  In an effort to address the training instabilities of GANs, we introduce a
class of dual-objective GANs with different value functions (objectives) for
the generator (G) and discriminator (D). In particular, we model each objective
using $\alpha$-loss, a tunable classification loss, to obtain
$(\alpha_D,\alpha_G)$-GANs, parameterized by $(\alpha_D,\alpha_G)\in
(0,\infty]^2$. For sufficiently large number of samples and capacities for G
and D, we show that the resulting non-zero sum game simplifies to minimizing an
$f$-divergence under appropriate conditions on $(\alpha_D,\alpha_G)$. In the
finite sample and capacity setting, we define estimation error to quantify the
gap in the generator's performance relative to the optimal setting with
infinite samples and obtain upper bounds on this error, showing it to be order
optimal under certain conditions. Finally, we highlight the value of tuning
$(\alpha_D,\alpha_G)$ in alleviating training instabilities for the synthetic
2D Gaussian mixture ring and the Stacked MNIST datasets.
