Inclusive KL Minimization: A Wasserstein-Fisher-Rao Gradient Flow Perspective

TL;DR

This paper develops a mathematical framework for inclusive KL divergence minimization using Wasserstein-Fisher-Rao gradient flows, unifying existing algorithms and providing theoretical insights into their behavior.

Contribution

It introduces a PDE-based gradient flow approach for inclusive KL inference, unifies existing sampling algorithms under this paradigm, and establishes a theoretical foundation for Wasserstein-Fisher-Rao flows.

Findings

Existing algorithms can be interpreted as inclusive KL inference methods.

A PDE-based gradient flow framework for inclusive KL divergence is developed.

Theoretical foundation for Wasserstein-Fisher-Rao gradient flows is provided.

Abstract

Otto's (2001) Wasserstein gradient flow of the exclusive KL divergence functional provides a powerful and mathematically principled perspective for analyzing learning and inference algorithms. In contrast, algorithms for the inclusive KL inference, i.e., minimizing $ \mathrm{KL}(\pi \| \mu) $ with respect to $ \mu $ for some target $ \pi $, are rarely analyzed using tools from mathematical analysis. This paper shows that a general-purpose approximate inclusive KL inference paradigm can be constructed using the theory of gradient flows derived from PDE analysis. We uncover that several existing learning algorithms can be viewed as particular realizations of the inclusive KL inference paradigm. For example, existing sampling algorithms such as Arbel et al. (2019) and Korba et al. (2021) can be viewed in a unified manner as inclusive-KL inference with approximate gradient estimators. Finally, we provide the theoretical foundation for the Wasserstein-Fisher-Rao gradient flows for minimizing the inclusive KL divergence.