mlsanityData to 7 Oct 2026

Mirror Descent with Relative Smoothness in Measure Spaces, with application to Sinkhorn and EM

arXiv 2206.08873 ↗First posted 17 Jun 2022math.OC cs.LG stat.MLcited by 33

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Abstract (arXiv)

Many problems in machine learning can be formulated as optimizing a convex functional over a vector space of measures. This paper studies the convergence of the mirror descent algorithm in this infinite-dimensional setting. Defining Bregman divergences through directional derivatives, we derive the convergence of the scheme for relatively smooth and convex pairs of functionals. Such assumptions allow to handle non-smooth functionals such as the Kullback--Leibler (KL) divergence. Applying our result to joint distributions and KL, we show that Sinkhorn's primal iterations for entropic optimal transport in the continuous setting correspond to a mirror descent, and we obtain a new proof of its (sub)linear convergence. We also show that Expectation Maximization (EM) can always formally be written as a mirror descent. When optimizing only on the latent distribution while fixing the mixtures parameters -- which corresponds to the Richardson--Lucy deconvolution scheme in signal processing -- we derive sublinear rates of convergence.

Key facts. Cited by 33 papers in the map (data through 7 Oct 2026). Builds on 16 of its 62 references that are in the map. First posted on arXiv 17 Jun 2022.

Builds on

16 of 62 references.

References · 16

  1. 2018A Mean Field View of the Landscape of Two-Layers Neural Networkscited by 463
  2. 2018On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transportcited by 422
  3. 2015Training generative neural networks via Maximum Mean Discrepancy optimizationcited by 149
  4. 2018Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problemcited by 123
  5. 2019Maximum Mean Discrepancy Gradient Flowcited by 123
  6. 2016Relatively-Smooth Convex Optimization by First-Order Methods, and Applicationscited by 108
  7. 2020A Non-Asymptotic Analysis for Stein Variational Gradient Descentcited by 61
  8. 2021Kernel Stein Discrepancy Descentcited by 57
  9. 2015Entropic and displacement interpolation: a computational approach using the Hilbert metriccited by 34
  10. 2019Sinkhorn Barycenters with Free Support via Frank-Wolfe Algorithmcited by 27
  11. 2020A gradient descent perspective on Sinkhorncited by 24
  12. 2021Convergence Rates of Gradient Methods for Convex Optimization in the Space of Measurescited by 19
4 more
  1. 2020Homeomorphic-Invariance of EM: Non-Asymptotic Convergence in KL Divergence for Exponential Families via Mirror Descentcited by 18
  2. 2019Probability Functional Descent: A Unifying Perspective on GANs, Variational Inference, and Reinforcement Learningcited by 11
  3. 2019Dual Space Preconditioning for Gradient Descentcited by 7
  4. 2019Sinkhorn Algorithm as a Special Case of Stochastic Mirror Descentcited by 7

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Citing papers per year: 2022: 1; 2023: 4; 2024: 12; 2025: 11; 2026: 5.

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  1. 2023A Fisher-Rao gradient flow for entropy-regularised Markov decision processes in Polish spacescited by 19
  2. 2023A connection between Tempering and Entropic Mirror Descentcited by 16
  3. 2025Flow Density Control: Generative Optimization Beyond Entropy-Regularized Fine-Tuningcited by 15
  4. 2024Fisher-Rao Gradient Flow: Geodesic Convexity and Functional Inequalitiescited by 13
  5. 2025Provable Maximum Entropy Manifold Exploration via Diffusion Modelscited by 12
  6. 2024Theoretical Guarantees for Variational Inference with Fixed-Variance Mixture of Gaussianscited by 11
  7. 2024Entropy annealing for policy mirror descent in continuous time and spacecited by 9
  8. 2024Annealed Sinkhorn for Optimal Transport: convergence, regularization path and debiasingcited by 6
  9. 2025Variational Inference with Mixtures of Isotropic Gaussianscited by 6
  10. 2025DiffEM: Learning from Corrupted Data with Diffusion Models via Expectation Maximizationcited by 6
  11. 2022Mirror Sinkhorn: Fast Online Optimization on Transport Polytopescited by 5
  12. 2025Sequential Monte Carlo approximations of Wasserstein--Fisher--Rao gradient flowscited by 5
21 more
  1. 2024Analysis of Kernel Mirror Prox for Measure Optimizationcited by 4
  2. 2024Computing Optimal Regularizers for Online Linear Optimizationcited by 4
  3. 2024Inclusive KL Gradient Flows: Otto-Wasserstein, Fisher-Rao-Gaussian, and Local-Estimator Dynamicscited by 4
  4. 2024A semiconcavity approach to stability of entropic plans and exponential convergence of Sinkhorn's algorithmcited by 4
  5. 2025PPO in the Fisher-Rao geometrycited by 4
  6. 2024A Fisher-Rao gradient flow for entropic mean-field min-max gamescited by 3
  7. 2025Variational Online Mirror Descent for Robust Learning in Schr"odinger Bridgecited by 2
  8. 2026A Unified Density Operator View of Flow Control and Mergingcited by 2
  9. 2023Sinkhorn Flow: A Continuous-Time Framework for Understanding and Generalizing the Sinkhorn Algorithmcited by 1
  10. 2024Mirror Descent-Ascent for mean-field min-max problemscited by 1
  11. 2024Fast convergence of the Expectation Maximization algorithm under a logarithmic Sobolev inequalitycited by 1
  12. 2024Dual Approximation Policy Optimizationcited by 1
  13. 2025The Geometry of Nonlinear Reinforcement Learningcited by 1
  14. 2026Robust Sublinear Convergence Rates for Iterative Bregman Projectionscited by 1
  15. 2023Convergences for Minimax Optimization Problems over Infinite-Dimensional Spaces Towards Stability in Adversarial Trainingcited by 0
  16. 2025Hessian stability and convergence rates for entropic and Sinkhorn potentials via semiconcavitycited by 0
  17. 2025Designing Algorithms for Entropic Optimal Transport from an Optimisation Perspectivecited by 0
  18. 2025Thermodynamic structure of the Sinkhorn flowcited by 0
  19. 2026On propagation of chaos for the Fisher-Rao gradient flow in entropic mean-field optimizationcited by 0
  20. 2026The Dually Flat Geometry of Planning as Inferencecited by 0
  21. 2026An analysis of Mirror-Descent Soft Actor-Criticcited by 0