# Weakly-Convex Concave Min-Max Optimization: Provable Algorithms and Applications in Machine Learning

arXiv: https://arxiv.org/abs/1810.02060 · HTML: https://mlsanity.com/paper/1810.02060/

## Key facts

- Cited by 63 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Oct 2018.

## Built on by (most cited first, 30 of 63)

- [Provably Faster Algorithms for Bilevel Optimization](https://mlsanity.com/paper/2106.04692/) (2021)
- [AUC Maximization in the Era of Big Data and AI: A Survey](https://mlsanity.com/paper/2203.15046/) (2022)
- [FedNest: Federated Bilevel, Minimax, and Compositional Optimization](https://mlsanity.com/paper/2205.02215/) (2022)
- [Faster Single-loop Algorithms for Minimax Optimization without Strong Concavity](https://mlsanity.com/paper/2112.05604/) (2021)
- [Federated Minimax Optimization: Improved Convergence Analyses and Algorithms](https://mlsanity.com/paper/2203.04850/) (2022)
- [Differentially Private SGDA for Minimax Problems](https://mlsanity.com/paper/2201.09046/) (2022)
- [What is a Good Metric to Study Generalization of Minimax Learners?](https://mlsanity.com/paper/2206.04502/) (2022)
- [Stochastic Constrained DRO with a Complexity Independent of Sample Size](https://mlsanity.com/paper/2210.05740/) (2022)
- [Two-Timescale Gradient Descent Ascent Algorithms for Nonconvex Minimax Optimization](https://mlsanity.com/paper/2408.11974/) (2024)
- [Large-scale Optimization of Partial AUC in a Range of False Positive Rates](https://mlsanity.com/paper/2203.01505/) (2022)
- [Wasserstein Convergence of Score-based Generative Models under Semiconvexity and Discontinuous Gradients](https://mlsanity.com/paper/2505.03432/) (2025)
- [Stability and Generalization for Markov Chain Stochastic Gradient Methods](https://mlsanity.com/paper/2209.08005/) (2022)
- [Unified Convergence Analysis for Adaptive Optimization with Moving Average Estimator](https://mlsanity.com/paper/2104.14840/) (2021)
- [AdaGDA: Faster Adaptive Gradient Descent Ascent Methods for Minimax Optimization](https://mlsanity.com/paper/2106.16101/) (2021)
- [Nonsmooth Nonconvex-Nonconcave Minimax Optimization: Primal-Dual Balancing and Iteration Complexity Analysis](https://mlsanity.com/paper/2209.10825/) (2022)
- [Zeroth-Order Alternating Gradient Descent Ascent Algorithms for a Class of Nonconvex-Nonconcave Minimax Problems](https://mlsanity.com/paper/2211.13668/) (2022)
- [Oracle Complexity of Single-Loop Switching Subgradient Methods for Non-Smooth Weakly Convex Functional Constrained Optimization](https://mlsanity.com/paper/2301.13314/) (2023)
- [Convergence Rates of Two-Time-Scale Gradient Descent-Ascent Dynamics for Solving Nonconvex Min-Max Problems](https://mlsanity.com/paper/2112.09579/) (2021)
- [Non-Smooth Weakly-Convex Finite-sum Coupled Compositional Optimization](https://mlsanity.com/paper/2310.03234/) (2023)
- [MaxMatch: Semi-Supervised Learning with Worst-Case Consistency](https://mlsanity.com/paper/2209.12611/) (2022)
- [Minimax Optimization with Smooth Algorithmic Adversaries](https://mlsanity.com/paper/2106.01488/) (2021)
- [Nonconvex-Nonconcave Min-Max Optimization with a Small Maximization Domain](https://mlsanity.com/paper/2110.03950/) (2021)
- [A first-order augmented Lagrangian method for constrained minimax optimization](https://mlsanity.com/paper/2301.02060/) (2023)
- [Federated Minimax Optimization with Client Heterogeneity](https://mlsanity.com/paper/2302.04249/) (2023)
- [On the Generalization Ability of Unsupervised Pretraining](https://mlsanity.com/paper/2403.06871/) (2024)
- [Derivative-free Alternating Projection Algorithms for General Nonconvex-Concave Minimax Problems](https://mlsanity.com/paper/2108.00473/) (2021)
- [Generalization Bounds of Nonconvex-(Strongly)-Concave Stochastic Minimax Optimization](https://mlsanity.com/paper/2205.14278/) (2022)
- [Tight Analysis of Extra-gradient and Optimistic Gradient Methods For Nonconvex Minimax Problems](https://mlsanity.com/paper/2210.09382/) (2022)
- [An Efficient Stochastic Algorithm for Decentralized Nonconvex-Strongly-Concave Minimax Optimization](https://mlsanity.com/paper/2212.02387/) (2022)
- [Decentralized gradient descent maximization method for composite nonconvex strongly-concave minimax problems](https://mlsanity.com/paper/2304.02441/) (2023)

## Abstract (arXiv)

Min-max problems have broad applications in machine learning, including learning with non-decomposable loss and learning with robustness to data distribution. Convex-concave min-max problem is an active topic of research with efficient algorithms and sound theoretical foundations developed. However, it remains a challenge to design provably efficient algorithms for non-convex min-max problems with or without smoothness. In this paper, we study a family of non-convex min-max problems, whose objective function is weakly convex in the variables of minimization and is concave in the variables of maximization. We propose a proximally guided stochastic subgradient method and a proximally guided stochastic variance-reduced method for the non-smooth and smooth instances, respectively, in this family of problems. We analyze the time complexities of the proposed methods for finding a nearly stationary point of the outer minimization problem corresponding to the min-max problem.