Coupled Wasserstein Gradient Flows for Min-Max and Cooperative Games
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_ | 1866929705259630592 |
|---|---|
| author | Conger, Lauren Hoffmann, Franca Mazumdar, Eric Ratliff, Lillian J. |
| author_facet | Conger, Lauren Hoffmann, Franca Mazumdar, Eric Ratliff, Lillian J. |
| contents | We propose a framework for two-player infinite-dimensional games with cooperative or competitive structure. These games take the form of coupled partial differential equations in which players optimize over a space of measures, driven by either a gradient descent or gradient descent-ascent in Wasserstein-2 space. We characterize the properties of the Nash equilibrium of the system, and relate it to the steady state of the dynamics. In the min-max setting, we show, under sufficient convexity conditions, that solutions converge exponentially fast and with explicit rate to the unique Nash equilibrium. Similar results are obtained for the cooperative setting. We apply this framework to distribution shift induced by interactions among a strategic population of agents and an algorithm, proving additional convergence results in the timescale-separated setting. We illustrate the performance of our model on (i) real data from an economics study on Colombia census data, (ii) feature modification in loan applications, and (iii) performative prediction. The numerical experiments demonstrate the importance of distribution-level, rather than moment-level, modeling. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_07403 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Coupled Wasserstein Gradient Flows for Min-Max and Cooperative Games Conger, Lauren Hoffmann, Franca Mazumdar, Eric Ratliff, Lillian J. Analysis of PDEs Optimization and Control 35G50, 91A25, 49J35 We propose a framework for two-player infinite-dimensional games with cooperative or competitive structure. These games take the form of coupled partial differential equations in which players optimize over a space of measures, driven by either a gradient descent or gradient descent-ascent in Wasserstein-2 space. We characterize the properties of the Nash equilibrium of the system, and relate it to the steady state of the dynamics. In the min-max setting, we show, under sufficient convexity conditions, that solutions converge exponentially fast and with explicit rate to the unique Nash equilibrium. Similar results are obtained for the cooperative setting. We apply this framework to distribution shift induced by interactions among a strategic population of agents and an algorithm, proving additional convergence results in the timescale-separated setting. We illustrate the performance of our model on (i) real data from an economics study on Colombia census data, (ii) feature modification in loan applications, and (iii) performative prediction. The numerical experiments demonstrate the importance of distribution-level, rather than moment-level, modeling. |
| title | Coupled Wasserstein Gradient Flows for Min-Max and Cooperative Games |
| topic | Analysis of PDEs Optimization and Control 35G50, 91A25, 49J35 |
| url | https://arxiv.org/abs/2411.07403 |