Moreau-Yosida f-divergences

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, χ 2 , reverse χ 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.