# Large deviation principles for convolutional Bayesian neural networks

arXiv: https://arxiv.org/abs/2603.06023 · HTML: https://mlsanity.com/paper/2603.06023/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 29 references that are in the map.
- First posted on arXiv 6 Mar 2026.

## Builds on (most cited first, 14 of 14)

- [Deep Neural Networks as Gaussian Processes](https://mlsanity.com/paper/1711.00165/) (2017)
- [Gaussian Process Behaviour in Wide Deep Neural Networks](https://mlsanity.com/paper/1804.11271/) (2018)
- [Bayesian Deep Convolutional Networks with Many Channels are Gaussian Processes](https://mlsanity.com/paper/1810.05148/) (2018)
- [Tensor Programs I: Wide Feedforward or Recurrent Neural Networks of Any Architecture are Gaussian Processes](https://mlsanity.com/paper/1910.12478/) (2019)
- [Deep Convolutional Networks as shallow Gaussian Processes](https://mlsanity.com/paper/1808.05587/) (2018)
- [Random Neural Networks in the Infinite Width Limit as Gaussian Processes](https://mlsanity.com/paper/2107.01562/) (2021)
- [The Neural Covariance SDE: Shaped Infinite Depth-and-Width Networks at Initialization](https://mlsanity.com/paper/2206.02768/) (2022)
- [Random Fully Connected Neural Networks as Perturbatively Solvable Hierarchies](https://mlsanity.com/paper/2204.01058/) (2022)
- [Quantitative CLTs in Deep Neural Networks](https://mlsanity.com/paper/2307.06092/) (2023)
- [Local Kernel Renormalization as a mechanism for feature learning in overparametrized Convolutional Neural Networks](https://mlsanity.com/paper/2307.11807/) (2023)
- [Wide Deep Neural Networks with Gaussian Weights are Very Close to Gaussian Processes](https://mlsanity.com/paper/2312.11737/) (2023)
- [Entropic bounds for conditionally Gaussian vectors and applications to neural networks](https://mlsanity.com/paper/2504.08335/) (2025)
- [Large Deviations of Gaussian Neural Networks with ReLU activation](https://mlsanity.com/paper/2405.16958/) (2024)
- [Proportional infinite-width infinite-depth limit for deep linear neural networks](https://mlsanity.com/paper/2411.15267/) (2024)

## Abstract (arXiv)

While suitably scaled CNNs with Gaussian initialization are known to converge to Gaussian processes as the number of channels diverges, little is known beyond this Gaussian limit. We establish a large deviation principle (LDP) for convolutional neural networks in the infinite-channel regime. We consider a broad class of multidimensional CNN architectures characterized by general receptive fields encoded through a patch-extractor function satisfying mild structural assumptions. Our main result establishes a large deviation principle (LDP) for the sequence of conditional covariance matrices under Gaussian prior distribution on the weights. We further derive an LDP for the posterior distribution obtained by conditioning on a finite number of observations. In addition, we provide a streamlined proof of the concentration of the conditional covariances and of the Gaussian equivalence of the network. To the best of our knowledge, this is the first large deviation principle established for convolutional neural networks.