  A unifying $\alpha$-parametrized generator loss function is introduced for a
dual-objective generative adversarial network (GAN), which uses a canonical (or
classical) discriminator loss function such as the one in the original GAN
(VanillaGAN) system. The generator loss function is based on a symmetric class
probability estimation type function, $\mathcal{L}_\alpha$, and the resulting
GAN system is termed $\mathcal{L}_\alpha$-GAN. Under an optimal discriminator,
it is shown that the generator's optimization problem consists of minimizing a
Jensen-$f_\alpha$-divergence, a natural generalization of the Jensen-Shannon
divergence, where $f_\alpha$ is a convex function expressed in terms of the
loss function $\mathcal{L}_\alpha$. It is also demonstrated that this
$\mathcal{L}_\alpha$-GAN problem recovers as special cases a number of GAN
problems in the literature, including VanillaGAN, Least Squares GAN (LSGAN),
Least $k$th order GAN (L$k$GAN) and the recently introduced
$(\alpha_D,\alpha_G)$-GAN with $\alpha_D=1$. Finally, experimental results are
conducted on three datasets, MNIST, CIFAR-10, and Stacked MNIST to illustrate
the performance of various examples of the $\mathcal{L}_\alpha$-GAN system.
