The generative adversarial network (GAN) is an important model developed for high-dimensional distribution learning in recent years. However, there is a pressing need for a comprehensive method to understand its error convergence rate. In this research, we focus on studying the error convergence rate of the GAN model that is based on a class of functions encompassing the discriminator and generator neural networks. These functions are VC type with bounded envelope function under our assumptions, enabling the application of the Talagrand inequality. By employing the Talagrand inequality and Borel-Cantelli lemma, we establish a tight convergence rate for the error of GAN. This method can also be applied on existing error estimations of GAN and yields improved convergence rates. In particular, the error defined with the neural network distance is a special case error in our definition.

翻译：生成对抗网络（GAN）是近年来为高维分布学习而发展的重要模型。然而，目前迫切需要一种全面理解其误差收敛率的方法。本研究聚焦于基于一类包含判别器和生成器神经网络函数的GAN模型的误差收敛率。在我们假设下，这些函数具有VC型性质且具有有界包络函数，从而能够应用Talagrand不等式。通过运用Talagrand不等式和Borel-Cantelli引理，我们建立了GAN误差的紧致收敛率。该方法亦可应用于现有GAN误差估计中，并得到改进的收敛率。特别地，以神经网络距离定义的误差是我们定义中误差的一个特例。