  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 $k$th-order GAN (L$k$GAN), is first introduced,
generalizing the least squares GANs (LSGANs) by using a $k$th order absolute
error distortion measure with $k \geq 1$ (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 $k$th-order Pearson-Vajda divergence. Another novel GAN
generator loss function is next proposed in terms of R\'{e}nyi cross-entropy
functionals with order $\alpha >0$, $\alpha\neq 1$. It is demonstrated that
this R\'{e}nyi-centric generalized loss function, which provably reduces to the
original GAN loss function as $\alpha\to1$, preserves the equilibrium point
satisfied by the original GAN based on the Jensen-R\'{e}nyi 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 $\alpha$, 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.
