Learning heavy-tailed distributions with Wasserstein-proximal-regularized $α$-divergences

• URL: http://arxiv.org/abs/2405.13962v1
• Date: Wed, 22 May 2024 19:58:13 GMT
• Title: Learning heavy-tailed distributions with Wasserstein-proximal-regularized $α$-divergences
• Authors: Ziyu Chen, Hyemin Gu, Markos A. Katsoulakis, Luc Rey-Bellet, Wei Zhu,
• Abstract summary: We propose Wasserstein proximals of $alpha$-divergences as suitable objective functionals for learning heavy-tailed distributions. Heuristically, $alpha$-divergences handle the heavy tails and Wasserstein proximals allow non-absolute continuity between distributions.
• Score: 12.19634962193403
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In this paper, we propose Wasserstein proximals of $\alpha$-divergences as suitable objective functionals for learning heavy-tailed distributions in a stable manner. First, we provide sufficient, and in some cases necessary, relations among data dimension, $\alpha$, and the decay rate of data distributions for the Wasserstein-proximal-regularized divergence to be finite. Finite-sample convergence rates for the estimation in the case of the Wasserstein-1 proximal divergences are then provided under certain tail conditions. Numerical experiments demonstrate stable learning of heavy-tailed distributions -- even those without first or second moment -- without any explicit knowledge of the tail behavior, using suitable generative models such as GANs and flow-based models related to our proposed Wasserstein-proximal-regularized $\alpha$-divergences. Heuristically, $\alpha$-divergences handle the heavy tails and Wasserstein proximals allow non-absolute continuity between distributions and control the velocities of flow-based algorithms as they learn the target distribution deep into the tails.

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