Moreau-Yosida $f$-divergences
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.
BibTeX
@InProceedings{pmlr-v139-terjek21a,
title = {Moreau-Yosida $f$-divergences},
author = {Terj{\'e}k, D{\'a}vid},
booktitle = {Proceedings of the 38th International Conference on Machine Learning},
pages = {10214--10224},
year = {2021},
editor = {Meila, Marina and Zhang, Tong},
volume = {139},
series = {Proceedings of Machine Learning Research},
month = {18--24 Jul},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v139/terjek21a/terjek21a.pdf},
url = {https://proceedings.mlr.press/v139/terjek21a.html},
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.}
}