# Convergence of GANs Training: A Game and Stochastic Control Methodology

arXiv: https://arxiv.org/abs/2112.00222 · HTML: https://mlsanity.com/paper/2112.00222/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 38 of its 78 references that are in the map.
- First posted on arXiv 1 Dec 2021.

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

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Improved Techniques for Training GANs](https://mlsanity.com/paper/1606.03498/) (2016)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Conditional Generative Adversarial Nets](https://mlsanity.com/paper/1411.1784/) (2014)
- [Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks](https://mlsanity.com/paper/1511.06434/) (2015)
- [Photo-Realistic Single Image Super-Resolution Using a Generative Adversarial Network](https://mlsanity.com/paper/1609.04802/) (2016)
- [An overview of gradient descent optimization algorithms](https://mlsanity.com/paper/1609.04747/) (2016)
- [Generative Adversarial Text to Image Synthesis](https://mlsanity.com/paper/1605.05396/) (2016)
- [Stochastic First- and Zeroth-order Methods for Nonconvex Stochastic Programming](https://mlsanity.com/paper/1309.5549/) (2013)
- [Which Training Methods for GANs do actually Converge?](https://mlsanity.com/paper/1801.04406/) (2018)
- [Gradient Descent Finds Global Minima of Deep Neural Networks](https://mlsanity.com/paper/1811.03804/) (2018)
- [Towards Principled Methods for Training Generative Adversarial Networks](https://mlsanity.com/paper/1701.04862/) (2017)
- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [Generating Videos with Scene Dynamics](https://mlsanity.com/paper/1609.02612/) (2016)
- [Generative Visual Manipulation on the Natural Image Manifold](https://mlsanity.com/paper/1609.03552/) (2016)
- [Deep Generative Image Models using a Laplacian Pyramid of Adversarial Networks](https://mlsanity.com/paper/1506.05751/) (2015)
- [Stochastic Modified Equations and Dynamics of Stochastic Gradient Algorithms I: Mathematical Foundations](https://mlsanity.com/paper/1811.01558/) (2018)
- [Quant GANs: Deep Generation of Financial Time Series](https://mlsanity.com/paper/1907.06673/) (2019)
- [Accelerated Gradient Methods for Nonconvex Nonlinear and Stochastic Programming](https://mlsanity.com/paper/1310.3787/) (2013)
- [Semantic Segmentation using Adversarial Networks](https://mlsanity.com/paper/1611.08408/) (2016)
- [Finding Mixed Nash Equilibria of Generative Adversarial Networks](https://mlsanity.com/paper/1811.02002/) (2018)
- [Robust Reinforcement Learning via Adversarial training with Langevin Dynamics](https://mlsanity.com/paper/2002.06063/) (2020)
- [Generating Realistic Stock Market Order Streams](https://mlsanity.com/paper/2006.04212/) (2020)
- [A mean-field analysis of two-player zero-sum games](https://mlsanity.com/paper/2002.06277/) (2020)
- [A Closer Look at the Optimization Landscapes of Generative Adversarial Networks](https://mlsanity.com/paper/1906.04848/) (2019)
- [Deep Hedging: Learning to Simulate Equity Option Markets](https://mlsanity.com/paper/1911.01700/) (2019)
- [Time Series Simulation by Conditional Generative Adversarial Net](https://mlsanity.com/paper/1904.11419/) (2019)
- [CorrGAN: Sampling Realistic Financial Correlation Matrices Using Generative Adversarial Networks](https://mlsanity.com/paper/1910.09504/) (2019)
- [Using generative adversarial networks to synthesize artificial financial datasets](https://mlsanity.com/paper/2002.02271/) (2020)
- [Game on Random Environment, Mean-field Langevin System and Neural Networks](https://mlsanity.com/paper/2004.02457/) (2020)

## Built on by (most cited first, 2 of 2)

- [On the Nonconvexity of Push-Forward Constraints and Its Consequences in Machine Learning](https://mlsanity.com/paper/2403.07471/) (2024)
- [Large deviations for interacting particle dynamics for finding mixed equilibria in zero-sum games](https://mlsanity.com/paper/2206.15177/) (2022)

## Abstract (arXiv)

Training generative adversarial networks (GANs) is known to be difficult, especially for financial time series. This paper first analyzes the well-posedness problem in GANs minimax games and the convexity issue in GANs objective functions. It then proposes a stochastic control framework for hyper-parameters tuning in GANs training. The weak form of dynamic programming principle and the uniqueness and the existence of the value function in the viscosity sense for the corresponding minimax game are established. In particular, explicit forms for the optimal adaptive learning rate and batch size are derived and are shown to depend on the convexity of the objective function, revealing a relation between improper choices of learning rate and explosion in GANs training. Finally, empirical studies demonstrate that training algorithms incorporating this adaptive control approach outperform the standard ADAM method in terms of convergence and robustness. From GANs training perspective, the analysis in this paper provides analytical support for the popular practice of ``clipping'', and suggests that the convexity and well-posedness issues in GANs may be tackled through appropriate choices of hyper-parameters.