mlsanityData to 14 Oct 2026

Anchored Langevin Algorithms

arXiv 2509.19455 ↗First posted 23 Sep 2025stat.ML cs.LG math.PRcited by 3

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Abstract (arXiv)

Standard first-order Langevin algorithms such as the unadjusted Langevin algorithm (ULA) are obtained by discretizing the Langevin diffusion and are widely used for sampling in machine learning because they scale to high dimensions and large datasets. However, they face two key limitations: (i) they require differentiable log-densities, excluding targets with non-differentiable components; and (ii) they generally fail to sample heavy-tailed targets. We propose anchored Langevin dynamics, a unified approach that accommodates non-differentiable targets and certain classes of heavy-tailed distributions. The method replaces the original potential with a smooth reference potential and modifies the Langevin diffusion via multiplicative scaling. We establish non-asymptotic guarantees in the 2-Wasserstein distance to the target distribution and provide an equivalent formulation derived via a random time change of the Langevin diffusion. We provide numerical experiments to illustrate the theory and practical performance of our proposed approach.

Key facts. Cited by 3 papers in the map (data through 14 Oct 2026). Builds on 42 of its 79 references that are in the map. First posted on arXiv 23 Sep 2025.

Builds on

42 of 79 references.

References · 42

  1. 2014Theoretical guarantees for approximate sampling from smooth and log-concave densitiescited by 208
  2. 2015Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithmcited by 197
  3. 2016High-dimensional Bayesian inference via the Unadjusted Langevin Algorithmcited by 146
  4. 2018Analysis of Langevin Monte Carlo via convex optimizationcited by 127
  5. 2018Log-concave sampling: Metropolis-Hastings algorithms are fastcited by 118
  6. 2017User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradientcited by 116
  7. 2021Priors in Bayesian Deep Learning: A Reviewcited by 101
  8. 2017Convergence of Langevin MCMC in KL-divergencecited by 88
  9. 2016Efficient Bayesian computation by proximal Markov chain Monte Carlo: when Langevin meets Moreaucited by 84
  10. 2014Consistency and fluctuations for stochastic gradient Langevin dynamicscited by 65
  11. 2022Towards a Theory of Non-Log-Concave Sampling: First-Order Stationarity Guarantees for Langevin Monte Carlocited by 63
  12. 2013Proximal Markov chain Monte Carlo algorithmscited by 46
30 more
  1. 2019Nonasymptotic estimates for Stochastic Gradient Langevin Dynamics under local conditions in nonconvex optimizationcited by 38
  2. 2021The Mirror Langevin Algorithm Converges with Vanishing Biascited by 37
  3. 2020Primal Dual Interpretation of the Proximal Stochastic Gradient Langevin Algorithmcited by 35
  4. 2019Proximal Langevin Algorithm: Rapid Convergence Under Isoperimetrycited by 34
  5. 2018Subgradient methods for sharp weakly convex functionscited by 33
  6. 2019Stochastic Runge-Kutta Accelerates Langevin Monte Carlo and Beyondcited by 33
  7. 2020Projected Stochastic Gradient Langevin Algorithms for Constrained Sampling and Non-Convex Learningcited by 33
  8. 2021Fast Approximation of the Sliced-Wasserstein Distance Using Concentration of Random Projectionscited by 33
  9. 2017Convergence Rate of Riemannian Hamiltonian Monte Carlo and Faster Polytope Volume Computationcited by 32
  10. 2014On the folded normal distributioncited by 31
  11. 2019Langevin Monte Carlo without smoothnesscited by 28
  12. 2021Sqrt(d) Dimension Dependence of Langevin Monte Carlocited by 25
  13. 2022SAPD+: An Accelerated Stochastic Method for Nonconvex-Concave Minimax Problemscited by 23
  14. 2018On stochastic gradient Langevin dynamics with dependent data streams in the logconcave casecited by 20
  15. 2019An Efficient Sampling Algorithm for Non-smooth Composite Potentialscited by 19
  16. 2020Accelerating Nonconvex Learning via Replica Exchange Langevin Diffusioncited by 16
  17. 2019On stochastic gradient Langevin dynamics with dependent data streams: the fully non-convex casecited by 14
  18. 2022Penalized Overdamped and Underdamped Langevin Monte Carlo Algorithms for Constrained Samplingcited by 13
  19. 2023Subgradient Langevin Methods for Sampling from Non-smooth Potentialscited by 11
  20. 2024Posterior and variational inference for deep neural networks with heavy-tailed weightscited by 10
  21. 2022Bregman Proximal Langevin Monte Carlo via Bregman--Moreau Envelopescited by 7
  22. 2019Implicit Langevin Algorithms for Sampling From Log-concave Densitiescited by 6
  23. 2021Gradient-Based Markov Chain Monte Carlo for Bayesian Inference With Non-Differentiable Priorscited by 6
  24. 2022Heavy-tailed Sampling via Transformed Unadjusted Langevin Algorithmcited by 6
  25. 2013Gradient-Based Adaptive Stochastic Search for Non-Differentiable Optimizationcited by 4
  26. 2023Distributionally Robust Learning with Weakly Convex Losses: Convergence Rates and Finite-Sample Guaranteescited by 4
  27. 2023Mean-Square Analysis of Discretized Ito Diffusions for Heavy-tailed Samplingcited by 4
  28. 2022Langevin algorithms for Markovian Neural Networks and Deep Stochastic controlcited by 2
  29. 2016Ergodicity of Markov chain Monte Carlo with reversible proposalcited by 1
  30. 2022Theoretical properties of Bayesian Student-t linear regressioncited by 1

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2026: 3.

3 papers.

  1. 2026The Tamed Subgradient Unadjusted Langevin Algorithm beyond Convexitycited by 1
  2. 2026Reflected Anchored Langevin Algorithmscited by 1
  3. 2026Accelerating Non-Smooth and Heavy-Tailed Samplingcited by 0