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Proximal Langevin Algorithm: Rapid Convergence Under Isoperimetry

arXiv 1911.01469 ↗First posted 4 Nov 2019stat.ML cs.DS cs.ITcited by 31

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Abstract (arXiv)

We study the Proximal Langevin Algorithm (PLA) for sampling from a probability distribution nu = e-f on ℝn under isoperimetry. We prove a convergence guarantee for PLA in Kullback-Leibler (KL) divergence when nu satisfies log-Sobolev inequality (LSI) and f has bounded second and third derivatives. This improves on the result for the Unadjusted Langevin Algorithm (ULA), and matches the fastest known rate for sampling under LSI (without Metropolis filter) with a better dependence on the LSI constant. We also prove convergence guarantees for PLA in R'enyi divergence of order q > 1 when the biased limit satisfies either LSI or Poincar'e inequality.

Key facts. Cited by 31 papers in the map (data through 7 Oct 2026). First posted on arXiv 4 Nov 2019.

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Citing papers per year: 2021: 2; 2022: 8; 2023: 8; 2024: 4; 2025: 7; 2026: 2.

31 papers.

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  1. 2022Improved analysis for a proximal algorithm for samplingcited by 72
  2. 2023Faster high-accuracy log-concave sampling via algorithmic warm startscited by 44
  3. 2023A Rigorous Link between Deep Ensembles and (Variational) Bayesian Methodscited by 31
  4. 2021A Proximal Algorithm for Sampling from Non-smooth Potentialscited by 23
  5. 2022A Proximal Algorithm for Samplingcited by 21
  6. 2021The query complexity of sampling from strongly log-concave distributions in one dimensioncited by 18
  7. 2022Convergence of Stein Variational Gradient Descent under a Weaker Smoothness Conditioncited by 16
  8. 2022Fisher information lower bounds for samplingcited by 12
  9. 2023Query lower bounds for log-concave samplingcited by 12
  10. 2024Mirror and Preconditioned Gradient Descent in Wasserstein Spacecited by 11
  11. 2023Algorithmic Aspects of the Log-Laplace Transform and a Non-Euclidean Proximal Samplercited by 9
  12. 2022Bregman Proximal Langevin Monte Carlo via Bregman--Moreau Envelopescited by 7
19 more
  1. 2024Parallel Simulation for Log-concave Sampling and Score-based Diffusion Modelscited by 6
  2. 2025Diffusion at Absolute Zero: Langevin Sampling Using Successive Moreau Envelopes [conference paper]cited by 6
  3. 2023Fast Conditional Mixing of MCMC Algorithms for Non-log-concave Distributionscited by 5
  4. 2025On the query complexity of sampling from non-log-concave distributionscited by 5
  5. 2022Federated Learning with a Sampling Algorithm under Isoperimetrycited by 4
  6. 2023ELF: Federated Langevin Algorithms with Primal, Dual and Bidirectional Compressioncited by 4
  7. 2023Non-Log-Concave and Nonsmooth Sampling via Langevin Monte Carlo Algorithmscited by 4
  8. 2023Learning Rate Free Sampling in Constrained Domainscited by 3
  9. 2024Fast Mixing of Data Augmentation Algorithms: Bayesian Probit, Logit, and Lasso Regressioncited by 3
  10. 2025Mirror Mean-Field Langevin Dynamicscited by 3
  11. 2025Beyond Scores: Proximal Diffusion Modelscited by 3
  12. 2022A Proximal Algorithm for Sampling from Non-convex Potentialscited by 2
  13. 2022A Dynamical System View of Langevin-Based Non-Convex Samplingcited by 2
  14. 2025Anchored Langevin Algorithmscited by 2
  15. 2025Convergence Analysis of the Wasserstein Proximal Algorithm beyond Geodesic Convexitycited by 1
  16. 2024An Improved Analysis of Langevin Algorithms with Prior Diffusion for Non-Log-Concave Samplingcited by 0
  17. 2025Mixing Times and Privacy Analysis for the Projected Langevin Algorithm under a Modulus of Continuitycited by 0
  18. 2026Functional Stochastic Localizationcited by 0
  19. 2026Omega: Operator-based Mixture Ensemble for Generative Assimilationcited by 0