Hellinger loss function for Generative Adversarial Networks

About

We propose Hellinger-type loss functions for training Generative Adversarial Networks (GANs), motivated by the boundedness, symmetry, and robustness properties of the Hellinger distance. We define an adversarial objective based on this divergence and study its statistical properties within a general parametric framework. We establish the existence, uniqueness, consistency, and joint asymptotic normality of the estimators obtained from the adversarial training procedure. In particular, we analyze the joint estimation of both generator and discriminator parameters, offering a comprehensive asymptotic characterization of the resulting estimators. We introduce two implementations of the Hellinger-type loss and we evaluate their empirical behavior in comparison with the classic (Maximum Likelihood-type) GAN loss. Through a controlled simulation study, we demonstrate that both proposed losses yield improved estimation accuracy and robustness under increasing levels of data contamination.

Giovanni Saraceno, Anand N. Vidyashankar, Claudio Agostinelli · 2025

Related benchmarks

Task	Dataset	Result
Joint parameter estimation	Contaminated Gaussian (ε=5%)	Median RMSE(µ̂, σ̂) (x100)0.00595	6
Joint parameter estimation	Contaminated Gaussian ε=10%	Median RMSE (µ̂, σ̂)0.00926	6
Joint parameter estimation	Contaminated Gaussian ε=20%	Median RMSE (µ, σ)0.05381	6
Parameter Estimation	Gaussian data eps=5% (synthetic)	Median MSE(sigma)0.00003	6
Parameter Estimation	Gaussian data eps=10% (synthetic)	Median MSE(sigma)0.007	6
Parameter Estimation	Gaussian data eps=20% (synthetic)	Median MSE(sigma)0.226	6
Parameter Estimation (mu)	Gaussian Distribution epsilon=5%	Median Best MSE (mu)0.00002	6
Parameter Estimation (mu)	Gaussian Distribution epsilon=10%	Median Best MSE(mu)0.6	6
Parameter Estimation (mu)	Gaussian Distribution epsilon=20%	Median MSE(mu)0.00203	6
Joint parameter estimation	Contaminated Gaussian ε=0%	Median RMSE (µ, σ)0.00514	6