Linear convergence of proximal descent schemes on the Wasserstein space
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909399988043776 |
|---|---|
| author | Lascu, Razvan-Andrei Majka, Mateusz B. Šiška, David Szpruch, Łukasz |
| author_facet | Lascu, Razvan-Andrei Majka, Mateusz B. Šiška, David Szpruch, Łukasz |
| contents | 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 $I(\cdot|π)$ 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 $\operatorname{KL}(\cdot|π)$ has a unique Wasserstein sub-gradient, and that the relative Fisher information is indeed finite. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_15067 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Linear convergence of proximal descent schemes on the Wasserstein space Lascu, Razvan-Andrei Majka, Mateusz B. Šiška, David Szpruch, Łukasz Optimization and Control Machine Learning Probability 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 $I(\cdot|π)$ 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 $\operatorname{KL}(\cdot|π)$ has a unique Wasserstein sub-gradient, and that the relative Fisher information is indeed finite. |
| title | Linear convergence of proximal descent schemes on the Wasserstein space |
| topic | Optimization and Control Machine Learning Probability |
| url | https://arxiv.org/abs/2411.15067 |