First-order Conditions for Optimization in the Wasserstein Space
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913711320465408 |
|---|---|
| author | Lanzetti, Nicolas Bolognani, Saverio Dörfler, Florian |
| author_facet | Lanzetti, Nicolas Bolognani, Saverio Dörfler, Florian |
| contents | We study first-order optimality conditions for constrained optimization in the Wasserstein space, whereby one seeks to minimize a real-valued function over the space of probability measures endowed with the Wasserstein distance. Our analysis combines recent insights on the geometry and the differential structure of the Wasserstein space with more classical calculus of variations. We show that simple rationales such as "setting the derivative to zero" and "gradients are aligned at optimality" carry over to the Wasserstein space. We deploy our tools to study and solve optimization problems in the setting of distributionally robust optimization and statistical inference. The generality of our methodology allows us to naturally deal with functionals, such as mean-variance, Kullback-Leibler divergence, and Wasserstein distance, which are traditionally difficult to study in a unified framework. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_12197 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | First-order Conditions for Optimization in the Wasserstein Space Lanzetti, Nicolas Bolognani, Saverio Dörfler, Florian Optimization and Control We study first-order optimality conditions for constrained optimization in the Wasserstein space, whereby one seeks to minimize a real-valued function over the space of probability measures endowed with the Wasserstein distance. Our analysis combines recent insights on the geometry and the differential structure of the Wasserstein space with more classical calculus of variations. We show that simple rationales such as "setting the derivative to zero" and "gradients are aligned at optimality" carry over to the Wasserstein space. We deploy our tools to study and solve optimization problems in the setting of distributionally robust optimization and statistical inference. The generality of our methodology allows us to naturally deal with functionals, such as mean-variance, Kullback-Leibler divergence, and Wasserstein distance, which are traditionally difficult to study in a unified framework. |
| title | First-order Conditions for Optimization in the Wasserstein Space |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2209.12197 |