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arXiv · 2610.33208

Lower expected Fisher rank implies better PAC generalization

Marco Miani·Albert Kjøller Jacobsen·Albert Lopez i Serrano·Søren Hauberg·Georgios Arvanitidis
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Abstract

Deep learning architectures tend to generalize well to unseen data, but explaining this capability is still an open area of research. Under the assumption of i.i.d. data, PAC-Bayes provides a bound on generalization error, and its tightness relies on a parameter distribution that covers a large volume of low-loss solutions. The Fisher information metric endows the parameter space with a geometry for which distances reflect functional changes, and whose zero-component captures functional invariances. We define and study a Fisher-aligned parameter distribution for which we can simultaneously guarantee (1) a low-loss, thanks to the exploration constrained on the functional invariances and (2) a large log-volume of parameters, proportional to the number of zero eigenvalues. This distribution results in a high-probability generalization error bound, which scales with the expected rank of the Fisher information matrix, shedding light on the relevance of this quantity.

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Marco Miani, Albert Kjøller Jacobsen, Albert Lopez i Serrano, Søren Hauberg, Georgios Arvanitidis. 2026-10-23. Lower expected Fisher rank implies better PAC generalization. https://arxiv.org/abs/2610.33208

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