  Nowozin \textit{et al} showed last year how to extend the GAN
\textit{principle} to all $f$-divergences. The approach is elegant but falls
short of a full description of the supervised game, and says little about the
key player, the generator: for example, what does the generator actually
converge to if solving the GAN game means convergence in some space of
parameters? How does that provide hints on the generator's design and compare
to the flourishing but almost exclusively experimental literature on the
subject?
  In this paper, we unveil a broad class of distributions for which such
convergence happens --- namely, deformed exponential families, a wide superset
of exponential families --- and show tight connections with the three other key
GAN parameters: loss, game and architecture. In particular, we show that
current deep architectures are able to factorize a very large number of such
densities using an especially compact design, hence displaying the power of
deep architectures and their concinnity in the $f$-GAN game. This result holds
given a sufficient condition on \textit{activation functions} --- which turns
out to be satisfied by popular choices. The key to our results is a variational
generalization of an old theorem that relates the KL divergence between regular
exponential families and divergences between their natural parameters. We
complete this picture with additional results and experimental insights on how
these results may be used to ground further improvements of GAN architectures,
via (i) a principled design of the activation functions in the generator and
(ii) an explicit integration of proper composite losses' link function in the
discriminator.
