  Variational representations of $f$-divergences are central to many machine
learning algorithms, with Lipschitz constrained variants recently gaining
attention. Inspired by this, we define the Moreau-Yosida approximation of
$f$-divergences with respect to the Wasserstein-$1$ metric. The corresponding
variational formulas provide a generalization of a number of recent results,
novel special cases of interest and a relaxation of the hard Lipschitz
constraint. Additionally, we prove that the so-called tight variational
representation of $f$-divergences can be to be taken over the quotient space of
Lipschitz functions, and give a characterization of functions achieving the
supremum in the variational representation. On the practical side, we propose
an algorithm to calculate the tight convex conjugate of $f$-divergences
compatible with automatic differentiation frameworks. As an application of our
results, we propose the Moreau-Yosida $f$-GAN, providing an implementation of
the variational formulas for the Kullback-Leibler, reverse Kullback-Leibler,
$\chi^2$, reverse $\chi^2$, squared Hellinger, Jensen-Shannon, Jeffreys,
triangular discrimination and total variation divergences as GANs trained on
CIFAR-10, leading to competitive results and a simple solution to the problem
of uniqueness of the optimal critic.
