Moreau-Yosida $f$-divergences
This work addresses a theoretical bottleneck in f-divergence representations for machine learning practitioners, offering incremental improvements with practical implementations.
The paper tackles the problem of variational representations of f-divergences in machine learning by defining the Moreau-Yosida approximation with respect to the Wasserstein-1 metric, leading to a generalization of recent results and a relaxation of Lipschitz constraints, with practical applications in GANs trained on CIFAR-10 showing competitive results.
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.