Inclusive KL Minimization: A Wasserstein-Fisher-Rao Gradient Flow Perspective
This work provides a theoretical foundation for inclusive KL inference, which is incremental as it builds on existing gradient flow concepts to unify and analyze algorithms in machine learning.
The paper tackles the lack of mathematical analysis for inclusive KL minimization algorithms by constructing a general-purpose inference paradigm using gradient flow theory from PDE analysis, showing that several existing sampling algorithms can be unified under this framework.
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}(π\| μ) $ with respect to $ μ$ for some target $ π$, 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.