# Universal Gradient Descent Ascent Method for Nonconvex-Nonconcave Minimax Optimization

arXiv: https://arxiv.org/abs/2212.12978 · HTML: https://mlsanity.com/paper/2212.12978/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 36 of its 62 references that are in the map.
- First posted on arXiv 26 Dec 2022.

## Builds on (most cited first, 30 of 36)

- [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)
- [Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)](https://mlsanity.com/paper/1511.07289/) (2015)
- [Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Lojasiewicz Condition](https://mlsanity.com/paper/1608.04636/) (2016)
- [Certifying Some Distributional Robustness with Principled Adversarial Training](https://mlsanity.com/paper/1710.10571/) (2017)
- [On Gradient Descent Ascent for Nonconvex-Concave Minimax Problems](https://mlsanity.com/paper/1906.00331/) (2019)
- [Large-Scale Methods for Distributionally Robust Optimization](https://mlsanity.com/paper/2010.05893/) (2020)
- [Deep Decentralized Multi-task Multi-Agent Reinforcement Learning under Partial Observability](https://mlsanity.com/paper/1703.06182/) (2017)
- [Solving a Class of Non-Convex Min-Max Games Using Iterative First Order Methods](https://mlsanity.com/paper/1902.08297/) (2019)
- [A Variational Inequality Perspective on Generative Adversarial Networks](https://mlsanity.com/paper/1802.10551/) (2018)
- [Optimistic mirror descent in saddle-point problems: Going the extra (gradient) mile](https://mlsanity.com/paper/1807.02629/) (2018)
- [Efficient Methods for Structured Nonconvex-Nonconcave Min-Max Optimization](https://mlsanity.com/paper/2011.00364/) (2020)
- [Accelerated Algorithms for Smooth Convex-Concave Minimax Problems with O(1/k2) Rate on Squared Gradient Norm](https://mlsanity.com/paper/2102.07922/) (2021)
- [Fast Extra Gradient Methods for Smooth Structured Nonconvex-Nonconcave Minimax Problems](https://mlsanity.com/paper/2106.02326/) (2021)
- [A Single-Loop Smoothed Gradient Descent-Ascent Algorithm for Nonconvex-Concave Min-Max Problems](https://mlsanity.com/paper/2010.15768/) (2020)
- [A Unified Single-loop Alternating Gradient Projection Algorithm for Nonconvex-Concave and Convex-Nonconcave Minimax Problems](https://mlsanity.com/paper/2006.02032/) (2020)
- [The limits of min-max optimization algorithms: convergence to spurious non-critical sets](https://mlsanity.com/paper/2006.09065/) (2020)
- [Faster Single-loop Algorithms for Minimax Optimization without Strong Concavity](https://mlsanity.com/paper/2112.05604/) (2021)
- [Global Convergence and Variance-Reduced Optimization for a Class of Nonconvex-Nonconcave Minimax Problems](https://mlsanity.com/paper/2002.09621/) (2020)
- [Calculus of the exponent of Kurdyka-Lojasiewicz inequality and its applications to linear convergence of first-order methods](https://mlsanity.com/paper/1602.02915/) (2016)
- [First-order Convergence Theory for Weakly-Convex-Weakly-Concave Min-max Problems](https://mlsanity.com/paper/1810.10207/) (2018)
- [On Solving Minimax Optimization Locally: A Follow-the-Ridge Approach](https://mlsanity.com/paper/1910.07512/) (2019)
- [Extragradient method with variance reduction for stochastic variational inequalities](https://mlsanity.com/paper/1703.00260/) (2017)
- [Accelerated Single-Call Methods for Constrained Min-Max Optimization](https://mlsanity.com/paper/2210.03096/) (2022)
- [On the Convergence Properties of Non-Euclidean Extragradient Methods for Variational Inequalities with Generalized Monotone Operators](https://mlsanity.com/paper/1311.2776/) (2013)
- [Towards Better Understanding of Adaptive Gradient Algorithms in Generative Adversarial Nets](https://mlsanity.com/paper/1912.11940/) (2019)
- [Optimistic Dual Extrapolation for Coherent Non-monotone Variational Inequalities](https://mlsanity.com/paper/2103.04410/) (2021)
- [Local Saddle Point Optimization: A Curvature Exploitation Approach](https://mlsanity.com/paper/1805.05751/) (2018)
- [Solving Nonconvex-Nonconcave Min-Max Problems exhibiting Weak Minty Solutions](https://mlsanity.com/paper/2201.12247/) (2022)
- [Accelerated Algorithms for Constrained Nonconvex-Nonconcave Min-Max Optimization and Comonotone Inclusion](https://mlsanity.com/paper/2206.05248/) (2022)

