$(f,\Gamma)$-Divergences: Interpolating between $f$-Divergences and Integral Probability Metrics

• URL: http://arxiv.org/abs/2011.05953v3
• Date: Wed, 15 Sep 2021 14:25:24 GMT
• Title: $(f,\Gamma)$-Divergences: Interpolating between $f$-Divergences and Integral Probability Metrics
• Authors: Jeremiah Birrell, Paul Dupuis, Markos A. Katsoulakis, Yannis Pantazis, Luc Rey-Bellet
• Abstract summary: We develop a framework for constructing information-theoretic divergences that subsume both $f$-divergences and integral probability metrics (IPMs) We show that they can be expressed as a two-stage mass-redistribution/mass-transport process. Using statistical learning as an example, we demonstrate their advantage in training generative adversarial networks (GANs) for heavy-tailed, not-absolutely continuous sample distributions.
• Score: 6.221019624345409
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We develop a rigorous and general framework for constructing information-theoretic divergences that subsume both $f$-divergences and integral probability metrics (IPMs), such as the $1$-Wasserstein distance. We prove under which assumptions these divergences, hereafter referred to as $(f,\Gamma)$-divergences, provide a notion of `distance' between probability measures and show that they can be expressed as a two-stage mass-redistribution/mass-transport process. The $(f,\Gamma)$-divergences inherit features from IPMs, such as the ability to compare distributions which are not absolutely continuous, as well as from $f$-divergences, namely the strict concavity of their variational representations and the ability to control heavy-tailed distributions for particular choices of $f$. When combined, these features establish a divergence with improved properties for estimation, statistical learning, and uncertainty quantification applications. Using statistical learning as an example, we demonstrate their advantage in training generative adversarial networks (GANs) for heavy-tailed, not-absolutely continuous sample distributions. We also show improved performance and stability over gradient-penalized Wasserstein GAN in image generation.

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