Canonical Regularisation of Wide Feature-Learning Neural Networks

• URL: http://arxiv.org/abs/2605.18180v1
• Date: Mon, 18 May 2026 10:23:06 GMT
• Title: Canonical Regularisation of Wide Feature-Learning Neural Networks
• Abstract summary: Feature-learning networks are far less studied than their kernel-regime counterparts.<n>We show that ridge regularisation biases gradient flow in feature-learning regime networks.<n>We also propose arc ridge as a minimax-robust, scalable surrogate to geodesic ridge.
• Score: 1.5581032465072868
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Wide neural networks in the feature-learning regime drive modern deep learning, and yet they remain far less studied than their kernel-regime counterparts. We consider a critical yet under-explored difference between these two regimes: the regulariser and prior implied by gradient flow training. This canonical regularisation property is well-studied in kernel regime networks -- of all the infinite global minima, gradient flow selects exactly the vanishing ridge solution -- and underpins the celebrated NN-GP correspondence, precisely allowing the modelling of noise during training. However, we prove ridge regularisation biases gradient flow in feature-learning regime networks, even in the infinitesimal limit of vanishing regularisation. Over training, ridge distorts the inductive bias of the network, with a particular damage done to pretrained networks where the implicit prior is informative. We resolve this by axiomatising the canonical regulariser as a regime-agnostic function-space energy and lift, which uniquely identifies ridge in the kernel regime, and crucially generalises to the feature-learning regime. By studying the Riemannian geometry of feature-learning networks, we derive geodesic ridge from our framework, generalising ridge to the feature-learning regime. Correspondingly, we prove the canonical function-space prior is a Riemannian Gibbs Process, generalising the more familiar Gaussian Process. As a practical contribution, we propose arc ridge as a minimax-robust, scalable surrogate to geodesic ridge, revealing a deep relationship between early stopping and canonical regularisation across learning regimes. Finally, we demonstrate the consequences of our theory empirically on both image processing and NLP transfer-learning problems.

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