A Gaussian Process View on Observation Noise and Initialization in Wide Neural Networks

Sergio Calvo Ordoñez, Jonathan Plenk, Richard Bergna, Alvaro Cartea, José Miguel Hernández-Lobato, Konstantina Palla, Kamil Ciosek
Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, PMLR 300:3889-3897, 2026.

Abstract

Performing gradient descent in a wide neural network is equivalent to computing the posterior mean of a Gaussian Process with the Neural Tangent Kernel (NTK-GP), for a specific prior mean and with zero observation noise. However, existing formulations have two limitations: (i) observation noise, since the NTK-GP assumes noiseless targets, leading to misspecification on noisy data; (ii) the equivalence does not extend to arbitrary prior means, which are essential for well-specified models. To address (i), we introduce a regularizer into the training objective, showing its correspondence to incorporating observation noise in the NTK-GP. To address (ii), we propose a \textit{shifted network} that enables arbitrary prior means and allows obtaining the posterior mean with gradient descent on a single network, without ensembling or kernel inversion. We validate our results with experiments across datasets and architectures, showing that this approach removes key obstacles to the practical use of NTK-GP equivalence in applied Gaussian process modeling.

Cite this Paper

BibTeX
@InProceedings{pmlr-v300-ordonez26a, title = { A Gaussian Process View on Observation Noise and Initialization in Wide Neural Networks }, author = {Ordo\~{n}ez, Sergio Calvo and Plenk, Jonathan and Bergna, Richard and Cartea, Alvaro and Hern{\'a}ndez-Lobato, Jos{\'e} Miguel and Palla, Konstantina and Ciosek, Kamil}, booktitle = {Proceedings of The 29th International Conference on Artificial Intelligence and Statistics}, pages = {3889--3897}, year = {2026}, editor = {Khan, Emtiyaz and Li, Yingzhen and Solin, Arno and Ramdas, Aaditya}, volume = {300}, series = {Proceedings of Machine Learning Research}, month = {02--05 May}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v300/main/assets/ordonez26a/ordonez26a.pdf}, url = {https://proceedings.mlr.press/v300/ordonez26a.html}, abstract = { Performing gradient descent in a wide neural network is equivalent to computing the posterior mean of a Gaussian Process with the Neural Tangent Kernel (NTK-GP), for a specific prior mean and with zero observation noise. However, existing formulations have two limitations: (i) observation noise, since the NTK-GP assumes noiseless targets, leading to misspecification on noisy data; (ii) the equivalence does not extend to arbitrary prior means, which are essential for well-specified models. To address (i), we introduce a regularizer into the training objective, showing its correspondence to incorporating observation noise in the NTK-GP. To address (ii), we propose a \textit{shifted network} that enables arbitrary prior means and allows obtaining the posterior mean with gradient descent on a single network, without ensembling or kernel inversion. We validate our results with experiments across datasets and architectures, showing that this approach removes key obstacles to the practical use of NTK-GP equivalence in applied Gaussian process modeling. } }
Endnote
%0 Conference Paper %T A Gaussian Process View on Observation Noise and Initialization in Wide Neural Networks %A Sergio Calvo Ordoñez %A Jonathan Plenk %A Richard Bergna %A Alvaro Cartea %A José Miguel Hernández-Lobato %A Konstantina Palla %A Kamil Ciosek %B Proceedings of The 29th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2026 %E Emtiyaz Khan %E Yingzhen Li %E Arno Solin %E Aaditya Ramdas %F pmlr-v300-ordonez26a %I PMLR %P 3889--3897 %U https://proceedings.mlr.press/v300/ordonez26a.html %V 300 %X Performing gradient descent in a wide neural network is equivalent to computing the posterior mean of a Gaussian Process with the Neural Tangent Kernel (NTK-GP), for a specific prior mean and with zero observation noise. However, existing formulations have two limitations: (i) observation noise, since the NTK-GP assumes noiseless targets, leading to misspecification on noisy data; (ii) the equivalence does not extend to arbitrary prior means, which are essential for well-specified models. To address (i), we introduce a regularizer into the training objective, showing its correspondence to incorporating observation noise in the NTK-GP. To address (ii), we propose a \textit{shifted network} that enables arbitrary prior means and allows obtaining the posterior mean with gradient descent on a single network, without ensembling or kernel inversion. We validate our results with experiments across datasets and architectures, showing that this approach removes key obstacles to the practical use of NTK-GP equivalence in applied Gaussian process modeling.
APA
Ordoñez, S.C., Plenk, J., Bergna, R., Cartea, A., Hernández-Lobato, J.M., Palla, K. & Ciosek, K.. (2026). A Gaussian Process View on Observation Noise and Initialization in Wide Neural Networks . Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 300:3889-3897 Available from https://proceedings.mlr.press/v300/ordonez26a.html.

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