mlsanityData to 14 Oct 2026

Towards a Theory of Non-Log-Concave Sampling: First-Order Stationarity Guarantees for Langevin Monte Carlo

arXiv 2202.05214 ↗First posted 10 Feb 2022math.ST stat.ML stat.THcited by 63

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Abstract (arXiv)

For the task of sampling from a density π propto exp(-V) on ℝd, where V is possibly non-convex but L-gradient Lipschitz, we prove that averaged Langevin Monte Carlo outputs a sample with varepsilon-relative Fisher information after O( L2 d2/varepsilon2) iterations. This is the sampling analogue of complexity bounds for finding an varepsilon-approximate first-order stationary points in non-convex optimization and therefore constitutes a first step towards the general theory of non-log-concave sampling. We discuss numerous extensions and applications of our result; in particular, it yields a new state-of-the-art guarantee for sampling from distributions which satisfy a Poincar'e inequality.

Key facts. Cited by 63 papers in the map (data through 14 Oct 2026). Builds on 22 of its 69 references that are in the map. First posted on arXiv 10 Feb 2022.

Builds on

22 of 69 references.

References · 22

  1. 2016Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Lojasiewicz Conditioncited by 539
  2. 2017Non-convex learning via Stochastic Gradient Langevin Dynamics: a nonasymptotic analysiscited by 243
  3. 2019Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Sufficescited by 227
  4. 2015Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithmcited by 197
  5. 2018Analysis of Langevin Monte Carlo via convex optimizationcited by 127
  6. 2018Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problemcited by 123
  7. 2020PAGE: A Simple and Optimal Probabilistic Gradient Estimator for Nonconvex Optimizationcited by 119
  8. 2017User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradientcited by 116
  9. 2017Global Convergence of Langevin Dynamics Based Algorithms for Nonconvex Optimizationcited by 97
  10. 2018Sharp convergence rates for Langevin dynamics in the nonconvex settingcited by 68
  11. 2018The promises and pitfalls of Stochastic Gradient Langevin Dynamicscited by 56
  12. 2020On the Convergence of Langevin Monte Carlo: The Interplay between Tail Growth and Smoothnesscited by 55
10 more
  1. 2017Assigning a value to a power likelihood in a general Bayesian modelcited by 46
  2. 2018Global Non-convex Optimization with Discretized Diffusionscited by 44
  3. 2019Improved Bounds for Discretization of Langevin Diffusions: Near-Optimal Rates without Convexitycited by 39
  4. 2018Non-asymptotic bounds for sampling algorithms without log-concavitycited by 35
  5. 2019Stochastic Runge-Kutta Accelerates Langevin Monte Carlo and Beyondcited by 33
  6. 2019Langevin Monte Carlo without smoothnesscited by 28
  7. 2018On the Theory of Variance Reduction for Stochastic Gradient Monte Carlocited by 19
  8. 2021The query complexity of sampling from strongly log-concave distributions in one dimensioncited by 18
  9. 2017Bayesian model comparison with the Hyv"arinen score: computation and consistencycited by 10
  10. 2022Heavy-tailed Sampling via Transformed Unadjusted Langevin Algorithmcited by 6

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2022: 11; 2023: 16; 2024: 16; 2025: 16; 2026: 4.

