mlsanityData to 14 Oct 2026

User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient

arXiv 1710.00095 ↗First posted 29 Sep 2017math.ST cs.LG math.PRcited by 116

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Abstract (arXiv)

In this paper, we study the problem of sampling from a given probability density function that is known to be smooth and strongly log-concave. We analyze several methods of approximate sampling based on discretizations of the (highly overdamped) Langevin diffusion and establish guarantees on its error measured in the Wasserstein-2 distance. Our guarantees improve or extend the state-of-the-art results in three directions. First, we provide an upper bound on the error of the first-order Langevin Monte Carlo (LMC) algorithm with optimized varying step-size. This result has the advantage of being horizon free (we do not need to know in advance the target precision) and to improve by a logarithmic factor the corresponding result for the constant step-size. Second, we study the case where accurate evaluations of the gradient of the log-density are unavailable, but one can have access to approximations of the aforementioned gradient. In such a situation, we consider both deterministic and stochastic approximations of the gradient and provide an upper bound on the sampling error of the first-order LMC that quantifies the impact of the gradient evaluation inaccuracies. Third, we establish upper bounds for two versions of the second-order LMC, which leverage the Hessian of the log-density. We provide nonasymptotic guarantees on the sampling error of these second-order LMCs. These guarantees reveal that the second-order LMC algorithms improve on the first-order LMC in ill-conditioned settings.

Key facts. Cited by 116 papers in the map (data through 14 Oct 2026). First posted on arXiv 29 Sep 2017.

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Citing papers per year: 2021: 21; 2022: 17; 2023: 25; 2024: 19; 2025: 16; 2026: 18.

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  2. 2023Restart Sampling for Improving Generative Processescited by 84
  3. 2021Estimating High Order Gradients of the Data Distribution by Denoisingcited by 64
  4. 2022Towards a Theory of Non-Log-Concave Sampling: First-Order Stationarity Guarantees for Langevin Monte Carlocited by 63
  5. 2024Optimal score estimation via empirical Bayes smoothingcited by 61
  6. 2024Convergence Analysis for General Probability Flow ODEs of Diffusion Models in Wasserstein Distancescited by 56
  7. 2021KALE Flow: A Relaxed KL Gradient Flow for Probabilities with Disjoint Supportcited by 37
  8. 2021The Mirror Langevin Algorithm Converges with Vanishing Biascited by 37
  9. 2024Conformalized-DeepONet: A Distribution-Free Framework for Uncertainty Quantification in Deep Operator Networkscited by 28
  10. 2021Time-independent Generalization Bounds for SGLD in Non-convex Settingscited by 27
  11. 2021QLSD: Quantised Langevin stochastic dynamics for Bayesian federated learningcited by 26
  12. 2023Improved Convergence of Score-Based Diffusion Models via Prediction-Correctioncited by 24
38 more
  1. 2022Langevin Monte Carlo for Contextual Banditscited by 22
  2. 2022Resolving the Mixing Time of the Langevin Algorithm to its Stationary Distribution for Log-Concave Samplingcited by 22
  3. 2024Zeroth-Order Sampling Methods for Non-Log-Concave Distributions: Alleviating Metastability by Denoising Diffusioncited by 19
  4. 2021The query complexity of sampling from strongly log-concave distributions in one dimensioncited by 18
  5. 2023Interacting Particle Langevin Algorithm for Maximum Marginal Likelihood Estimationcited by 17
  6. 2022Convergence of Stein Variational Gradient Descent under a Weaker Smoothness Conditioncited by 16
  7. 2021On Convergence of Federated Averaging Langevin Dynamicscited by 14
  8. 2024Sampling from Bayesian Neural Network Posteriors with Symmetric Minibatch Splitting Langevin Dynamicscited by 13
  9. 2021When is the Convergence Time of Langevin Algorithms Dimension Independent? A Composite Optimization Viewpointcited by 12
  10. 2022Low-Precision Stochastic Gradient Langevin Dynamicscited by 12
  11. 2023Kinetic Langevin MCMC Sampling Without Gradient Lipschitz Continuity -- the Strongly Convex Casecited by 12
  12. 2023Query lower bounds for log-concave samplingcited by 12
  13. 2021DG-LMC: A Turn-key and Scalable Synchronous Distributed MCMC Algorithm via Langevin Monte Carlo within Gibbscited by 11
  14. 2023Fast sampling from constrained spaces using the Metropolis-adjusted Mirror Langevin algorithmcited by 11
  15. 2021Non-asymptotic estimates for TUSLA algorithm for non-convex learning with applications to neural networks with ReLU activation functioncited by 10
  16. 2023Langevin Monte Carlo for strongly log-concave distributions: Randomized midpoint revisitedcited by 10
  17. 2023Particle-based Variational Inference with Generalized Wasserstein Gradient Flowcited by 10
  18. 2024Accelerating Approximate Thompson Sampling with Underdamped Langevin Monte Carlocited by 10
  19. 2024Constrained Sampling with Primal-Dual Langevin Monte Carlocited by 10
  20. 2024Parallelized Midpoint Randomization for Langevin Monte Carlocited by 9
  21. 2025Assessing the Quality of Denoising Diffusion Models in Wasserstein Distance: Noisy Score and Optimal Boundscited by 9
  22. 2021Wasserstein distance estimates for the distributions of numerical approximations to ergodic stochastic differential equationscited by 8
  23. 2022Quantum Algorithms for Sampling Log-Concave Distributions and Estimating Normalizing Constantscited by 8
  24. 2023Coin Sampling: Gradient-Based Bayesian Inference without Learning Ratescited by 8
  25. 2021Stochastic Gradient Langevin Dynamics with Variance Reductioncited by 7
  26. 2021Convergence Analysis of Schr"odinger-F"ollmer Sampler without Convexitycited by 7
  27. 2022Utilising the CLT Structure in Stochastic Gradient based Sampling : Improved Analysis and Faster Algorithmscited by 7
  28. 2024Concentration properties of fractional posterior in 1-bit matrix completioncited by 7
  29. 2024Kinetic Interacting Particle Langevin Monte Carlocited by 7
  30. 2022Heavy-tailed Sampling via Transformed Unadjusted Langevin Algorithmcited by 6
  31. 2023Entropy-MCMC: Sampling from Flat Basins with Easecited by 6
  32. 2025Adaptive Stochastic Coefficients for Accelerating Diffusion Samplingcited by 6
  33. 2021Batch Normalization Preconditioning for Neural Network Trainingcited by 5
  34. 2022Langevin dynamics based algorithm e-THvarepsilonO POULA for stochastic optimization problems with discontinuous stochastic gradientcited by 5
  35. 2022Convergence of the Inexact Langevin Algorithm in KL Divergence with Application to Score-based Generative Modelscited by 5
  36. 2023Fast Conditional Mixing of MCMC Algorithms for Non-log-concave Distributionscited by 5
  37. 2024Faster Sampling via Stochastic Gradient Proximal Samplercited by 5
  38. 2025From Global to Local: A Scalable Benchmark for Local Posterior Samplingcited by 5

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