Optimal generalisation and learning transition in extensive-width shallow neural networks near interpolation

arXiv:2501.18530v3 Announce Type: replace-cross Abstract: We consider a teacher-student model of supervised learning with a fully-trained two-layer neural network whose width k and input dimension d are large and proportional. We provide an effective theory for approximating the Bayes-optimal generalisation error of the network for any activation function in the regime of sample size n scaling quadratically with the input dimension, i.e., around the interpolation threshold where the number of trainable parameters kd+k and of data n are comparable. Our analysis tackles generic weight distributions. We uncover a discontinuous phase transition separating a "universal" phase from a "specialisation" phase. In the first, the generalisation error is independent of the weight distribution and decays slowly with the sampling rate n/d^2, with the student learning only some non-linear combinations of the teacher weights. In the latter, the error is weight distribution-dependent and decays faster due to the alignment of the student towards the teacher network. We thus unveil the existence of a highly predictive solution near interpolation, which is however potentially hard to find by practical algorithms.

Source: arXiv cs.LG | 2026-07-27