Dynamics of Fourier Modes in Torus Generative Adversarial Networks
This work addresses the training instability issue in GANs, which is a critical problem for researchers and practitioners in machine learning, but it is incremental as it builds on existing theoretical frameworks.
The authors tackled the problem of unstable training in Generative Adversarial Networks (GANs) by analyzing convergence and stability using Fourier series decomposition, showing that Nash equilibria are spiral attractors, which theoretically explains the slow and unstable training observed.
Generative Adversarial Networks (GANs) are powerful Machine Learning models capable of generating fully synthetic samples of a desired phenomenon with a high resolution. Despite their success, the training process of a GAN is highly unstable and typically it is necessary to implement several accessory heuristics to the networks to reach an acceptable convergence of the model. In this paper, we introduce a novel method to analyze the convergence and stability in the training of Generative Adversarial Networks. For this purpose, we propose to decompose the objective function of the adversary min-max game defining a periodic GAN into its Fourier series. By studying the dynamics of the truncated Fourier series for the continuous Alternating Gradient Descend algorithm, we are able to approximate the real flow and to identify the main features of the convergence of the GAN. This approach is confirmed empirically by studying the training flow in a $2$-parametric GAN aiming to generate an unknown exponential distribution. As byproduct, we show that convergent orbits in GANs are small perturbations of periodic orbits so the Nash equillibria are spiral attractors. This theoretically justifies the slow and unstable training observed in GANs.