# Provably convergent quasistatic dynamics for mean-field two-player zero-sum games

arXiv: https://arxiv.org/abs/2202.10947 · HTML: https://mlsanity.com/paper/2202.10947/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 33 references that are in the map.
- First posted on arXiv 15 Feb 2022.

## Builds on (most cited first, 19 of 19)

- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [A Mean Field View of the Landscape of Two-Layers Neural Networks](https://mlsanity.com/paper/1804.06561/) (2018)
- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [Generalization and Equilibrium in Generative Adversarial Nets (GANs)](https://mlsanity.com/paper/1703.00573/) (2017)
- [Mean Field Analysis of Neural Networks: A Central Limit Theorem](https://mlsanity.com/paper/1808.09372/) (2018)
- [A Mean-field Analysis of Deep ResNet and Beyond: Towards Provable Optimization Via Overparameterization From Depth](https://mlsanity.com/paper/2003.05508/) (2020)
- [Machine Learning from a Continuous Viewpoint](https://mlsanity.com/paper/1912.12777/) (2019)
- [Finding Mixed Nash Equilibria of Generative Adversarial Networks](https://mlsanity.com/paper/1811.02002/) (2018)
- [Mean Field Analysis of Deep Neural Networks](https://mlsanity.com/paper/1903.04440/) (2019)
- [Accelerated Information Gradient flow](https://mlsanity.com/paper/1909.02102/) (2019)
- [On the Banach spaces associated with multi-layer ReLU networks: Function representation, approximation theory and gradient descent dynamics](https://mlsanity.com/paper/2007.15623/) (2020)
- [Alternating the Population and Control Neural Networks to Solve High-Dimensional Stochastic Mean-Field Games](https://mlsanity.com/paper/2002.10113/) (2020)
- [Mean Field Limit of the Learning Dynamics of Multilayer Neural Networks](https://mlsanity.com/paper/1902.02880/) (2019)
- [Global convergence of neuron birth-death dynamics](https://mlsanity.com/paper/1902.01843/) (2019)
- [Wasserstein Proximal of GANs](https://mlsanity.com/paper/2102.06862/) (2021)
- [An Online Learning Approach to Generative Adversarial Networks](https://mlsanity.com/paper/1706.03269/) (2017)
- [Kernelized Wasserstein Natural Gradient](https://mlsanity.com/paper/1910.09652/) (2019)
- [Simultaneous Transport Evolution for Minimax Equilibria on Measures](https://mlsanity.com/paper/2202.06460/) (2022)
- [Entropy dissipation via Information Gamma calculus: Non-reversible stochastic differential equations](https://mlsanity.com/paper/2011.08058/) (2020)

## Built on by (most cited first, 9 of 9)

- [An Exponentially Converging Particle Method for the Mixed Nash Equilibrium of Continuous Games](https://mlsanity.com/paper/2211.01280/) (2022)
- [Two-Scale Gradient Descent Ascent Dynamics Finds Mixed Nash Equilibria of Continuous Games: A Mean-Field Perspective](https://mlsanity.com/paper/2212.08791/) (2022)
- [Symmetric Mean-field Langevin Dynamics for Distributional Minimax Problems](https://mlsanity.com/paper/2312.01127/) (2023)
- [A Fisher-Rao gradient flow for entropic mean-field min-max games](https://mlsanity.com/paper/2405.15834/) (2024)
- [PAPAL: A Provable PArticle-based Primal-Dual ALgorithm for Mixed Nash Equilibrium](https://mlsanity.com/paper/2303.00970/) (2023)
- [Convergence of two-timescale gradient descent ascent dynamics: finite-dimensional and mean-field perspectives](https://mlsanity.com/paper/2501.17122/) (2025)
- [Convergence of Time-Averaged Mean Field Gradient Descent Dynamics for Continuous Multi-Player Zero-Sum Games](https://mlsanity.com/paper/2505.07642/) (2025)
- [Towards Optimal Randomized Strategies in Adversarial Example Game](https://mlsanity.com/paper/2306.16738/) (2023)
- [Local exponential stability of mean-field Langevin descent-ascent and associated particle system](https://mlsanity.com/paper/2602.01564/) (2026)

## Abstract (arXiv)

In this paper, we study the problem of finding mixed Nash equilibrium for mean-field two-player zero-sum games. Solving this problem requires optimizing over two probability distributions. We consider a quasistatic Wasserstein gradient flow dynamics in which one probability distribution follows the Wasserstein gradient flow, while the other one is always at the equilibrium. Theoretical analysis are conducted on this dynamics, showing its convergence to the mixed Nash equilibrium under mild conditions. Inspired by the continuous dynamics of probability distributions, we derive a quasistatic Langevin gradient descent method with inner-outer iterations, and test the method on different problems, including training mixture of GANs.