# Finding Mixed Nash Equilibria of Generative Adversarial Networks

arXiv: https://arxiv.org/abs/1811.02002 · HTML: https://mlsanity.com/paper/1811.02002/

## Key facts

- Cited by 49 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Oct 2018.

## Built on by (most cited first, 30 of 49)

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- [SAN: Inducing Metrizability of GAN with Discriminative Normalized Linear Layer](https://mlsanity.com/paper/2301.12811/) (2023)
- [Provable Maximum Entropy Manifold Exploration via Diffusion Models](https://mlsanity.com/paper/2506.15385/) (2025)
- [Adaptive Importance Sampling meets Mirror Descent: a Bias-variance tradeoff](https://mlsanity.com/paper/2110.15590/) (2021)
- [Two-Scale Gradient Descent Ascent Dynamics Finds Mixed Nash Equilibria of Continuous Games: A Mean-Field Perspective](https://mlsanity.com/paper/2212.08791/) (2022)
- [Complex Momentum for Optimization in Games](https://mlsanity.com/paper/2102.08431/) (2021)
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- [Provably convergent quasistatic dynamics for mean-field two-player zero-sum games](https://mlsanity.com/paper/2202.10947/) (2022)
- [Verifier-Constrained Flow Expansion for Discovery Beyond the Data](https://mlsanity.com/paper/2602.15984/) (2026)
- [DO-GAN: A Double Oracle Framework for Generative Adversarial Networks](https://mlsanity.com/paper/2102.08577/) (2021)
- [Game of GANs: Game-Theoretical Models for Generative Adversarial Networks](https://mlsanity.com/paper/2106.06976/) (2021)
- [Stochastic Re-weighted Gradient Descent via Distributionally Robust Optimization](https://mlsanity.com/paper/2306.09222/) (2023)
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- [Transferable Reward Learning by Dynamics-Agnostic Discriminator Ensemble](https://mlsanity.com/paper/2206.00238/) (2022)
- [Wasserstein Gradient Flow over Variational Parameter Space for Variational Inference](https://mlsanity.com/paper/2310.16705/) (2023)
- [Expected Variational Inequalities](https://mlsanity.com/paper/2502.18605/) (2025)
- [Analysis of Kernel Mirror Prox for Measure Optimization](https://mlsanity.com/paper/2403.00147/) (2024)
- [Inclusive KL Gradient Flows: Otto-Wasserstein, Fisher-Rao-Gaussian, and Local-Estimator Dynamics](https://mlsanity.com/paper/2411.00214/) (2024)
- [On Characterizing GAN Convergence Through Proximal Duality Gap](https://mlsanity.com/paper/2105.04801/) (2021)
- [A Fisher-Rao gradient flow for entropic mean-field min-max games](https://mlsanity.com/paper/2405.15834/) (2024)
- [Convergence of GANs Training: A Game and Stochastic Control Methodology](https://mlsanity.com/paper/2112.00222/) (2021)
- [Simultaneous Transport Evolution for Minimax Equilibria on Measures](https://mlsanity.com/paper/2202.06460/) (2022)
- [PAPAL: A Provable PArticle-based Primal-Dual ALgorithm for Mixed Nash Equilibrium](https://mlsanity.com/paper/2303.00970/) (2023)
- [Mirror Descent-Ascent for mean-field min-max problems](https://mlsanity.com/paper/2402.08106/) (2024)
- [Convergence of two-timescale gradient descent ascent dynamics: finite-dimensional and mean-field perspectives](https://mlsanity.com/paper/2501.17122/) (2025)
- [Convergence of Time-Averaged Mean Field Gradient Descent Dynamics for Continuous Multi-Player Zero-Sum Games](https://mlsanity.com/paper/2505.07642/) (2025)
- [Fast 2DGS: Efficient Image Representation with Deep Gaussian Prior](https://mlsanity.com/paper/2512.12774/) (2025)

## Abstract (arXiv)

We reconsider the training objective of Generative Adversarial Networks (GANs) from the mixed Nash Equilibria (NE) perspective. Inspired by the classical prox methods, we develop a novel algorithmic framework for GANs via an infinite-dimensional two-player game and prove rigorous convergence rates to the mixed NE, resolving the longstanding problem that no provably convergent algorithm exists for general GANs. We then propose a principled procedure to reduce our novel prox methods to simple sampling routines, leading to practically efficient algorithms. Finally, we provide experimental evidence that our approach outperforms methods that seek pure strategy equilibria, such as SGD, Adam, and RMSProp, both in speed and quality.