  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.
