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Improved Bounds for Discretization of Langevin Diffusions: Near-Optimal Rates without Convexity

arXiv 1907.11331 ↗First posted 25 Jul 2019math.PR math.ST stat.COcited by 38

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Abstract (arXiv)

We present an improved analysis of the Euler-Maruyama discretization of the Langevin diffusion. Our analysis does not require global contractivity, and yields polynomial dependence on the time horizon. Compared to existing approaches, we make an additional smoothness assumption, and improve the existing rate from O(η) to O(η2) in terms of the KL divergence. This result matches the correct order for numerical SDEs, without suffering from exponential time dependence. When applied to algorithms for sampling and learning, this result simultaneously improves all those methods based on Dalayan's approach.

Key facts. Cited by 38 papers in the map (data through 7 Oct 2026). First posted on arXiv 25 Jul 2019.

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Citing papers per year: 2021: 4; 2022: 7; 2023: 10; 2024: 8; 2025: 8; 2026: 1.

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  1. 2023Nearly d-Linear Convergence Bounds for Diffusion Models via Stochastic Localizationcited by 230
  2. 2022Towards a Theory of Non-Log-Concave Sampling: First-Order Stationarity Guarantees for Langevin Monte Carlocited by 63
  3. 2021Separation of Scales and a Thermodynamic Description of Feature Learning in Some CNNscited by 53
  4. 2023Towards a Complete Analysis of Langevin Monte Carlo: Beyond Poincar'e Inequalitycited by 31
  5. 2023Improved Discretization Analysis for Underdamped Langevin Monte Carlocited by 26
  6. 2021Sqrt(d) Dimension Dependence of Langevin Monte Carlocited by 25
  7. 2022A Proximal Algorithm for Samplingcited by 21
  8. 2022Resolving the Mixing Time of the Langevin Algorithm to its Stationary Distribution for Log-Concave Samplingcited by 21
  9. 2023Time-Independent Information-Theoretic Generalization Bounds for SGLDcited by 12
  10. 2021Stationary Behavior of Constant Stepsize SGD Type Algorithms: An Asymptotic Characterizationcited by 10
  11. 2023Langevin Monte Carlo for strongly log-concave distributions: Randomized midpoint revisitedcited by 10
  12. 2023Particle-based Variational Inference with Generalized Wasserstein Gradient Flowcited by 10
26 more
  1. 2024Parallelized Midpoint Randomization for Langevin Monte Carlocited by 9
  2. 2025Composition and Alignment of Diffusion Models using Constrained Learningcited by 8
  3. 2021Convergence Analysis of Schr"odinger-F"ollmer Sampler without Convexitycited by 7
  4. 2023Taming under isoperimetrycited by 7
  5. 2024Tamed Langevin sampling under weaker conditionscited by 7
  6. 2024The Poisson Midpoint Method for Langevin Dynamics: Provably Efficient Discretization for Diffusion Modelscited by 6
  7. 2023Fast Conditional Mixing of MCMC Algorithms for Non-log-concave Distributionscited by 5
  8. 2024A Separation in Heavy-Tailed Sampling: Gaussian vs. Stable Oracles for Proximal Samplerscited by 5
  9. 2023Non-Log-Concave and Nonsmooth Sampling via Langevin Monte Carlo Algorithmscited by 4
  10. 2024Convergence of Unadjusted Langevin in High Dimensions: Delocalization of Biascited by 4
  11. 2025The Performance Of The Unadjusted Langevin Algorithm Without Smoothness Assumptionscited by 4
  12. 2022A sharp uniform-in-time error estimate for Stochastic Gradient Langevin Dynamicscited by 3
  13. 2024R'enyi-infinity constrained sampling with d3 membership queriescited by 3
  14. 2022A Proximal Algorithm for Sampling from Non-convex Potentialscited by 2
  15. 2022A Dynamical System View of Langevin-Based Non-Convex Samplingcited by 2
  16. 2024Randomized Midpoint Method for Log-Concave Sampling under Constraintscited by 2
  17. 2025Convergence of two-timescale gradient descent ascent dynamics: finite-dimensional and mean-field perspectivescited by 2
  18. 2025kTULA: A Langevin sampling algorithm with improved KL bounds under super-linear log-gradientscited by 2
  19. 2024Inflationary Flows: Calibrated Bayesian Inference with Diffusion-Based Modelscited by 1
  20. 2025Langevin Monte-Carlo Provably Learns Depth Two Neural Nets at Any Size and Datacited by 1
  21. 2025Restricted Spectral Gap Decomposition for Simulated Tempering Targeting Mixture Distributionscited by 1
  22. 2026DC-LA: Difference-of-Convex Langevin Algorithmcited by 1
  23. 2022Asynchronous Bayesian Learning over a Networkcited by 0
  24. 2023Stochastic Langevin Monte Carlo for (weakly) log-concave posterior distributionscited by 0
  25. 2025Poisson Midpoint Method for Log Concave Sampling: Beyond the Strong Error Lower Boundscited by 0
  26. 2025When Langevin Monte Carlo Meets Randomization: New Sampling Algorithms with Non-asymptotic Error Bounds beyond Log-Concavity and Gradient Lipschitznesscited by 0