Beyond NNGP: Large Deviations and Feature Learning in Bayesian Neural Networks

Katerina Papagiannouli; Dario Trevisan; Giuseppe Pio Zitto

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arXiv · 2602.22925

Beyond NNGP: Large Deviations and Feature Learning in Bayesian Neural Networks

Katerina Papagiannouli·Dario Trevisan·Giuseppe Pio Zitto

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Abstract

We study wide Bayesian neural networks focusing on the rare but statistically dominant fluctuations that govern posterior concentration, beyond Gaussian-process limits. Large-deviation theory provides explicit variational objectives-rate functions-on predictors, providing an emerging notion of complexity and feature learning directly at the functional level. We show that the posterior output rate function is obtained by a joint optimization over predictors and internal kernels, in contrast with fixed-kernel (NNGP) theory. Numerical experiments demonstrate that the resulting predictions accurately describe finite-width behavior for moderately sized networks, capturing non-Gaussian tails, posterior deformation, and data-dependent kernel selection effects.

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Katerina Papagiannouli, Dario Trevisan, Giuseppe Pio Zitto. 2026-02-26. Beyond NNGP: Large Deviations and Feature Learning in Bayesian Neural Networks. https://arxiv.org/abs/2602.22925

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