# Linear convergence of proximal descent schemes on the Wasserstein space

arXiv: https://arxiv.org/abs/2411.15067 · HTML: https://mlsanity.com/paper/2411.15067/

## 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 (most cited first, 21 of 21)

- [A Mean Field View of the Landscape of Two-Layers Neural Networks](https://mlsanity.com/paper/1804.06561/) (2018)
- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [Maximum Mean Discrepancy Gradient Flow](https://mlsanity.com/paper/1906.04370/) (2019)
- [Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problem](https://mlsanity.com/paper/1802.08089/) (2018)
- [Trainability and Accuracy of Neural Networks: An Interacting Particle System Approach](https://mlsanity.com/paper/1805.00915/) (2018)
- [Large-Scale Wasserstein Gradient Flows](https://mlsanity.com/paper/2106.00736/) (2021)
- [Mean-Field Langevin Dynamics: Exponential Convergence and Annealing](https://mlsanity.com/paper/2202.01009/) (2022)
- [Efficiency of minimizing compositions of convex functions and smooth maps](https://mlsanity.com/paper/1605.00125/) (2016)
- [Convex Analysis of the Mean Field Langevin Dynamics](https://mlsanity.com/paper/2201.10469/) (2022)
- [Stochastic Particle Gradient Descent for Infinite Ensembles](https://mlsanity.com/paper/1712.05438/) (2017)
- [Mean-Field Langevin Dynamics and Energy Landscape of Neural Networks](https://mlsanity.com/paper/1905.07769/) (2019)
- [Policy Optimization as Wasserstein Gradient Flows](https://mlsanity.com/paper/1808.03030/) (2018)
- [Convergence of flow-based generative models via proximal gradient descent in Wasserstein space](https://mlsanity.com/paper/2310.17582/) (2023)
- [Uniform-in-time propagation of chaos for mean field Langevin dynamics](https://mlsanity.com/paper/2212.03050/) (2022)
- [Convergence of Policy Gradient for Entropy Regularized MDPs with Neural Network Approximation in the Mean-Field Regime](https://mlsanity.com/paper/2201.07296/) (2022)
- [Entropic Fictitious Play for Mean Field Optimization Problem](https://mlsanity.com/paper/2202.05841/) (2022)
- [Wasserstein Proximal Coordinate Gradient Algorithms](https://mlsanity.com/paper/2405.04628/) (2024)
- [Non-geodesically-convex optimization in the Wasserstein space](https://mlsanity.com/paper/2406.00502/) (2024)
- [Forward-Euler time-discretization for Wasserstein gradient flows can be wrong](https://mlsanity.com/paper/2406.08209/) (2024)
- [Generative Modeling by Minimizing the Wasserstein-2 Loss](https://mlsanity.com/paper/2406.13619/) (2024)
- [Proximal Mean Field Learning in Shallow Neural Networks](https://mlsanity.com/paper/2210.13879/) (2022)

## Built on by (most cited first, 4 of 4)

- [Convergence Analysis of the Wasserstein Proximal Algorithm beyond Geodesic Convexity](https://mlsanity.com/paper/2501.14993/) (2025)
- [A note on convergence of Wasserstein policy optimization](https://mlsanity.com/paper/2605.22622/) (2026)
- [Wasserstein Saddle-Free Newton: Saddle Escape and Local Spectral Convergence](https://mlsanity.com/paper/2605.17963/) (2026)
- [On propagation of chaos for the Fisher-Rao gradient flow in entropic mean-field optimization](https://mlsanity.com/paper/2602.15094/) (2026)

## 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.