Generative adversarial networks (GANs) comprise a generator, trained to learn the underlying distribution of the desired data, and a discriminator, trained to distinguish real samples from those output by the generator. A majority of GAN literature focuses on understanding the optimality of the discriminator through integral probability metric (IPM) or divergence based analysis. In this paper, we propose a unified approach to analyzing the generator optimization through variational approach. In $f$-divergence-minimizing GANs, we show that the optimal generator is the one that matches the score of its output distribution with that of the data distribution, while in IPM GANs, we show that this optimal generator matches score-like functions, involving the flow-field of the kernel associated with a chosen IPM constraint space. Further, the IPM-GAN optimization can be seen as one of smoothed score-matching, where the scores of the data and the generator distributions are convolved with the kernel associated with the constraint. The proposed approach serves to unify score-based training and existing GAN flavors, leveraging results from normalizing flows, while also providing explanations for empirical phenomena such as the stability of non-saturating GAN losses. Based on these results, we propose novel alternatives to $f$-GAN and IPM-GAN training based on score and flow matching, and discriminator-guided Langevin sampling.

翻译：生成对抗网络（GANs）由生成器与判别器组成：生成器学习目标数据的潜在分布，判别器则区分真实样本与生成器输出的样本。现有GAN文献主要聚焦于通过积分概率度量（IPM）或散度分析来研究判别器的最优性。本文提出一种基于变分方法的统一框架来分析生成器优化过程。在最小化f散度的GAN中，我们发现最优生成器需使其输出分布的分数与数据分布的分数相匹配；而在IPM-GAN中，这种最优生成器匹配的是类分数函数，这些函数涉及与所选IPM约束空间相关的核的流场。进一步，IPM-GAN优化可视为平滑分数匹配过程，其中数据分布与生成器分布的分数通过约束相关的核进行卷积操作。该方法统一了基于分数的训练与现有GAN范式，利用归一化流的研究成果，同时为非饱和GAN损失函数稳定性等经验现象提供了理论解释。基于这些结论，我们提出了基于分数匹配、流匹配及判别器引导的Langevin采样的f-GAN与IPM-GAN训练的新型替代方案。