Error Analysis in Generative Adversarial Networks: A Theoretical Framework

Abstract

The generative adversarial network (GAN) is an important model developed for high-dimensional distribution learning in recent years. However, there is a need for a comprehensive understanding of its generalization error and convergence behavior. In this research, we study the generalization error of GANs based on classes of functions encompassing discriminator and generator neural networks with piecewise linear activation functions. Under our assumptions, the resulting function classes are VC type with bounded envelope functions, enabling the application of Talagrand’s inequality. By employing Talagrand’s inequality and the Borel–Cantelli lemma, we establish almost sure convergence rates for the generalization error of GANs under fixed discriminator and generator architectures. We further apply the same empirical-process technique to several existing GAN error formulations and obtain corresponding almost sure convergence bounds. The resulting sample-size dependence is compared with existing results under the same fixed-architecture setting, while network-complexity quantities are treated consistently as fixed constants with respect to the sample sizes. In particular, under suitable range and symmetry conditions on the discriminator class, the error defined by the neural network distance is a special case of our general measuring-function objective.