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.

翻译：为了应对生成对抗网络（GAN）的训练不稳定性，我们提出了一类双目标GAN，其中生成器（G）和判别器（D）采用不同的值函数（目标函数）。具体而言，我们使用可调分类损失函数$\alpha$-loss对每个目标进行建模，从而得到由参数$(\alpha_D,\alpha_G)\in [0,\infty)^2$控制的$(\alpha_D,\alpha_G)$-GAN。在G和D的样本量与容量足够大的情况下，我们证明该非零和博弈在$(\alpha_D,\alpha_G)$满足适当条件时简化为最小化一个$f$-散度。在有限样本与有限容量设定中，我们定义了估计误差来量化生成器性能相对于无限样本最优设定的差距，并给出了该误差的上界，证明其在特定条件下达到阶最优。最后，我们通过合成二维高斯混合环数据集和堆叠MNIST数据集，展示了调整$(\alpha_D,\alpha_G)$在缓解训练不稳定性方面的价值。