# Convergence of two-timescale gradient descent ascent dynamics: finite-dimensional and mean-field perspectives

arXiv: https://arxiv.org/abs/2501.17122 · HTML: https://mlsanity.com/paper/2501.17122/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 55 references that are in the map.
- First posted on arXiv 28 Jan 2025.

## Builds on (most cited first, 27 of 27)

- [Training GANs with Optimism](https://mlsanity.com/paper/1711.00141/) (2017)
- [Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1507.05021/) (2015)
- [Underdamped Langevin MCMC: A non-asymptotic analysis](https://mlsanity.com/paper/1707.03663/) (2017)
- [Cycles in adversarial regularized learning](https://mlsanity.com/paper/1709.02738/) (2017)
- [The Mechanics of n-Player Differentiable Games](https://mlsanity.com/paper/1802.05642/) (2018)
- [Independent Policy Gradient Methods for Competitive Reinforcement Learning](https://mlsanity.com/paper/2101.04233/) (2021)
- [Interaction Matters: A Note on Non-asymptotic Local Convergence of Generative Adversarial Networks](https://mlsanity.com/paper/1802.06132/) (2018)
- [Sharp convergence rates for Langevin dynamics in the nonconvex setting](https://mlsanity.com/paper/1805.01648/) (2018)
- [Learning time-scales in two-layers neural networks](https://mlsanity.com/paper/2303.00055/) (2023)
- [Finding Mixed Nash Equilibria of Generative Adversarial Networks](https://mlsanity.com/paper/1811.02002/) (2018)
- [Unified Reinforcement Q-Learning for Mean Field Game and Control Problems](https://mlsanity.com/paper/2006.13912/) (2020)
- [Projection Robust Wasserstein Distance and Riemannian Optimization](https://mlsanity.com/paper/2006.07458/) (2020)
- [Communication-Efficient Algorithms for Decentralized and Stochastic Optimization](https://mlsanity.com/paper/1701.03961/) (2017)
- [Differentiable Game Mechanics](https://mlsanity.com/paper/1905.04926/) (2019)
- [Improved Bounds for Discretization of Langevin Diffusions: Near-Optimal Rates without Convexity](https://mlsanity.com/paper/1907.11331/) (2019)
- [A Tale of Two-Timescale Reinforcement Learning with the Tightest Finite-Time Bound](https://mlsanity.com/paper/1911.09157/) (2019)
- [Finite Regret and Cycles with Fixed Step-Size via Alternating Gradient Descent-Ascent](https://mlsanity.com/paper/1907.04392/) (2019)
- [Local Saddle Point Optimization: A Curvature Exploitation Approach](https://mlsanity.com/paper/1805.05751/) (2018)
- [A mean-field analysis of two-player zero-sum games](https://mlsanity.com/paper/2002.06277/) (2020)
- [Saddle-point dynamics: conditions for asymptotic stability of saddle points](https://mlsanity.com/paper/1510.02145/) (2015)
- [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)
- [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)
- [Coupled Wasserstein Gradient Flows for Min-Max and Cooperative Games](https://mlsanity.com/paper/2411.07403/) (2024)
- [On the Convergence of Min-Max Langevin Dynamics and Algorithm](https://mlsanity.com/paper/2412.20471/) (2024)

## Built on by (most cited first, 2 of 2)

- [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)

## Abstract (arXiv)

The two-timescale gradient descent-ascent (GDA) is a canonical gradient algorithm designed to find Nash equilibria in min-max games. We analyze the two-timescale GDA by investigating the effects of learning rate ratios on convergence behavior in both finite-dimensional and mean-field settings. In particular, for finite-dimensional quadratic min-max games, we obtain long-time convergence in near quasi-static regimes through the hypocoercivity method. For mean-field GDA dynamics, we investigate convergence under a finite-scale ratio using a mixed synchronous-reflection coupling technique.