Inclusive KL Minimization: A Wasserstein-Fisher-Rao Gradient Flow Perspective (2411.00214v1)

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.

• The paper introduces a novel gradient flow framework for inclusive KL minimization that unifies inference and sampling algorithms using Wasserstein and Fisher-Rao geometries.
• It applies kernelized integral operators to approximate gradient flows, effectively handling non-smooth regions in the divergence landscape.
• The study demonstrates equivalence with MMD and KSD based flows, paving the way for enhanced generative models and Bayesian computation.

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

This paper presents a sophisticated exploration of inclusive Kullback-Leibler (KL) minimization through the lens of gradient flows in spaces defined by Wasserstein and Fisher-Rao geometries. The work builds on prior foundational research by Otto, bringing a new perspective to the mathematical treatment of adaptive learning and inference algorithms, especially in the context of minimizing the inclusive KL divergence.

Key Contributions and Theoretical Insights

1. Gradient Flow Framework for Inclusive KL Minimization: The paper establishes a comprehensive framework for understanding inclusive KL inference by leveraging the gradient flow perspective derived from partial differential equations (PDEs). Traditionally, the focus has been predominantly on exclusive KL minimization, grounded in well-founded mathematical analysis. Here, the authors extend such rigor to inclusive KL minimization, highlighting the underlying Wasserstein and Fisher-Rao gradient structures. They demonstrate that several inference and sampling algorithms, including expectation propagation and maximum mean discrepancy (MMD) based sampling, can be unified under this inclusive-KL gradient flow framework.
2. Approximate Gradient Flows via Integral Operators: One of the salient aspects of this work is the application of integral operators to approximate the gradient flows of inclusive KL divergence, which traditionally suffers from non-smooth nature around zero mass regions. By applying kernel operations, the authors derive kernelized variants of the Wasserstein gradient flow equations, which are implementable using existing algorithms for MMD minimization.
3. Fisher-Rao and Wasserstein-Fisher-Rao Gradient Flows: The authors also delve into the Fisher-Rao gradient flows, uncovering intriguing properties such as straight-line mass transport within Fisher-Rao geometry—highlighting its distinct analytical features. They expand this to consider the Wasserstein-Fisher-Rao (WFR) metric, integrating mass transport and creation/destruction mechanisms, promising exponential decay rates for inclusive KL divergence—a significant observation offering new algorithmic pathways for unbalanced transport in generative models.

Implications and Future Directions

Algorithmic Equivalence and Practical Implementations: A critical contribution is proving the equivalence between the approximate inclusive KL gradient flows and known MMD and KSD based flows. This not only offers a unifying theoretical underpinning but also validates practical algorithm implementations in existing machine learning frameworks for inference and sampling tasks.

Potential for Expanding Generative Models: While focused on Bayesian inference, the principles outlined could extend to generative adversarial networks (GANs) and score-based modeling, offering a potential to rigorously formalize these domains through gradient flow dynamics—particularly advantageous when considering non-Gaussian anisotropic noise and complex manifolds.

Conclusion

This paper advances the understanding of gradient flow methods for inclusive KL divergence minimization with far-reaching implications in Bayesian computation and machine learning. By adopting a rigorous mathematical approach, it bridges a gap in theoretical foundations while affirming the practical validity of algorithms essential for modern AI applications. Researchers are encouraged to explore further integration of inclusive KL inference in generative models and expand on the interrelation between different gradient geometries to innovate in artificial intelligence solutions.