# Iterative Minimax Games with Coupled Linear Constraints

arXiv: https://arxiv.org/abs/2212.04672 · HTML: https://mlsanity.com/paper/2212.04672/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 49 of its 76 references that are in the map.
- First posted on arXiv 9 Dec 2022.

## Builds on (most cited first, 30 of 49)

- [Generative Adversarial Imitation Learning](https://mlsanity.com/paper/1606.03476/) (2016)
- [ZOO: Zeroth Order Optimization based Black-box Attacks to Deep Neural Networks without Training Substitute Models](https://mlsanity.com/paper/1708.03999/) (2017)
- [On Gradient Descent Ascent for Nonconvex-Concave Minimax Problems](https://mlsanity.com/paper/1906.00331/) (2019)
- [Training GANs with Optimism](https://mlsanity.com/paper/1711.00141/) (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)
- [A Unified Analysis of Extra-gradient and Optimistic Gradient Methods for Saddle Point Problems: Proximal Point Approach](https://mlsanity.com/paper/1901.08511/) (2019)
- [Near-Optimal Algorithms for Minimax Optimization](https://mlsanity.com/paper/2002.02417/) (2020)
- [The Limit Points of (Optimistic) Gradient Descent in Min-Max Optimization](https://mlsanity.com/paper/1807.03907/) (2018)
- [Efficient Methods for Structured Nonconvex-Nonconcave Min-Max Optimization](https://mlsanity.com/paper/2011.00364/) (2020)
- [Lower complexity bounds of first-order methods for convex-concave bilinear saddle-point problems](https://mlsanity.com/paper/1808.02901/) (2018)
- [Efficient Algorithms for Smooth Minimax Optimization](https://mlsanity.com/paper/1907.01543/) (2019)
- [Weakly-Convex Concave Min-Max Optimization: Provable Algorithms and Applications in Machine Learning](https://mlsanity.com/paper/1810.02060/) (2018)
- [Fast Extra Gradient Methods for Smooth Structured Nonconvex-Nonconcave Minimax Problems](https://mlsanity.com/paper/2106.02326/) (2021)
- [Error bounds, quadratic growth, and linear convergence of proximal methods](https://mlsanity.com/paper/1602.06661/) (2016)
- [Negative Momentum for Improved Game Dynamics](https://mlsanity.com/paper/1807.04740/) (2018)
- [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)
- [Hybrid Block Successive Approximation for One-Sided Non-Convex Min-Max Problems: Algorithms and Applications](https://mlsanity.com/paper/1902.08294/) (2019)
- [On the Convergence and Robustness of Training GANs with Regularized Optimal Transport](https://mlsanity.com/paper/1802.08249/) (2018)
- [The Complexity of Nonconvex-Strongly-Concave Minimax Optimization](https://mlsanity.com/paper/2103.15888/) (2021)
- [Efficient Search of First-Order Nash Equilibria in Nonconvex-Concave Smooth Min-Max Problems](https://mlsanity.com/paper/2002.07919/) (2020)
- [Global Convergence and Variance-Reduced Optimization for a Class of Nonconvex-Nonconcave Minimax Problems](https://mlsanity.com/paper/2002.09621/) (2020)
- [Complexity Lower Bounds for Nonconvex-Strongly-Concave Min-Max Optimization](https://mlsanity.com/paper/2104.08708/) (2021)
- [Robust Optimization over Multiple Domains](https://mlsanity.com/paper/1805.07588/) (2018)
- [Near-optimal Local Convergence of Alternating Gradient Descent-Ascent for Minimax Optimization](https://mlsanity.com/paper/2102.09468/) (2021)
- [Accelerated Schemes For A Class of Variational Inequalities](https://mlsanity.com/paper/1403.4164/) (2014)
- [An accelerated inexact proximal point method for solving nonconvex-concave min-max problems](https://mlsanity.com/paper/1905.13433/) (2019)
- [Optimistic Dual Extrapolation for Coherent Non-monotone Variational Inequalities](https://mlsanity.com/paper/2103.04410/) (2021)
- [Solving Nonconvex-Nonconcave Min-Max Problems exhibiting Weak Minty Solutions](https://mlsanity.com/paper/2201.12247/) (2022)

## Abstract (arXiv)

The study of nonconvex minimax games has gained significant momentum in machine learning and decision science communities due to their fundamental connections to adversarial training scenarios. This work develops a primal-dual alternating proximal gradient (PDAPG) algorithm framework for resolving iterative minimax games featuring nonsmooth nonconvex objectives subject to coupled linear constraints. We establish rigorous convergence guarantees for both nonconvex-strongly concave and nonconvex-concave game configurations, demonstrating that PDAPG achieves an varepsilon-stationary solution within Oleft( varepsilon -2 right) iterations for strongly concave settings and Oleft( varepsilon -4 right) iterations for concave scenarios. Our analysis provides the first known iteration complexity bounds for this class of constrained minimax games, particularly addressing the critical challenge of coupled linear constraints that induce inherent interdependencies among strategy variables. The proposed game-theoretic framework advances existing solution methodologies by simultaneously handling nonsmooth components and coordinated constraint structures through alternating primal-dual updates.