# A Unified Analysis of Variational Inequality Methods: Variance Reduction, Sampling, Quantization and Coordinate Descent

arXiv: https://arxiv.org/abs/2201.12206 · HTML: https://mlsanity.com/paper/2201.12206/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Jan 2022.

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

- [Smooth Monotone Stochastic Variational Inequalities and Saddle Point Problems: A Survey](https://mlsanity.com/paper/2208.13592/) (2022)
- [Enhancing Stability of Physics-Informed Neural Network Training Through Saddle-Point Reformulation](https://mlsanity.com/paper/2507.16008/) (2025)
- [SARAH-based Variance-reduced Algorithm for Stochastic Finite-sum Cocoercive Variational Inequalities](https://mlsanity.com/paper/2210.05994/) (2022)
- [Unified analysis of SGD-type methods](https://mlsanity.com/paper/2303.16502/) (2023)

## Abstract (arXiv)

In this paper, we present a unified analysis of methods for such a wide class of problems as variational inequalities, which includes minimization problems and saddle point problems. We develop our analysis on the modified Extra-Gradient method (the classic algorithm for variational inequalities) and consider the strongly monotone and monotone cases, which corresponds to strongly-convex-strongly-concave and convex-concave saddle point problems. The theoretical analysis is based on parametric assumptions about Extra-Gradient iterations. Therefore, it can serve as a strong basis for combining the already existing type methods and also for creating new algorithms. In particular, to show this we develop new robust methods, which include methods with quantization, coordinate methods, distributed randomized local methods, and others. Most of these approaches have never been considered in the generality of variational inequalities and have previously been used only for minimization problems. The robustness of the new methods is also confirmed by numerical experiments with GANs.