mlsanityData to 7 Oct 2026

Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problem

arXiv 1802.08089 ↗First posted 22 Feb 2018math.OC cs.IT cs.LGcited by 123

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Abstract (arXiv)

We study sampling as optimization in the space of measures. We focus on gradient flow-based optimization with the Langevin dynamics as a case study. We investigate the source of the bias of the unadjusted Langevin algorithm (ULA) in discrete time, and consider how to remove or reduce the bias. We point out the difficulty is that the heat flow is exactly solvable, but neither its forward nor backward method is implementable in general, except for Gaussian data. We propose the symmetrized Langevin algorithm (SLA), which should have a smaller bias than ULA, at the price of implementing a proximal gradient step in space. We show SLA is in fact consistent for Gaussian target measure, whereas ULA is not. We also illustrate various algorithms explicitly for Gaussian target measure, including gradient descent, proximal gradient, and Forward-Backward, and show they are all consistent.

Key facts. Cited by 123 papers in the map (data through 7 Oct 2026). First posted on arXiv 22 Feb 2018.

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Citing papers per year: 2021: 14; 2022: 22; 2023: 21; 2024: 24; 2025: 25; 2026: 17.

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  1. 2022Variational inference via Wasserstein gradient flowscited by 84
  2. 2024Provably Robust Score-Based Diffusion Posterior Sampling for Plug-and-Play Image Reconstructioncited by 73
  3. 2022Improved analysis for a proximal algorithm for samplingcited by 72
  4. 2022Towards a Theory of Non-Log-Concave Sampling: First-Order Stationarity Guarantees for Langevin Monte Carlocited by 63
  5. 2021Kernel Stein Discrepancy Descentcited by 57
  6. 2021Optimizing Functionals on the Space of Probabilities with Input Convex Neural Networkscited by 53
  7. 2021Variational Wasserstein gradient flowcited by 53
  8. 2022Convex Analysis of the Mean Field Langevin Dynamicscited by 53
  9. 2025Inference-time Scaling of Diffusion Models through Classical Searchcited by 48
  10. 2023Faster high-accuracy log-concave sampling via algorithmic warm startscited by 44
  11. 2021The Mirror Langevin Algorithm Converges with Vanishing Biascited by 37
  12. 2023Forward-backward Gaussian variational inference via JKO in the Bures-Wasserstein Spacecited by 36
38 more
  1. 2022Mirror Descent with Relative Smoothness in Measure Spaces, with application to Sinkhorn and EMcited by 33
  2. 2024Diffusion Models as Constrained Samplers for Optimization with Unknown Constraintscited by 33
  3. 2022Birth-death dynamics for sampling: Global convergence, approximations and their asymptoticscited by 30
  4. 2021Sqrt(d) Dimension Dependence of Langevin Monte Carlocited by 25
  5. 2021A Proximal Algorithm for Sampling from Non-smooth Potentialscited by 23
  6. 2024Implicit Diffusion: Efficient Optimization through Stochastic Samplingcited by 22
  7. 2021A Probabilistic Interpretation of Self-Paced Learning with Applications to Reinforcement Learningcited by 21
  8. 2021A Convergence Theory for SVGD in the Population Limit under Talagrand's Inequality T1cited by 21
  9. 2021Efficient Gradient Flows in Sliced-Wasserstein Spacecited by 21
  10. 2022A Proximal Algorithm for Samplingcited by 21
  11. 2022Resolving the Mixing Time of the Langevin Algorithm to its Stationary Distribution for Log-Concave Samplingcited by 21
  12. 2022Particle-based Variational Inference with Preconditioned Functional Gradient Flowcited by 20
  13. 2023Sampling via Gradient Flows in the Space of Probability Measurescited by 20
  14. 2023Algorithms for mean-field variational inference via polyhedral optimization in the Wasserstein spacecited by 20
  15. 2022Mean-field Variational Inference via Wasserstein Gradient Flowcited by 19
  16. 2024A Unified and General Framework for Continual Learningcited by 19
  17. 2021The query complexity of sampling from strongly log-concave distributions in one dimensioncited by 18
  18. 2022Regularized Stein Variational Gradient Flowcited by 18
  19. 2022Sampling with Mollified Interaction Energy Descentcited by 17
  20. 2022Improved Convergence Rate of Stochastic Gradient Langevin Dynamics with Variance Reduction and its Application to Optimizationcited by 16
  21. 2022Convergence of Stein Variational Gradient Descent under a Weaker Smoothness Conditioncited by 16
  22. 2023An Explicit Expansion of the Kullback-Leibler Divergence along its Fisher-Rao Gradient Flowcited by 16
  23. 2023A connection between Tempering and Entropic Mirror Descentcited by 16
  24. 2024Convergence of coordinate ascent variational inference for log-concave measures via optimal transportcited by 13
  25. 2024Fisher-Rao Gradient Flow: Geodesic Convexity and Functional Inequalitiescited by 13
  26. 2021When is the Convergence Time of Langevin Algorithms Dimension Independent? A Composite Optimization Viewpointcited by 12
  27. 2022Fisher information lower bounds for samplingcited by 12
  28. 2023Query lower bounds for log-concave samplingcited by 12
  29. 2023Fast sampling from constrained spaces using the Metropolis-adjusted Mirror Langevin algorithmcited by 11
  30. 2024Theoretical Guarantees for Variational Inference with Fixed-Variance Mixture of Gaussianscited by 11
  31. 2025Mixing Time of the Proximal Sampler in Relative Fisher Information via Strong Data Processing Inequalitycited by 11
  32. 2022Deterministic Langevin Monte Carlo with Normalizing Flows for Bayesian Inferencecited by 10
  33. 2024Provable Convergence and Limitations of Geometric Tempering for Langevin Dynamicscited by 10
  34. 2024Fast Convergence of Phi-Divergence Along the Unadjusted Langevin Algorithm and Proximal Samplercited by 10
  35. 2024Constrained Sampling with Primal-Dual Langevin Monte Carlocited by 10
  36. 2025Constrained Diffusers for Safe Planning and Controlcited by 9
  37. 2021Sampling from Log-Concave Distributions with Infinity-Distance Guaranteescited by 8
  38. 2023Coin Sampling: Gradient-Based Bayesian Inference without Learning Ratescited by 8

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