mlsanityData to 7 Oct 2026

Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices

arXiv 1903.08568 ↗First posted 20 Mar 2019cs.DS cs.LG math.PRcited by 224

Not on the map. Why?

Cited by 38 in Diffusion models, but placed by its own abstract, which didn’t fit.

Abstract (arXiv)

We study the Unadjusted Langevin Algorithm (ULA) for sampling from a probability distribution nu = e-f on ℝn. We prove a convergence guarantee in Kullback-Leibler (KL) divergence assuming nu satisfies a log-Sobolev inequality and the Hessian of f is bounded. Notably, we do not assume convexity or bounds on higher derivatives. We also prove convergence guarantees in R'enyi divergence of order q > 1 assuming the limit of ULA satisfies either the log-Sobolev or Poincar'e inequality. We also prove a bound on the bias of the limiting distribution of ULA assuming third-order smoothness of f, without requiring isoperimetry.

Key facts. Cited by 224 papers in the map (data through 7 Oct 2026). First posted on arXiv 20 Mar 2019.

Builds on

No reference list yet.

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 12; 2022: 38; 2023: 44; 2024: 50; 2025: 44; 2026: 36.

224 papers.

Sort:

By area:

  1. 2022Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptionscited by 440
  2. 2023Nearly d-Linear Convergence Bounds for Diffusion Models via Stochastic Localizationcited by 230
  3. 2022Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptionscited by 215
  4. 2022Convergence for score-based generative modeling with polynomial complexitycited by 176
  5. 2023Policy Representation via Diffusion Probability Model for Reinforcement Learningcited by 105
  6. 2024Principled Probabilistic Imaging using Diffusion Models as Plug-and-Play Priorscited by 93
  7. 2022Variational inference via Wasserstein gradient flowscited by 84
  8. 2023Restoration-Degradation Beyond Linear Diffusions: A Non-Asymptotic Analysis For DDIM-Type Samplerscited by 73
  9. 2022Improved analysis for a proximal algorithm for samplingcited by 72
  10. 2021Differential Privacy Dynamics of Langevin Diffusion and Noisy Gradient Descentcited by 71
  11. 2022Towards a Theory of Non-Log-Concave Sampling: First-Order Stationarity Guarantees for Langevin Monte Carlocited by 63
  12. 2022Statistical Efficiency of Score Matching: The View from Isoperimetrycited by 63
38 more
  1. 2022Privacy of Noisy Stochastic Gradient Descent: More Iterations without More Privacy Losscited by 60
  2. 2024Optimal score estimation via empirical Bayes smoothingcited by 60
  3. 2022Forget Unlearning: Towards True Data-Deletion in Machine Learningcited by 58
  4. 2021Separation of Scales and a Thermodynamic Description of Feature Learning in Some CNNscited by 53
  5. 2022Convex Analysis of the Mean Field Langevin Dynamicscited by 53
  6. 2023Provable Probabilistic Imaging using Score-Based Generative Priorscited by 51
  7. 2024Langevin Unlearning: A New Perspective of Noisy Gradient Descent for Machine Unlearningcited by 48
  8. 2022Differentially Private Learning Needs Hidden State (Or Much Faster Convergence)cited by 45
  9. 2023Faster high-accuracy log-concave sampling via algorithmic warm startscited by 44
  10. 2023Reverse Diffusion Monte Carlocited by 41
  11. 2021The Mirror Langevin Algorithm Converges with Vanishing Biascited by 37
  12. 2023Towards a Complete Analysis of Langevin Monte Carlo: Beyond Poincar'e Inequalitycited by 31
  13. 2022Birth-death dynamics for sampling: Global convergence, approximations and their asymptoticscited by 30
  14. 2025Accelerated Diffusion Models via Speculative Samplingcited by 30
  15. 2023Improved dimension dependence of a proximal algorithm for samplingcited by 29
  16. 2023Improved Discretization Analysis for Underdamped Langevin Monte Carlocited by 26
  17. 2021Sqrt(d) Dimension Dependence of Langevin Monte Carlocited by 25
  18. 2024Diffusion Soup: Model Merging for Text-to-Image Diffusion Modelscited by 25
  19. 2022Convergence of the Riemannian Langevin Algorithmcited by 23
  20. 2023Improved Convergence of Score-Based Diffusion Models via Prediction-Correctioncited by 23
  21. 2024Provable Benefit of Annealed Langevin Monte Carlo for Non-log-concave Samplingcited by 23
  22. 2022Langevin Monte Carlo for Contextual Banditscited by 22
  23. 2024Implicit Diffusion: Efficient Optimization through Stochastic Samplingcited by 22
  24. 2024CPR: Retrieval Augmented Generation for Copyright Protectioncited by 22
  25. 2021A Convergence Theory for SVGD in the Population Limit under Talagrand's Inequality T1cited by 21
  26. 2022A Proximal Algorithm for Samplingcited by 21
  27. 2022Resolving the Mixing Time of the Langevin Algorithm to its Stationary Distribution for Log-Concave Samplingcited by 21
  28. 2024Diffusive Gibbs Samplingcited by 21
  29. 2024Training Neural Samplers with Reverse Diffusive KL Divergencecited by 21
  30. 2022Particle-based Variational Inference with Preconditioned Functional Gradient Flowcited by 20
  31. 2023Sampling via Gradient Flows in the Space of Probability Measurescited by 20
  32. 2025Diffusion Models and the Manifold Hypothesis: Log-Domain Smoothing is Geometry Adaptivecited by 20
  33. 2021Neural Variational Gradient Descentcited by 19
  34. 2023Convergence of mean-field Langevin dynamics: Time and space discretization, stochastic gradient, and variance reductioncited by 19
  35. 2024Zeroth-Order Sampling Methods for Non-Log-Concave Distributions: Alleviating Metastability by Denoising Diffusioncited by 19
  36. 2022Regularized Stein Variational Gradient Flowcited by 18
  37. 2024In-and-Out: Algorithmic Diffusion for Sampling Convex Bodiescited by 18
  38. 2023Interacting Particle Langevin Algorithm for Maximum Marginal Likelihood Estimationcited by 17

50 most cited of 224 All 224 (JSON)