# Least kth-Order and R'enyi Generative Adversarial Networks

arXiv: https://arxiv.org/abs/2006.02479 · HTML: https://mlsanity.com/paper/2006.02479/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Jun 2020.

## Built on by (most cited first, 6 of 6)

- [α-GAN: Convergence and Estimation Guarantees](https://mlsanity.com/paper/2205.06393/) (2022)
- [Renyi Fair Information Bottleneck for Image Classification](https://mlsanity.com/paper/2203.04950/) (2022)
- [A Unifying Generator Loss Function for Generative Adversarial Networks](https://mlsanity.com/paper/2308.07233/) (2023)
- [(αD,αG)-GANs: Addressing GAN Training Instabilities via Dual Objectives](https://mlsanity.com/paper/2302.14320/) (2023)
- [Generalized Dual Discriminator GANs](https://mlsanity.com/paper/2507.17684/) (2025)
- [On the R'enyi Cross-Entropy](https://mlsanity.com/paper/2206.14329/) (2022)

## Abstract (arXiv)

We investigate the use of parametrized families of information-theoretic measures to generalize the loss functions of generative adversarial networks (GANs) with the objective of improving performance. A new generator loss function, called least kth-order GAN (LkGAN), is first introduced, generalizing the least squares GANs (LSGANs) by using a kth order absolute error distortion measure with k ≥ 1 (which recovers the LSGAN loss function when k=2). It is shown that minimizing this generalized loss function under an (unconstrained) optimal discriminator is equivalent to minimizing the kth-order Pearson-Vajda divergence. Another novel GAN generator loss function is next proposed in terms of R'enyi cross-entropy functionals with order α >0, αneq 1. It is demonstrated that this R'enyi-centric generalized loss function, which provably reduces to the original GAN loss function as α→1, preserves the equilibrium point satisfied by the original GAN based on the Jensen-R'enyi divergence, a natural extension of the Jensen-Shannon divergence. Experimental results indicate that the proposed loss functions, applied to the MNIST and CelebA datasets, under both DCGAN and StyleGAN architectures, confer performance benefits by virtue of the extra degrees of freedom provided by the parameters k and α, respectively. More specifically, experiments show improvements with regard to the quality of the generated images as measured by the Fr'echet Inception Distance (FID) score and training stability. While it was applied to GANs in this study, the proposed approach is generic and can be used in other applications of information theory to deep learning, e.g., the issues of fairness or privacy in artificial intelligence.