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  1. 2024Principled Probabilistic Imaging using Diffusion Models as Plug-and-Play Priorscited by 97
  2. 2023Provable Probabilistic Imaging using Score-Based Generative Priorscited by 51
  3. 2023Forward-backward Gaussian variational inference via JKO in the Bures-Wasserstein Spacecited by 36
  4. 2023Towards a Complete Analysis of Langevin Monte Carlo: Beyond Poincar'e Inequalitycited by 31
  5. 2024Provable Benefit of Annealed Langevin Monte Carlo for Non-log-concave Samplingcited by 23
  6. 2022A Proximal Algorithm for Samplingcited by 22
  7. 2022Resolving the Mixing Time of the Langevin Algorithm to its Stationary Distribution for Log-Concave Samplingcited by 22
  8. 2022Regularized Stein Variational Gradient Flowcited by 19
  9. 2024Zeroth-Order Sampling Methods for Non-Log-Concave Distributions: Alleviating Metastability by Denoising Diffusioncited by 19
  10. 2025Split Gibbs Discrete Diffusion Posterior Samplingcited by 18
  11. 2022Sampling with Mollified Interaction Energy Descentcited by 17
  12. 2022Improved Convergence Rate of Stochastic Gradient Langevin Dynamics with Variance Reduction and its Application to Optimizationcited by 16
38 more
  1. 2022Convergence of Stein Variational Gradient Descent under a Weaker Smoothness Conditioncited by 16
  2. 2022Fisher information lower bounds for samplingcited by 12
  3. 2023Query lower bounds for log-concave samplingcited by 12
  4. 2025Mixing Time of the Proximal Sampler in Relative Fisher Information via Strong Data Processing Inequalitycited by 12
  5. 2024Theoretical Guarantees for Variational Inference with Fixed-Variance Mixture of Gaussianscited by 11
  6. 2025Complexity Analysis of Normalizing Constant Estimation: from Jarzynski Equality to Annealed Importance Sampling and beyondcited by 11
  7. 2023Convergence Rates for Non-Log-Concave Sampling and Log-Partition Estimationcited by 10
  8. 2023Langevin Monte Carlo for strongly log-concave distributions: Randomized midpoint revisitedcited by 10
  9. 2023Particle-based Variational Inference with Generalized Wasserstein Gradient Flowcited by 10
  10. 2022Utilising the CLT Structure in Stochastic Gradient based Sampling : Improved Analysis and Faster Algorithmscited by 7
  11. 2023Taming under isoperimetrycited by 7
  12. 2024Taming Score-Based Diffusion Priors for Infinite-Dimensional Nonlinear Inverse Problemscited by 7
  13. 2024Tamed Langevin sampling under weaker conditionscited by 7
  14. 2022Constrained Langevin Algorithms with L-mixing External Random Variablescited by 6
  15. 2023Provably Fast Finite Particle Variants of SVGD via Virtual Particle Stochastic Approximationcited by 6
  16. 2024A Separation in Heavy-Tailed Sampling: Gaussian vs. Stable Oracles for Proximal Samplerscited by 6
  17. 2024The Poisson Midpoint Method for Langevin Dynamics: Provably Efficient Discretization for Diffusion Modelscited by 6
  18. 2023Fast Conditional Mixing of MCMC Algorithms for Non-log-concave Distributionscited by 5
  19. 2024Non-geodesically-convex optimization in the Wasserstein spacecited by 5
  20. 2025On the query complexity of sampling from non-log-concave distributionscited by 5
  21. 2022Federated Learning with a Sampling Algorithm under Isoperimetrycited by 4
  22. 2023Mean-Square Analysis of Discretized Ito Diffusions for Heavy-tailed Samplingcited by 4
  23. 2023Non-Log-Concave and Nonsmooth Sampling via Langevin Monte Carlo Algorithmscited by 4
  24. 2025DIGing--SGLD: Decentralized and Scalable Langevin Sampling over Time--Varying Networkscited by 4
  25. 2023Learning Rate Free Sampling in Constrained Domainscited by 3
  26. 2024Fast Mixing of Data Augmentation Algorithms: Bayesian Probit, Logit, and Lasso Regressioncited by 3
  27. 2024A Particle Algorithm for Mean-Field Variational Inferencecited by 3
  28. 2025kTULA: A Langevin sampling algorithm with improved KL bounds under super-linear log-gradientscited by 3
  29. 2025Efficient Approximate Posterior Sampling with Annealed Langevin Monte Carlocited by 3
  30. 2025Anchored Langevin Algorithmscited by 3
  31. 2022A Dynamical System View of Langevin-Based Non-Convex Samplingcited by 2
  32. 2023Conditionally Strongly Log-Concave Generative Modelscited by 2
  33. 2024Kernelized Normalizing Constant Estimation: Bridging Bayesian Quadrature and Bayesian Optimizationcited by 2
  34. 2026High-accuracy log-concave sampling with stochastic queriescited by 2
  35. 2023Efficient Sampling of Stochastic Differential Equations with Positive Semi-Definite Modelscited by 1
  36. 2023Subsampling Error in Stochastic Gradient Langevin Diffusionscited by 1
  37. 2024A phase transition in sampling from Restricted Boltzmann Machinescited by 1
  38. 2024Sampling with Adaptive Variance for Multimodal Distributionscited by 1

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