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Coupled Wasserstein Gradient Flows for Min-Max and Cooperative Games
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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.
Forward citations
Cited by 7 Pith papers
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Local exponential stability of mean-field Langevin descent-ascent and associated particle system
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The Proximal Sampler has exponential convergence in relative Fisher information for strongly log-concave targets, matching the rate of continuous-time Langevin dynamics.
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A Stiefel-manifold dynamical approximation for Wasserstein gradient flows represents the evolving transport map by a moving linear subspace and controls the Wasserstein error through an adaptive background space.
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A min-max gradient flow with a dynamically adapted KL penalty is proposed as a particle method for approximating optimal transport couplings, with claimed convergence to the optimal plan.
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