Researchers Establish Large Deviation Principles for Convolutional Bayesian Neural Networks
Mathematicians have derived the first large deviation principle (LDP) for convolutional neural networks (CNNs) in the infinite-channel regime, characterizing probabilistic behavior well beyond the known Gaussian limit. The work addresses a gap in the theoretical understanding of CNNs, which were previously known to converge to Gaussian processes under scaling but lacked rigorous large-deviation analysis. The results provide sharper probabilistic guarantees for Bayesian CNN inference, with implications for understanding rare events and tail behavior in deep learning models.
A preprint posted to arXiv establishes, for the first time, a large deviation principle for convolutional neural networks operating in the infinite-channel limit. Prior work had shown that suitably scaled CNNs with Gaussian weight initialization converge to Gaussian processes as the number of channels grows, but behavior beyond this Gaussian approximation remained poorly understood. The authors consider a broad class of multidimensional CNN architectures defined by general receptive fields encoded through a patch-extractor function under mild structural assumptions. Their main theorem proves an LDP for the sequence of conditional covariance matrices under a Gaussian prior on the weights, and they further extend this to the posterior distribution obtained by conditioning on finitely many observations. The paper also provides a streamlined proof of the concentration of conditional covariances and the Gaussian equivalence of the network. The work was first submitted in March 2026 and updated in June 2026 with simplified notation, suggesting ongoing refinement of the theoretical framework.
What's missing
The paper does not discuss computational or practical implications of the LDP results for finite-width or finite-channel networks used in practice. As a preprint, the work has not yet undergone formal peer review.
What different sources said
- arXiv stat.MLCenter
Large deviation principles for convolutional Bayesian neural networks
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