A simple mean field model of feature learning

• URL: http://arxiv.org/abs/2510.15174v1
• Date: Thu, 16 Oct 2025 22:28:44 GMT
• Title: A simple mean field model of feature learning
• Authors: Niclas Göring, Chris Mingard, Yoonsoo Nam, Ard Louis,
• Abstract summary: We derive a tractable, self-consistent mean-field (MF) theory for two-layer non-linear networks trained with a gradient Langevin dynamics (SGLD)<n>At infinite width, this theory reduces to kernel ridge regression, but at finite width it predicts symmetry breaking phase transition where networks abruptly align with target functions.<n>While the basic MF theory provides theoretical insight into the emergence of FL in the finite-width regime, semi-quantitatively predicting the onset of FL with noise or sample size, it substantially underestimates the improvements in generalisation after the transition.
• Score: 2.3215806943173676
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Feature learning (FL), where neural networks adapt their internal representations during training, remains poorly understood. Using methods from statistical physics, we derive a tractable, self-consistent mean-field (MF) theory for the Bayesian posterior of two-layer non-linear networks trained with stochastic gradient Langevin dynamics (SGLD). At infinite width, this theory reduces to kernel ridge regression, but at finite width it predicts a symmetry breaking phase transition where networks abruptly align with target functions. While the basic MF theory provides theoretical insight into the emergence of FL in the finite-width regime, semi-quantitatively predicting the onset of FL with noise or sample size, it substantially underestimates the improvements in generalisation after the transition. We trace this discrepancy to a key mechanism absent from the plain MF description: \textit{self-reinforcing input feature selection}. Incorporating this mechanism into the MF theory allows us to quantitatively match the learning curves of SGLD-trained networks and provides mechanistic insight into FL.

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