mlsanityData to 14 Oct 2026

Linear convergence of proximal descent schemes on the Wasserstein space

arXiv 2411.15067 ↗First posted 22 Nov 2024math.OC cs.LG math.PRcited by 4

Not on the map. Why?

Abstract (arXiv)

We investigate proximal descent methods, inspired by the minimizing movement scheme introduced by Jordan, Kinderlehrer and Otto, for optimizing entropy-regularized functionals on the Wasserstein space. We establish linear convergence under flat convexity assumptions, thereby relaxing the common reliance on geodesic convexity. Our analysis circumvents the need for discrete-time adaptations of the Evolution Variational Inequality (EVI). Instead, we leverage a uniform logarithmic Sobolev inequality (LSI) and the entropy ``sandwich" lemma, extending the analysis from arXiv:2201.10469 and arXiv:2202.01009. The major challenge in the proof via LSI is to show that the relative Fisher information is well-defined at every step of the scheme. Since the relative entropy is not Wasserstein differentiable, we prove that along the scheme the iterates belong to a certain class of Sobolev regularity, and hence the relative entropy has a unique Wasserstein sub-gradient, and that the relative Fisher information is indeed finite.

Key facts. Cited by 4 papers in the map (data through 14 Oct 2026). Builds on 21 of its 62 references that are in the map. First posted on arXiv 22 Nov 2024.

Builds on

21 of 62 references.

References · 21

  1. 2018A Mean Field View of the Landscape of Two-Layers Neural Networkscited by 466
  2. 2018On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transportcited by 426
  3. 2019Maximum Mean Discrepancy Gradient Flowcited by 127
  4. 2018Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problemcited by 123
  5. 2018Trainability and Accuracy of Neural Networks: An Interacting Particle System Approachcited by 106
  6. 2021Large-Scale Wasserstein Gradient Flowscited by 77
  7. 2022Mean-Field Langevin Dynamics: Exponential Convergence and Annealingcited by 61
  8. 2016Efficiency of minimizing compositions of convex functions and smooth mapscited by 59
  9. 2022Convex Analysis of the Mean Field Langevin Dynamicscited by 55
  10. 2017Stochastic Particle Gradient Descent for Infinite Ensemblescited by 52
  11. 2019Mean-Field Langevin Dynamics and Energy Landscape of Neural Networkscited by 50
  12. 2018Policy Optimization as Wasserstein Gradient Flowscited by 34
9 more
  1. 2023Convergence of flow-based generative models via proximal gradient descent in Wasserstein spacecited by 34
  2. 2022Uniform-in-time propagation of chaos for mean field Langevin dynamicscited by 23
  3. 2022Convergence of Policy Gradient for Entropy Regularized MDPs with Neural Network Approximation in the Mean-Field Regimecited by 21
  4. 2022Entropic Fictitious Play for Mean Field Optimization Problemcited by 12
  5. 2024Wasserstein Proximal Coordinate Gradient Algorithmscited by 7
  6. 2024Non-geodesically-convex optimization in the Wasserstein spacecited by 5
  7. 2024Forward-Euler time-discretization for Wasserstein gradient flows can be wrongcited by 3
  8. 2024Generative Modeling by Minimizing the Wasserstein-2 Losscited by 3
  9. 2022Proximal Mean Field Learning in Shallow Neural Networkscited by 1

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2025: 1; 2026: 3.

4 papers.

  1. 2025Convergence Analysis of the Wasserstein Proximal Algorithm beyond Geodesic Convexitycited by 2
  2. 2026A note on convergence of Wasserstein policy optimizationcited by 2
  3. 2026Wasserstein Saddle-Free Newton: Saddle Escape and Local Spectral Convergencecited by 1
  4. 2026On propagation of chaos for the Fisher-Rao gradient flow in entropic mean-field optimizationcited by 0