Hellinger-Type Losses for Generative Adversarial Networks
G. Saraceno${}^a$, A. N. Vidyashankar${}^b$ and C. Agostinelli${}^c$
${}^a$Department of Statistical Sciences, University of Padua, Italy, ${}^b$Department of Statistics, George Mason University, USA, ${}^c$Department of Mathematics, University of Trento, Italy
Generative adversarial networks (GANs) are commonly studied through the convergence of the induced distributions, while less attention has been given to the joint statistical behavior of the generator and discriminator estimators, and to their robustness under contamination. In this study, we investigate GAN training from a statistical perspective by introducing Hellinger-type adversarial loss functions, motivated by the boundedness and symmetry of the Hellinger distance. Within a parametric framework, we formulate adversarial training as a joint minimax estimation problem for the parameters of the generator and discriminator. We consider a complete Hellinger formulation and a tractable approximated version, and we study their statistical properties. In particular, we prove the existence and uniqueness of the minimax solution, and establish consistency and joint asymptotic normality. We also derive profiled asymptotic results for the generator parameter. To investigate robustness, we compute the influence functions, thereby linking adversarial learning to robustness theory and M-estimation. Finally, we also briefly discuss how the same construction can be extended to other $f$-divergence-based adversarial loss functions. The theoretical analysis is complemented by a simulation study in a Gaussian parametric setting, where we introduce controlled contamination to examine how different adversarial losses affect estimation accuracy and training dynamics. Additionally, we provide an illustration on a higher-dimensional image generation problem using the Fashion-MNIST dataset. Theoretical details and simulation results are available in [1]. Keywords: GANs, Robust statistics, Generative models.