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Convergence of Langevin MCMC in KL-divergence

arXiv 1705.09048 ↗First posted 25 May 2017stat.MLcited by 86

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Abstract (arXiv)

Langevin diffusion is a commonly used tool for sampling from a given distribution. In this work, we establish that when the target density p^* is such that log p^* is L smooth and m strongly convex, discrete Langevin diffusion produces a distribution p with KL(p||p^*)≤ ε in tildeO(fracdε) steps, where d is the dimension of the sample space. We also study the convergence rate when the strong-convexity assumption is absent. By considering the Langevin diffusion as a gradient flow in the space of probability distributions, we obtain an elegant analysis that applies to the stronger property of convergence in KL-divergence and gives a conceptually simpler proof of the best-known convergence results in weaker metrics.

Key facts. Cited by 86 papers in the map (data through 7 Oct 2026). First posted on arXiv 25 May 2017.

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Citing papers per year: 2021: 13; 2022: 13; 2023: 21; 2024: 17; 2025: 14; 2026: 8.

86 papers.

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  2. 2022Convergence for score-based generative modeling with polynomial complexitycited by 176
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  5. 2025Inference-time Scaling of Diffusion Models through Classical Searchcited by 48
  6. 2021Minimax Mixing Time of the Metropolis-Adjusted Langevin Algorithm for Log-Concave Samplingcited by 44
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  10. 2021Sqrt(d) Dimension Dependence of Langevin Monte Carlocited by 25
  11. 2024Diffusion Soup: Model Merging for Text-to-Image Diffusion Modelscited by 25
  12. 2021A Free Lunch from the Noise: Provable and Practical Exploration for Representation Learningcited by 24
38 more
  1. 2021A Proximal Algorithm for Sampling from Non-smooth Potentialscited by 23
  2. 2022Discrete Langevin Sampler via Wasserstein Gradient Flowcited by 23
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  6. 2022Convergence of Stein Variational Gradient Descent under a Weaker Smoothness Conditioncited by 16
  7. 2023Exploring the Optimal Choice for Generative Processes in Diffusion Models: Ordinary vs Stochastic Differential Equationscited by 15
  8. 2022Denoising MCMC for Accelerating Diffusion-Based Generative Modelscited by 13
  9. 2024Error bounds for particle gradient descent, and extensions of the log-Sobolev and Talagrand inequalitiescited by 13
  10. 2021When is the Convergence Time of Langevin Algorithms Dimension Independent? A Composite Optimization Viewpointcited by 12
  11. 2023When does Metropolized Hamiltonian Monte Carlo provably outperform Metropolis-adjusted Langevin algorithm?cited by 12
  12. 2023Fast sampling from constrained spaces using the Metropolis-adjusted Mirror Langevin algorithmcited by 11
  13. 2025Mixing Time of the Proximal Sampler in Relative Fisher Information via Strong Data Processing Inequalitycited by 11
  14. 2024Fast Convergence of Phi-Divergence Along the Unadjusted Langevin Algorithm and Proximal Samplercited by 10
  15. 2023Accelerating optimization over the space of probability measurescited by 9
  16. 2021Sampling from Log-Concave Distributions with Infinity-Distance Guaranteescited by 8
  17. 2023Coin Sampling: Gradient-Based Bayesian Inference without Learning Ratescited by 8
  18. 2023Bridging the Gap Between Variational Inference and Wasserstein Gradient Flowscited by 8
  19. 2024Think Twice Before You Act: Improving Inverse Problem Solving With MCMCcited by 8
  20. 2025Assessing the Quality of Denoising Diffusion Models in Wasserstein Distance: Noisy Score and Optimal Boundscited by 8
  21. 2021Stochastic Gradient Langevin Dynamics with Variance Reductioncited by 7
  22. 2021Convergence Analysis of Schr"odinger-F"ollmer Sampler without Convexitycited by 7
  23. 2022Utilising the CLT Structure in Stochastic Gradient based Sampling : Improved Analysis and Faster Algorithmscited by 7
  24. 2023Langevin Thompson Sampling with Logarithmic Communication: Bandits and Reinforcement Learningcited by 5
  25. 2023Fast Conditional Mixing of MCMC Algorithms for Non-log-concave Distributionscited by 5
  26. 2023Convergence guarantee for consistency modelscited by 5
  27. 2024Characterizing Dependence of Samples along the Langevin Dynamics and Algorithms via Contraction of Phi-Mutual Informationcited by 5
  28. 2024Faster Sampling via Stochastic Gradient Proximal Samplercited by 5
  29. 2025On the query complexity of sampling from non-log-concave distributionscited by 5
  30. 2023ELF: Federated Langevin Algorithms with Primal, Dual and Bidirectional Compressioncited by 4
  31. 2023Particle Mean Field Variational Bayescited by 4
  32. 2023Non-Log-Concave and Nonsmooth Sampling via Langevin Monte Carlo Algorithmscited by 4
  33. 2023Neural Sampling in Hierarchical Exponential-family Energy-based Modelscited by 4
  34. 2024Convergence of Unadjusted Langevin in High Dimensions: Delocalization of Biascited by 4
  35. 2025Accelerating Langevin Monte Carlo Sampling: A Large Deviations Analysiscited by 4
  36. 2025Fast Score-Based Sampling via Log-Concave Reductionscited by 4
  37. 2021Unadjusted Langevin algorithm for sampling a mixture of weakly smooth potentialscited by 3
  38. 2022Differentially Private Sampling from Rashomon Sets, and the Universality of Langevin Diffusion for Convex Optimizationcited by 3

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