Generative adversarial networks (GANs) are unsupervised learning methods for training a generator distribution to produce samples that approximate those drawn from a target distribution. Many such methods can be formulated as minimization of a metric or divergence. Recent works have proven the statistical consistency of GANs that are based on integral probability metrics (IPMs), e.g., WGAN which is based on the 1-Wasserstein metric. IPMs are defined by optimizing a linear functional (difference of expectations) over a space of discriminators. A much larger class of GANs, which allow for the use of nonlinear objective functionals, can be constructed using $(f,\Gamma)$-divergences; these generalize and interpolate between IPMs and $f$-divergences (e.g., KL or $\alpha$-divergences). Instances of $(f,\Gamma)$-GANs have been shown to exhibit improved performance in a number of applications. In this work we study the statistical consistency of $(f,\Gamma)$-GANs for general $f$ and $\Gamma$. Specifically, we derive finite-sample concentration inequalities. These derivations require novel arguments due to nonlinearity of the objective functional. We demonstrate that our new results reduce to the known results for IPM-GANs in the appropriate limit while also significantly extending the domain of applicability of this theory.

翻译：生成对抗网络（GAN）是一种无监督学习方法，用于训练生成器分布以产生近似从目标分布中抽取的样本。许多此类方法可被表述为某种度量或散度的最小化。近期研究已证明了基于积分概率度量（IPM）的GAN的统计一致性，例如基于1-Wasserstein度量的WGAN。IPM通过在判别器空间上优化一个线性泛函（期望之差）来定义。利用$(f,\Gamma)$-散度可以构建一个更广泛的GAN类别，它允许使用非线性目标泛函；这类方法概括并内插了IPM与$f$-散度（例如KL散度或$\alpha$-散度）。(f,Γ)-GAN的实例已在多项应用中展现出改进的性能。本文研究了一般$f$和$\Gamma$下$(f,\Gamma)$-GAN的统计一致性。具体而言，我们推导了有限样本的集中不等式。由于目标泛函的非线性，这些推导需要新颖的论证方法。我们证明，在适当的极限下，我们的新结果可约化为已知的IPM-GAN结果，同时显著扩展了该理论的适用范围。