On the natural gradient of the evidence lower bound

This article studies the Fisher-Rao gradient, also referred to as the natural gradient, of the evidence lower bound (ELBO) which plays a central role in generative machine learning. It reveals that the gap between the evidence and its lower bound, the ELBO, has essentially a vanishing natural gradient within unconstrained optimization. As a result, maximization of the ELBO is equivalent to minimization of the Kullback-Leibler divergence from a target distribution, the primary objective function of learning. Building on this insight, we derive a condition under which this equivalence persists even when optimization is constrained to a model. This condition yields a geometric characterization, which we formalize through the notion of a cylindrical model .

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Subjects

Evidence lower bound

variational gap

natural gradient

information geometry

variational inference

DDC Class

006.3: Artificial Intelligence

519: Applied Mathematics, Probabilities

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https://creativecommons.org/licenses/by/4.0/

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publishedVersion

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24-0606.pdf

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Main Article

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1.15 MB

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Adobe PDF