  Generative adversarial networks (GANs) are often billed as "universal
distribution learners", but precisely what distributions they can represent and
learn is still an open question. Heavy-tailed distributions are prevalent in
many different domains such as financial risk-assessment, physics, and
epidemiology. We observe that existing GAN architectures do a poor job of
matching the asymptotic behavior of heavy-tailed distributions, a problem that
we show stems from their construction. Additionally, when faced with the
infinite moments and large distances between outlier points that are
characteristic of heavy-tailed distributions, common loss functions produce
unstable or near-zero gradients. We address these problems with the Pareto GAN.
A Pareto GAN leverages extreme value theory and the functional properties of
neural networks to learn a distribution that matches the asymptotic behavior of
the marginal distributions of the features. We identify issues with standard
loss functions and propose the use of alternative metric spaces that enable
stable and efficient learning. Finally, we evaluate our proposed approach on a
variety of heavy-tailed datasets.
