# What is Local Optimality in Nonconvex-Nonconcave Minimax Optimization?

arXiv: https://arxiv.org/abs/1902.00618 · HTML: https://mlsanity.com/paper/1902.00618/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Feb 2019.

## Built on by (most cited first, 20 of 20)

- [Robust Fast Adaptation from Adversarially Explicit Task Distribution Generation](https://mlsanity.com/paper/2407.19523/) (2024)
- [Solving a class of non-convex min-max games using adaptive momentum methods](https://mlsanity.com/paper/2104.12676/) (2021)
- [Solving a Class of Non-Convex Minimax Optimization in Federated Learning](https://mlsanity.com/paper/2310.03613/) (2023)
- [A Decentralized Adaptive Momentum Method for Solving a Class of Min-Max Optimization Problems](https://mlsanity.com/paper/2106.06075/) (2021)
- [Solving Min-Max Optimization with Hidden Structure via Gradient Descent Ascent](https://mlsanity.com/paper/2101.05248/) (2021)
- [Direct-Search for a Class of Stochastic Min-Max Problems](https://mlsanity.com/paper/2102.11386/) (2021)
- [A First Order Meta Stackelberg Method for Robust Federated Learning](https://mlsanity.com/paper/2306.13800/) (2023)
- [Multi-level Monte-Carlo Gradient Methods for Stochastic Optimization with Biased Oracles](https://mlsanity.com/paper/2408.11084/) (2024)
- [Learning Equilibria in Adversarial Team Markov Games: A Nonconvex-Hidden-Concave Min-Max Optimization Problem](https://mlsanity.com/paper/2410.05673/) (2024)
- [Constrained Policy Optimization for Controlled Self-Learning in Conversational AI Systems](https://mlsanity.com/paper/2209.08429/) (2022)
- [Resource Constrained Model Compression via Minimax Optimization for Spiking Neural Networks](https://mlsanity.com/paper/2308.04672/) (2023)
- [A Fully Parameter-Free Second-Order Algorithm for Convex-Concave Minimax Problems](https://mlsanity.com/paper/2407.03571/) (2024)
- [Simultaneous Transport Evolution for Minimax Equilibria on Measures](https://mlsanity.com/paper/2202.06460/) (2022)
- [Competitive Gradient Optimization](https://mlsanity.com/paper/2205.14232/) (2022)
- [Rethinking Teacher-Student Curriculum Learning through the Cooperative Mechanics of Experience](https://mlsanity.com/paper/2404.03084/) (2024)
- [A Mean-Field Analysis of Neural Stochastic Gradient Descent-Ascent for Functional Minimax Optimization](https://mlsanity.com/paper/2404.12312/) (2024)
- [A first-order method for nonconvex-nonconcave minimax problems under a local Kurdyka-Lojasiewicz condition](https://mlsanity.com/paper/2507.01932/) (2025)
- [Enhancing Stability of Physics-Informed Neural Network Training Through Saddle-Point Reformulation](https://mlsanity.com/paper/2507.16008/) (2025)
- [Iterative Minimax Games with Coupled Linear Constraints](https://mlsanity.com/paper/2212.04672/) (2022)
- [Gradient Norm Regularization Second-Order Algorithms for Solving Nonconvex-Strongly Concave Minimax Problems](https://mlsanity.com/paper/2411.15769/) (2024)

## Abstract (arXiv)

Minimax optimization has found extensive applications in modern machine learning, in settings such as generative adversarial networks (GANs), adversarial training and multi-agent reinforcement learning. As most of these applications involve continuous nonconvex-nonconcave formulations, a very basic question arises---"what is a proper definition of local optima?" Most previous work answers this question using classical notions of equilibria from simultaneous games, where the min-player and the max-player act simultaneously. In contrast, most applications in machine learning, including GANs and adversarial training, correspond to sequential games, where the order of which player acts first is crucial (since minimax is in general not equal to maximin due to the nonconvex-nonconcave nature of the problems). The main contribution of this paper is to propose a proper mathematical definition of local optimality for this sequential setting---local minimax, as well as to present its properties and existence results. Finally, we establish a strong connection to a basic local search algorithm---gradient descent ascent (GDA): under mild conditions, all stable limit points of GDA are exactly local minimax points up to some degenerate points.