# Near-Optimal Decentralized Momentum Method for Nonconvex-PL Minimax Problems

arXiv: https://arxiv.org/abs/2304.10902 · HTML: https://mlsanity.com/paper/2304.10902/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 38 of its 57 references that are in the map.
- First posted on arXiv 21 Apr 2023.

## Builds on (most cited first, 30 of 38)

- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Lojasiewicz Condition](https://mlsanity.com/paper/1608.04636/) (2016)
- [Can Decentralized Algorithms Outperform Centralized Algorithms? A Case Study for Decentralized Parallel Stochastic Gradient Descent](https://mlsanity.com/paper/1705.09056/) (2017)
- [SAGA: A Fast Incremental Gradient Method With Support for Non-Strongly Convex Composite Objectives](https://mlsanity.com/paper/1407.0202/) (2014)
- [A Unified Theory of Decentralized SGD with Changing Topology and Local Updates](https://mlsanity.com/paper/2003.10422/) (2020)
- [Lower Bounds for Non-Convex Stochastic Optimization](https://mlsanity.com/paper/1912.02365/) (2019)
- [Momentum-Based Variance Reduction in Non-Convex SGD](https://mlsanity.com/paper/1905.10018/) (2019)
- [SPIDER: Near-Optimal Non-Convex Optimization via Stochastic Path Integrated Differential Estimator](https://mlsanity.com/paper/1807.01695/) (2018)
- [On Gradient Descent Ascent for Nonconvex-Concave Minimax Problems](https://mlsanity.com/paper/1906.00331/) (2019)
- [Decentralized Stochastic Optimization and Gossip Algorithms with Compressed Communication](https://mlsanity.com/paper/1902.00340/) (2019)
- [Solving a Class of Non-Convex Min-Max Games Using Iterative First Order Methods](https://mlsanity.com/paper/1902.08297/) (2019)
- [D2: Decentralized Training over Decentralized Data](https://mlsanity.com/paper/1803.07068/) (2018)
- [NEXT: In-Network Nonconvex Optimization](https://mlsanity.com/paper/1602.00591/) (2016)
- [Distributed Stochastic Gradient Tracking Methods](https://mlsanity.com/paper/1805.11454/) (2018)
- [Distributionally Robust Federated Averaging](https://mlsanity.com/paper/2102.12660/) (2021)
- [Communication Compression for Decentralized Training](https://mlsanity.com/paper/1803.06443/) (2018)
- [Stochastic Recursive Gradient Descent Ascent for Stochastic Nonconvex-Strongly-Concave Minimax Problems](https://mlsanity.com/paper/2001.03724/) (2020)
- [Hybrid Block Successive Approximation for One-Sided Non-Convex Min-Max Problems: Algorithms and Applications](https://mlsanity.com/paper/1902.08294/) (2019)
- [Faster Single-loop Algorithms for Minimax Optimization without Strong Concavity](https://mlsanity.com/paper/2112.05604/) (2021)
- [A Decentralized Parallel Algorithm for Training Generative Adversarial Nets](https://mlsanity.com/paper/1910.12999/) (2019)
- [Accelerated Zeroth-Order and First-Order Momentum Methods from Mini to Minimax Optimization](https://mlsanity.com/paper/2008.08170/) (2020)
- [Local Stochastic Gradient Descent Ascent: Convergence Analysis and Communication Efficiency](https://mlsanity.com/paper/2102.13152/) (2021)
- [Federated Minimax Optimization: Improved Convergence Analyses and Algorithms](https://mlsanity.com/paper/2203.04850/) (2022)
- [Extragradient method with variance reduction for stochastic variational inequalities](https://mlsanity.com/paper/1703.00260/) (2017)
- [Variance-Reduced Decentralized Stochastic Optimization with Accelerated Convergence](https://mlsanity.com/paper/1912.04230/) (2019)
- [A Hybrid Stochastic Optimization Framework for Stochastic Composite Nonconvex Optimization](https://mlsanity.com/paper/1907.03793/) (2019)
- [textttDeepSqueeze: Decentralization Meets Error-Compensated Compression](https://mlsanity.com/paper/1907.07346/) (2019)
- [A Hybrid Variance-Reduced Method for Decentralized Stochastic Non-Convex Optimization](https://mlsanity.com/paper/2102.06752/) (2021)
- [Proximal Gradient Descent-Ascent: Variable Convergence under KL Geometry](https://mlsanity.com/paper/2102.04653/) (2021)
- [Fast decentralized non-convex finite-sum optimization with recursive variance reduction](https://mlsanity.com/paper/2008.07428/) (2020)

## Built on by (most cited first, 4 of 4)

- [Solving a Class of Non-Convex Minimax Optimization in Federated Learning](https://mlsanity.com/paper/2310.03613/) (2023)
- [Diffusion Stochastic Optimization for Min-Max Problems](https://mlsanity.com/paper/2401.14585/) (2024)
- [Nonconvex Decentralized Stochastic Bilevel Optimization under Heavy-Tailed Noise](https://mlsanity.com/paper/2509.15543/) (2025)
- [Enhancing Privacy in Decentralized Min-Max Optimization: A Differentially Private Approach](https://mlsanity.com/paper/2508.07505/) (2025)

## Abstract (arXiv)

Minimax optimization plays an important role in many machine learning tasks such as generative adversarial networks (GANs) and adversarial training. Although recently a wide variety of optimization methods have been proposed to solve the minimax problems, most of them ignore the distributed setting where the data is distributed on multiple workers. Meanwhile, the existing decentralized minimax optimization methods rely on the strictly assumptions such as (strongly) concavity and variational inequality conditions. In the paper, thus, we propose an efficient decentralized momentum-based gradient descent ascent (DM-GDA) method for the distributed nonconvex-PL minimax optimization, which is nonconvex in primal variable and is nonconcave in dual variable and satisfies the Polyak-Lojasiewicz (PL) condition. In particular, our DM-GDA method simultaneously uses the momentum-based techniques to update variables and estimate the stochastic gradients. Moreover, we provide a solid convergence analysis for our DM-GDA method, and prove that it obtains a near-optimal gradient complexity of O(ε-3) for finding an ε-stationary solution of the nonconvex-PL stochastic minimax problems, which reaches the lower bound of nonconvex stochastic optimization. To the best of our knowledge, we first study the decentralized algorithm for Nonconvex-PL stochastic minimax optimization over a network.