Generative adversarial networks (GANs), modeled as a zero-sum game between a generator (G) and a discriminator (D), allow generating synthetic data with formal guarantees. Noting that D is a classifier, we begin by reformulating the GAN value function using class probability estimation (CPE) losses. We prove a two-way correspondence between CPE loss GANs and $f$-GANs which minimize $f$-divergences. We also show that all symmetric $f$-divergences are equivalent in convergence. In the finite sample and model capacity setting, we define and obtain bounds on estimation and generalization errors. We specialize these results to $\alpha$-GANs, defined using $\alpha$-loss, a tunable CPE loss family parametrized by $\alpha\in(0,\infty]$. We next introduce a class of dual-objective GANs to address training instabilities of GANs by modeling each player's objective using $\alpha$-loss to obtain $(\alpha_D,\alpha_G)$-GANs. We show that the resulting non-zero sum game simplifies to minimizing an $f$-divergence under appropriate conditions on $(\alpha_D,\alpha_G)$. Generalizing this dual-objective formulation using CPE losses, we define and obtain upper bounds on an appropriately defined estimation error. Finally, we highlight the value of tuning $(\alpha_D,\alpha_G)$ in alleviating training instabilities for the synthetic 2D Gaussian mixture ring as well as the large publicly available Celeb-A and LSUN Classroom image datasets.

翻译：生成对抗网络（GANs）以生成器（G）与判别器（D）之间的零和博弈为模型，能够生成具有形式化保障的合成数据。注意到D本质上是一个分类器，我们首先利用类概率估计（CPE）损失重构GAN价值函数，并证明基于CPE损失的GAN与最小化$f$-散度的$f$-GAN之间存在双向对应关系。同时证明所有对称$f$-散度在收敛性上等价。在有限样本与模型容量条件下，我们定义并推导了估计误差与泛化误差的界限。将这些结论特化至$\alpha$-GANs——即采用可调CPE损失族（由参数$\alpha\in(0,\infty]$控制的$\alpha$-损失）定义的GAN模型。进一步，我们提出一类双目标GAN框架，通过分别使用$\alpha$-损失构建博弈双方目标函数得到$(\alpha_D,\alpha_G)$-GANs，以解决GAN训练不稳定性问题。研究表明，在$(\alpha_D,\alpha_G)$满足特定条件时，该非零和博弈可简化为最小化$f$-散度。通过将这种基于CPE损失的双目标公式进行泛化，我们定义了适当形式的估计误差并推导其上界。最后，通过合成二维高斯混合环形分布以及大规模公开Celeb-A和LSUN教室图像数据集实验，验证了调整$(\alpha_D,\alpha_G)$参数在缓解训练不稳定性方面的重要价值。