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Coupled Wasserstein Gradient Flows for Min-Max and Cooperative Games

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arxiv 2411.07403 v2 pith:NQWDAFSL submitted 2024-11-11 math.AP math.OC

Lauren Conger , Franca Hoffmann , Eric Mazumdar , Lillian J. Ratliff This is my paper · ORCID
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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.

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Cited by 5 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Local exponential stability of mean-field Langevin descent-ascent and associated particle system

    cs.LG 2026-02 conditional novelty 7.0 of 10

    Starting close to the mixed Nash equilibrium, mean-field Langevin descent-ascent converges exponentially fast in Wasserstein distance.

  2. On gradient descent-ascent flows in metric spaces

    math.FA 2025-06 conditional novelty 7.0 of 10

    A metric-space theory of gradient descent-ascent flows is developed via evolution variational inequalities, yielding existence, uniqueness, and exponential convergence for Wasserstein GDA under strong convexity-concavity.

  3. Mixing Time of the Proximal Sampler in Relative Fisher Information via Strong Data Processing Inequality

    cs.IT 2025-02 accept novelty 7.0 of 10

    The Proximal Sampler has exponential convergence in relative Fisher information for strongly log-concave targets, matching the rate of continuous-time Langevin dynamics.

  4. A Dynamical Approximation Scheme on the Stiefel manifold for Wasserstein Gradient Flows

    math.NA 2026-07 conditional novelty 6.0 of 10

    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.

  5. Nesterov acceleration in optimizing over probability measures

    math.OC 2026-07 reject novelty 5.0 of 10

    Nesterov and heavy-ball acceleration are extended to probability-measure optimization through dual lifting, with claimed non-asymptotic rates matching the Euclidean case.

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