Understanding GANs in the LQG Setting: Formulation, Generalization and Stability

Generative Adversarial Networks (GANs) have become a popular method to learn a probability model from data. In this paper, we provide an understanding of basic issues surrounding GANs including their formulation, generalization and stability on a simple LQG benchmark where the generator is Linear, the discriminator is Quadratic and the data has a high-dimensional Gaussian distribution. Even in this simple benchmark, the GAN problem has not been well-understood as we observe that existing state-of-the-art GAN architectures may fail to learn a proper generative distribution owing to (1) stability issues (i.e., convergence to bad local solutions or not converging at all), (2) approximation issues (i.e., having improper global GAN optimizers caused by inappropriate GAN's loss functions), and (3) generalizability issues (i.e., requiring large number of samples for training). In this setup, we propose a GAN architecture which recovers the maximum-likelihood solution and demonstrates fast generalization. Moreover, we analyze global stability of different computational approaches for the proposed GAN and highlight their pros and cons. Finally, through experiments on MNIST and CIFAR-10 datasets, we outline extensions of our model-based approach to design GANs in more complex setups than the considered Gaussian benchmark.