On Minimax Optimality of GANs for Robust Mean Estimation

Generative adversarial networks (GANs) have become one of the most popular generative modeling techniques in machine learning. In this work, we study the statistical and robust properties of GANs for Gaussian mean estimation under Huber’s contamination model, where an epsilon proportion of training data may be arbitrarily corrupted. We prove that f-GAN, when equipped with appropriate discriminators, achieve optimal minimax rate, hence extending the recent result of Gao et al. (2019a). In contrast, we show that other GAN variants such as MMD-GAN (with Gaussian kernel) and W-GAN may fail to achieve minimax optimality. We further adapt f-GAN to the sparse and the unknown covariance settings. We perform numerical simulations to confirm our theoretical findings and reveal new insights on the importance of discriminators.