mlsanityData to 7 Oct 2026

Two-Scale Gradient Descent Ascent Dynamics Finds Mixed Nash Equilibria of Continuous Games: A Mean-Field Perspective

arXiv 2212.08791 ↗First posted 17 Dec 2022math.OC cs.LG math.APcited by 13

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Abstract (arXiv)

Finding the mixed Nash equilibria (MNE) of a two-player zero sum continuous game is an important and challenging problem in machine learning. A canonical algorithm to finding the MNE is the noisy gradient descent ascent method which in the infinite particle limit gives rise to the em Mean-Field Gradient Descent Ascent (GDA) dynamics on the space of probability measures. In this paper, we first study the convergence of a two-scale Mean-Field GDA dynamics for finding the MNE of the entropy-regularized objective. More precisely we show that for each finite temperature (or regularization parameter), the two-scale Mean-Field GDA with a suitable em finite scale ratio converges exponentially to the unique MNE without assuming the convexity or concavity of the interaction potential. The key ingredient of our proof lies in the construction of new Lyapunov functions that dissipate exponentially along the Mean-Field GDA. We further study the simulated annealing of the Mean-Field GDA dynamics. We show that with a temperature schedule that decays logarithmically in time the annealed Mean-Field GDA converges to the MNE of the original unregularized objective.

Key facts. Cited by 13 papers in the map (data through 7 Oct 2026). Builds on 33 of its 63 references that are in the map. First posted on arXiv 17 Dec 2022.

Builds on

33 of 63 references.

References · 33

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  9. 2018Trainability and Accuracy of Neural Networks: An Interacting Particle System Approachcited by 105
  10. 2018The Limit Points of (Optimistic) Gradient Descent in Min-Max Optimizationcited by 98
  11. 2018Last-Iterate Convergence: Zero-Sum Games and Constrained Min-Max Optimizationcited by 97
  12. 2022Mean-Field Langevin Dynamics: Exponential Convergence and Annealingcited by 60
21 more
  1. 2020A Mean-field Analysis of Deep ResNet and Beyond: Towards Provable Optimization Via Overparameterization From Depthcited by 56
  2. 2022Convex Analysis of the Mean Field Langevin Dynamicscited by 53
  3. 2019Mean-Field Langevin Dynamics and Energy Landscape of Neural Networkscited by 49
  4. 2018Finding Mixed Nash Equilibria of Generative Adversarial Networkscited by 48
  5. 2021Faster Single-loop Algorithms for Minimax Optimization without Strong Concavitycited by 47
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Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 3; 2024: 3; 2025: 5; 2026: 2.

13 papers.

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  1. 2023Symmetric Mean-field Langevin Dynamics for Distributional Minimax Problemscited by 11
  2. 2025Negative Stepsizes Make Gradient-Descent-Ascent Convergecited by 9
  3. 2023Strategic Distribution Shift of Interacting Agents via Coupled Gradient Flowscited by 5
  4. 2024A Fisher-Rao gradient flow for entropic mean-field min-max gamescited by 3
  5. 2023PAPAL: A Provable PArticle-based Primal-Dual ALgorithm for Mixed Nash Equilibriumcited by 2
  6. 2025Convergence of two-timescale gradient descent ascent dynamics: finite-dimensional and mean-field perspectivescited by 2
  7. 2025Convergence of Time-Averaged Mean Field Gradient Descent Dynamics for Continuous Multi-Player Zero-Sum Gamescited by 2
  8. 2025Non-convex entropic mean-field optimization via Best Response flowcited by 2
  9. 2024Mirror Descent-Ascent for mean-field min-max problemscited by 1
  10. 2024Paths to Equilibrium in Gamescited by 1
  11. 2025Grouped Satisficing Paths in Pure Strategy Games: a Topological Perspectivecited by 0
  12. 2026Local exponential stability of mean-field Langevin descent-ascent and associated particle systemcited by 0
1 more
  1. 2026Policy Gradient for Continuous-Time Robust Markov Decision Processescited by 0