Moreau-Yosida f-divergences

Terjék, Dávid (2021) Moreau-Yosida f-divergences. In: 38th International Conference on Machine Learning, ICML 2021. Proceedings of Machine Learning Research (139). ML Research Press, Maastricht, pp. 10214-10224. ISBN 9781713845065

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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 fdivergences 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, χ2, reverse χ2, 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.

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