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

arXiv: https://arxiv.org/abs/2212.08791 · HTML: https://mlsanity.com/paper/2212.08791/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 33 of its 63 references that are in the map.
- First posted on arXiv 17 Dec 2022.

## Builds on (most cited first, 30 of 33)

- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [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)
- [Breaking the Curse of Dimensionality with Convex Neural Networks](https://mlsanity.com/paper/1412.8690/) (2014)
- [On Gradient Descent Ascent for Nonconvex-Concave Minimax Problems](https://mlsanity.com/paper/1906.00331/) (2019)
- [Non-convex learning via Stochastic Gradient Langevin Dynamics: a nonasymptotic analysis](https://mlsanity.com/paper/1702.03849/) (2017)
- [Optimistic mirror descent in saddle-point problems: Going the extra (gradient) mile](https://mlsanity.com/paper/1807.02629/) (2018)
- [Trainability and Accuracy of Neural Networks: An Interacting Particle System Approach](https://mlsanity.com/paper/1805.00915/) (2018)
- [The Limit Points of (Optimistic) Gradient Descent in Min-Max Optimization](https://mlsanity.com/paper/1807.03907/) (2018)
- [Last-Iterate Convergence: Zero-Sum Games and Constrained Min-Max Optimization](https://mlsanity.com/paper/1807.04252/) (2018)
- [Mean-Field Langevin Dynamics: Exponential Convergence and Annealing](https://mlsanity.com/paper/2202.01009/) (2022)
- [A Mean-field Analysis of Deep ResNet and Beyond: Towards Provable Optimization Via Overparameterization From Depth](https://mlsanity.com/paper/2003.05508/) (2020)
- [Convex Analysis of the Mean Field Langevin Dynamics](https://mlsanity.com/paper/2201.10469/) (2022)
- [Mean-Field Langevin Dynamics and Energy Landscape of Neural Networks](https://mlsanity.com/paper/1905.07769/) (2019)
- [Finding Mixed Nash Equilibria of Generative Adversarial Networks](https://mlsanity.com/paper/1811.02002/) (2018)
- [Faster Single-loop Algorithms for Minimax Optimization without Strong Concavity](https://mlsanity.com/paper/2112.05604/) (2021)
- [Accelerating Langevin Sampling with Birth-death](https://mlsanity.com/paper/1905.09863/) (2019)
- [Mean Field Analysis of Deep Neural Networks](https://mlsanity.com/paper/1903.04440/) (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)
- [Exponential Family Estimation via Adversarial Dynamics Embedding](https://mlsanity.com/paper/1904.12083/) (2019)
- [Birth-death dynamics for sampling: Global convergence, approximations and their asymptotics](https://mlsanity.com/paper/2211.00450/) (2022)
- [A mean-field analysis of two-player zero-sum games](https://mlsanity.com/paper/2002.06277/) (2020)
- [Mean Field Limit of the Learning Dynamics of Multilayer Neural Networks](https://mlsanity.com/paper/1902.02880/) (2019)
- [Last-iterate convergence rates for min-max optimization](https://mlsanity.com/paper/1906.02027/) (2019)
- [Global convergence of neuron birth-death dynamics](https://mlsanity.com/paper/1902.01843/) (2019)
- [An Exponentially Converging Particle Method for the Mixed Nash Equilibrium of Continuous Games](https://mlsanity.com/paper/2211.01280/) (2022)
- [Convergence Rates of Two-Time-Scale Gradient Descent-Ascent Dynamics for Solving Nonconvex Min-Max Problems](https://mlsanity.com/paper/2112.09579/) (2021)
- [Provably convergent quasistatic dynamics for mean-field two-player zero-sum games](https://mlsanity.com/paper/2202.10947/) (2022)
- [Dual Training of Energy-Based Models with Overparametrized Shallow Neural Networks](https://mlsanity.com/paper/2107.05134/) (2021)

## Built on by (most cited first, 13 of 13)

- [Symmetric Mean-field Langevin Dynamics for Distributional Minimax Problems](https://mlsanity.com/paper/2312.01127/) (2023)
- [Negative Stepsizes Make Gradient-Descent-Ascent Converge](https://mlsanity.com/paper/2505.01423/) (2025)
- [Strategic Distribution Shift of Interacting Agents via Coupled Gradient Flows](https://mlsanity.com/paper/2307.01166/) (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)
- [Mirror Descent-Ascent for mean-field min-max problems](https://mlsanity.com/paper/2402.08106/) (2024)
- [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)
- [Non-convex entropic mean-field optimization via Best Response flow](https://mlsanity.com/paper/2505.22760/) (2025)
- [Paths to Equilibrium in Games](https://mlsanity.com/paper/2403.18079/) (2024)
- [Grouped Satisficing Paths in Pure Strategy Games: a Topological Perspective](https://mlsanity.com/paper/2509.23157/) (2025)
- [Local exponential stability of mean-field Langevin descent-ascent and associated particle system](https://mlsanity.com/paper/2602.01564/) (2026)
- [Policy Gradient for Continuous-Time Robust Markov Decision Processes](https://mlsanity.com/paper/2606.04335/) (2026)

## 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.