## Built on by (most cited first, 12 of 12)

- [Two-Timescale Gradient Descent Ascent Algorithms for Nonconvex Minimax Optimization](https://mlsanity.com/paper/2408.11974/) (2024)
- [Worst-case generation via minimax optimization in Wasserstein space](https://mlsanity.com/paper/2512.08176/) (2025)
- [Second-Order Min-Max Optimization with Lazy Hessians](https://mlsanity.com/paper/2410.09568/) (2024)
- [Min-Max Optimisation for Nonconvex-Nonconcave Functions Using a Random Zeroth-Order Extragradient Algorithm](https://mlsanity.com/paper/2504.07388/) (2025)
- [Dissipative Gradient Descent Ascent Method: A Control Theory Inspired Algorithm for Min-max Optimization](https://mlsanity.com/paper/2403.09090/) (2024)
- [A Single-Loop Robust Policy Gradient Method for Robust Markov Decision Processes](https://mlsanity.com/paper/2406.00274/) (2024)
- [Generative Conditional Distributions by Neural (Entropic) Optimal Transport](https://mlsanity.com/paper/2406.02317/) (2024)
- [A first-order method for nonconvex-nonconcave minimax problems under a local Kurdyka-Lojasiewicz condition](https://mlsanity.com/paper/2507.01932/) (2025)
- [EXOTIC: An Exact, Optimistic, Tree-Based Algorithm for Min-Max Optimization](https://mlsanity.com/paper/2508.12479/) (2025)
- [ℓ1-norm rank-one symmetric matrix factorization has no spurious second-order stationary points](https://mlsanity.com/paper/2410.05025/) (2024)
- [Gradient Norm Regularization Second-Order Algorithms for Solving Nonconvex-Strongly Concave Minimax Problems](https://mlsanity.com/paper/2411.15769/) (2024)
- [A first-order method for constrained nonconvex-nonconcave minimax optimization](https://mlsanity.com/paper/2510.01168/) (2025)

## Abstract (arXiv)

Nonconvex-nonconcave minimax optimization has received intense attention over the last decade due to its broad applications in machine learning. Most existing algorithms rely on one-sided information, such as the convexity (resp. concavity) of the primal (resp. dual) functions, or other specific structures, such as the Polyak-Lojasiewicz (PL) and Kurdyka-Lojasiewicz (KL) conditions. However, verifying these regularity conditions is challenging in practice. To meet this challenge, we propose a novel universally applicable single-loop algorithm, the doubly smoothed gradient descent ascent method (DS-GDA), which naturally balances the primal and dual updates. That is, DS-GDA with the same hyperparameters is able to uniformly solve nonconvex-concave, convex-nonconcave, and nonconvex-nonconcave problems with one-sided KL properties, achieving convergence with O(ε-4) complexity. Sharper (even optimal) iteration complexity can be obtained when the KL exponent is known. Specifically, under the one-sided KL condition with exponent θ∈(0,1), DS-GDA converges with an iteration complexity of O(ε^-2max2θ,1). They all match the corresponding best results in the literature. Moreover, we show that DS-GDA is practically applicable to general nonconvex-nonconcave problems even without any regularity conditions, such as the PL condition, KL condition, or weak Minty variational inequalities condition. For various challenging nonconvex-nonconcave examples in the literature, including ``Forsaken'', ``Bilinearly-coupled minimax'', ``Sixth-order polynomial'', and ``PolarGame'', the proposed DS-GDA can all get rid of limit cycles. To the best of our knowledge, this is the first first-order algorithm to achieve convergence on all of these formidable problems.