# Global Convergence to the Equilibrium of GANs using Variational Inequalities

arXiv: https://arxiv.org/abs/1808.01531 · HTML: https://mlsanity.com/paper/1808.01531/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Aug 2018.

## Built on by (most cited first, 15 of 15)

- [Stochastic Variance Reduction for Variational Inequality Methods](https://mlsanity.com/paper/2102.08352/) (2021)
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- [Causal Inference in Network Economics](https://mlsanity.com/paper/2109.11344/) (2021)
- [Revisiting GANs by Best-Response Constraint: Perspective, Methodology, and Application](https://mlsanity.com/paper/2205.10146/) (2022)
- [Recursive Reasoning in Minimax Games: A Level k Gradient Play Method](https://mlsanity.com/paper/2210.16482/) (2022)
- [Exploiting the Hidden Tasks of GANs: Making Implicit Subproblems Explicit](https://mlsanity.com/paper/2101.11863/) (2021)
- [Finding mixed-strategy equilibria of continuous-action games without gradients using randomized policy networks](https://mlsanity.com/paper/2211.15936/) (2022)
- [ApproxED: Approximate exploitability descent via learned best responses](https://mlsanity.com/paper/2301.08830/) (2023)
- [Local convergence of simultaneous min-max algorithms to differential equilibrium on Riemannian manifold](https://mlsanity.com/paper/2405.13392/) (2024)
- [Privacy Amplification Persists under Unlimited Synthetic Data Release](https://mlsanity.com/paper/2602.04895/) (2026)

## Abstract (arXiv)

In optimization, the negative gradient of a function denotes the direction of steepest descent. Furthermore, traveling in any direction orthogonal to the gradient maintains the value of the function. In this work, we show that these orthogonal directions that are ignored by gradient descent can be critical in equilibrium problems. Equilibrium problems have drawn heightened attention in machine learning due to the emergence of the Generative Adversarial Network (GAN). We use the framework of Variational Inequalities to analyze popular training algorithms for a fundamental GAN variant: the Wasserstein Linear-Quadratic GAN. We show that the steepest descent direction causes divergence from the equilibrium, and convergence to the equilibrium is achieved through following a particular orthogonal direction. We call this successful technique Crossing-the-Curl, named for its mathematical derivation as well as its intuition: identify the game's axis of rotation and move "across" space in the direction towards smaller "curling".