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.
@article{arxiv.2602.22925,
title = {Beyond NNGP: Large Deviations and Feature Learning in Bayesian Neural Networks},
author = {Katerina Papagiannouli and Dario Trevisan and Giuseppe Pio Zitto},
journal= {arXiv preprint arXiv:2602.22925},
year = {2026}
}