# A Decentralized Proximal Point-type Method for Saddle Point Problems

arXiv: https://arxiv.org/abs/1910.14380 · HTML: https://mlsanity.com/paper/1910.14380/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Oct 2019.

## Built on by (most cited first, 17 of 17)

- [Federated Minimax Optimization: Improved Convergence Analyses and Algorithms](https://mlsanity.com/paper/2203.04850/) (2022)
- [Decentralized Local Stochastic Extra-Gradient for Variational Inequalities](https://mlsanity.com/paper/2106.08315/) (2021)
- [Distributed Methods with Compressed Communication for Solving Variational Inequalities, with Theoretical Guarantees](https://mlsanity.com/paper/2110.03313/) (2021)
- [Communication-Efficient Distributionally Robust Decentralized Learning](https://mlsanity.com/paper/2205.15614/) (2022)
- [A Decentralized Adaptive Momentum Method for Solving a Class of Min-Max Optimization Problems](https://mlsanity.com/paper/2106.06075/) (2021)
- [Distributed Adversarial Training to Robustify Deep Neural Networks at Scale](https://mlsanity.com/paper/2206.06257/) (2022)
- [Distributed Saddle-Point Problems Under Similarity](https://mlsanity.com/paper/2107.10706/) (2021)
- [Nonconvex-Nonconcave Min-Max Optimization with a Small Maximization Domain](https://mlsanity.com/paper/2110.03950/) (2021)
- [INTERACT: Achieving Low Sample and Communication Complexities in Decentralized Bilevel Learning over Networks](https://mlsanity.com/paper/2207.13283/) (2022)
- [Federated Minimax Optimization with Client Heterogeneity](https://mlsanity.com/paper/2302.04249/) (2023)
- [DIAMOND: Taming Sample and Communication Complexities in Decentralized Bilevel Optimization](https://mlsanity.com/paper/2212.02376/) (2022)
- [Similarity, Compression and Local Steps: Three Pillars of Efficient Communications for Distributed Variational Inequalities](https://mlsanity.com/paper/2302.07615/) (2023)
- [Decentralized gradient descent maximization method for composite nonconvex strongly-concave minimax problems](https://mlsanity.com/paper/2304.02441/) (2023)
- [PRECISION: Decentralized Constrained Min-Max Learning with Low Communication and Sample Complexities](https://mlsanity.com/paper/2303.02532/) (2023)
- [Decentralized Personalized Federated Learning for Min-Max Problems](https://mlsanity.com/paper/2106.07289/) (2021)
- [Federated Composite Saddle Point Optimization](https://mlsanity.com/paper/2305.15643/) (2023)
- [Enhancing Privacy in Decentralized Min-Max Optimization: A Differentially Private Approach](https://mlsanity.com/paper/2508.07505/) (2025)

## Abstract (arXiv)

In this paper, we focus on solving a class of constrained non-convex non-concave saddle point problems in a decentralized manner by a group of nodes in a network. Specifically, we assume that each node has access to a summand of a global objective function and nodes are allowed to exchange information only with their neighboring nodes. We propose a decentralized variant of the proximal point method for solving this problem. We show that when the objective function is ρ-weakly convex-weakly concave the iterates converge to approximate stationarity with a rate of O(1/√T) where the approximation error depends linearly on √ρ. We further show that when the objective function satisfies the Minty VI condition (which generalizes the convex-concave case) we obtain convergence to stationarity with a rate of O(1/√T). To the best of our knowledge, our proposed method is the first decentralized algorithm with theoretical guarantees for solving a non-convex non-concave decentralized saddle point problem. Our numerical results for training a general adversarial network (GAN) in a decentralized manner match our theoretical guarantees.