mlsanityData to 14 Oct 2026

The Mirror Langevin Algorithm Converges with Vanishing Bias

arXiv 2109.12077 ↗First posted 24 Sep 2021cs.DS cs.LG math.STcited by 37

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Abstract (arXiv)

The technique of modifying the geometry of a problem from Euclidean to Hessian metric has proved to be quite effective in optimization, and has been the subject of study for sampling. The Mirror Langevin Diffusion (MLD) is a sampling analogue of mirror flow in continuous time, and it has nice convergence properties under log-Sobolev or Poincare inequalities relative to the Hessian metric, as shown by Chewi et al. (2020). In discrete time, a simple discretization of MLD is the Mirror Langevin Algorithm (MLA) studied by Zhang et al. (2020), who showed a biased convergence bound with a non-vanishing bias term (does not go to zero as step size goes to zero). This raised the question of whether we need a better analysis or a better discretization to achieve a vanishing bias. Here we study the basic Mirror Langevin Algorithm and show it indeed has a vanishing bias. We apply mean-square analysis based on Li et al. (2019) and Li et al. (2021) to show the mixing time bound for MLA under the modified self-concordance condition introduced by Zhang et al. (2020).

Key facts. Cited by 37 papers in the map (data through 14 Oct 2026). Builds on 12 of its 22 references that are in the map. First posted on arXiv 24 Sep 2021.

Builds on

12 of 22 references.

References · 12

  1. 2019Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Sufficescited by 227
  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. 2018Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problemcited by 123
  5. 2017User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradientcited by 116
  6. 2018Mirrored Langevin Dynamicscited by 61
  7. 2020Wasserstein Control of Mirror Langevin Monte Carlocited by 55
  8. 2020Efficient constrained sampling via the mirror-Langevin algorithmcited by 55
  9. 2020Exponential ergodicity of mirror-Langevin diffusionscited by 37
  10. 2019Stochastic Runge-Kutta Accelerates Langevin Monte Carlo and Beyondcited by 33
  11. 2021Sqrt(d) Dimension Dependence of Langevin Monte Carlocited by 25
  12. 2019Strong Self-Concordance and Samplingcited by 11

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Citing papers per year: 2022: 5; 2023: 9; 2024: 7; 2025: 10; 2026: 6.

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  1. 2023Mirror Diffusion Models for Constrained and Watermarked Generationcited by 42
  2. 2022Convergence of the Riemannian Langevin Algorithmcited by 25
  3. 2024Zeroth-Order Sampling Methods for Non-Log-Concave Distributions: Alleviating Metastability by Denoising Diffusioncited by 19
  4. 2024In-and-Out: Algorithmic Diffusion for Sampling Convex Bodiescited by 19
  5. 2023Optimal Preconditioning and Fisher Adaptive Langevin Samplingcited by 14
  6. 2022Private Convex Optimization in General Normscited by 13
  7. 2023Query lower bounds for log-concave samplingcited by 12
  8. 2023Fast sampling from constrained spaces using the Metropolis-adjusted Mirror Langevin algorithmcited by 11
  9. 2022Condition-number-independent convergence rate of Riemannian Hamiltonian Monte Carlo with numerical integratorscited by 10
  10. 2023Algorithmic Aspects of the Log-Laplace Transform and a Non-Euclidean Proximal Samplercited by 10
  11. 2024Neural Approximate Mirror Maps for Constrained Diffusion Modelscited by 9
  12. 2024Quantum State Generation with Structure-Preserving Diffusion Modelcited by 8
25 more
  1. 2025Mirror Flow Matching with Heavy-Tailed Priors for Generative Modeling on Convex Domainscited by 8
  2. 2022Bregman Proximal Langevin Monte Carlo via Bregman--Moreau Envelopescited by 7
  3. 2025Non-Reversible Langevin Algorithms for Constrained Samplingcited by 7
  4. 2024Efficient Sampling on Riemannian Manifolds via Langevin MCMCcited by 6
  5. 2023Variational Principles for Mirror Descent and Mirror Langevin Dynamicscited by 5
  6. 2023Mean-Square Analysis of Discretized Ito Diffusions for Heavy-tailed Samplingcited by 4
  7. 2023Non-Log-Concave and Nonsmooth Sampling via Langevin Monte Carlo Algorithmscited by 4
  8. 2025Accelerating Langevin Monte Carlo Sampling: A Large Deviations Analysiscited by 4
  9. 2023Learning Rate Free Sampling in Constrained Domainscited by 3
  10. 2024R'enyi-infinity constrained sampling with d3 membership queriescited by 3
  11. 2025Mirror Mean-Field Langevin Dynamicscited by 3
  12. 2025Anchored Langevin Algorithmscited by 3
  13. 2025Oracle-based Uniform Sampling from Convex Bodiescited by 3
  14. 2022A Dynamical System View of Langevin-Based Non-Convex Samplingcited by 2
  15. 2024Randomized Midpoint Method for Log-Concave Sampling under Constraintscited by 2
  16. 2025Efficient Bayesian Computation Using Plug-and-Play Priors for Poisson Inverse Problemscited by 2
  17. 2025Accelerating Constrained Sampling: A Large Deviations Approachcited by 2
  18. 2026Decentralized Proximal Stochastic Gradient Langevin Dynamicscited by 2
  19. 2026Safe, Scalable, and Accurate Bayes Posterior Sampling for Large-Data Generalized Linear Mixed Modelscited by 1
  20. 2026Mirror Langevin diffusions: Convergence rates and Markov chain approximationscited by 1
  21. 2025Langevin Multiplicative Weights Update with Applications in Polynomial Portfolio Managementcited by 0
  22. 2025Bregman geometry-aware split Gibbs sampling for Bayesian Poisson inverse problemscited by 0
  23. 2026Functional Stochastic Localizationcited by 0
  24. 2026Efficient Diffusion Models under Nonconvex Equality and Inequality constraints via Landingcited by 0
  25. 2026Generating Physically Consistent Molecules with Energy-Based Modelscited by 0