## Moreau-Yosida $f$-divergences

### Dávid Terjék

##### Virtual

Keywords: [ Convex Optimization ]

[ Abstract ]
[
[ Paper ]
Tue 20 Jul 9 p.m. PDT — 11 p.m. PDT

Spotlight presentation: Optimization (Convex) 2
Tue 20 Jul 7 p.m. PDT — 8 p.m. PDT

Abstract: 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.

Chat is not available.