 19
  12. 2023Renormalizing Diffusion Modelscited by 16
  13. 2021Lazy OCO: Online Convex Optimization on a Switching Budgetcited by 14
  14. 2021On Convergence of Federated Averaging Langevin Dynamicscited by 14
  15. 2023Chain of Log-Concave Markov Chainscited by 14
  16. 2024Sampling from Bayesian Neural Network Posteriors with Symmetric Minibatch Splitting Langevin Dynamicscited by 13
  17. 2021When is the Convergence Time of Langevin Algorithms Dimension Independent? A Composite Optimization Viewpointcited by 12
  18. 2022Fisher information lower bounds for samplingcited by 12
  19. 2023Kinetic Langevin MCMC Sampling Without Gradient Lipschitz Continuity -- the Strongly Convex Casecited by 12
  20. 2023Query lower bounds for log-concave samplingcited by 12
  21. 2023Momentum Particle Maximum Likelihoodcited by 12
  22. 2024Fast parallel sampling under isoperimetrycited by 12
  23. 2024More Efficient Randomized Exploration for Reinforcement Learning via Approximate Samplingcited by 12
  24. 2021Asymptotic bias of inexact Markov Chain Monte Carlo methods in high dimensioncited by 11
  25. 2025LanPaint: Training-Free Diffusion Inpainting with Asymptotically Exact and Fast Conditional Samplingcited by 11
  26. 2025Complexity Analysis of Normalizing Constant Estimation: from Jarzynski Equality to Annealed Importance Sampling and beyondcited by 11
  27. 2022Condition-number-independent convergence rate of Riemannian Hamiltonian Monte Carlo with numerical integratorscited by 10
  28. 2023Algorithmic Aspects of the Log-Laplace Transform and a Non-Euclidean Proximal Samplercited by 10
  29. 2023Langevin Monte Carlo for strongly log-concave distributions: Randomized midpoint revisitedcited by 10
  30. 2023Accelerating optimization over the space of probability measurescited by 10
  31. 2024Accelerating Approximate Thompson Sampling with Underdamped Langevin Monte Carlocited by 10
  32. 2022A Kernel-Expanded Stochastic Neural Networkcited by 9
  33. 2024Parallelized Midpoint Randomization for Langevin Monte Carlocited by 9
  34. 2021Wasserstein distance estimates for the distributions of numerical approximations to ergodic stochastic differential equationscited by 8
  35. 2021An Introduction to Hamiltonian Monte Carlo Method for Samplingcited by 8
  36. 2021Multimeasurement Generative Modelscited by 8
  37. 2022Gradient Norm Minimization of Nesterov Acceleration: o(1/k3)cited by 8
  38. 2022Quantum Algorithms for Sampling Log-Concave Distributions and Estimating Normalizing Constantscited by 8

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