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

We consider a teacher-student model of supervised learning with a fully-trained 2-layer neural network whose width $k$ and input dimension $d$ are large and proportional. We compute the Bayes-optimal generalisation error of the network for any activation function in the regime where the number of training data $n$ scales quadratically with the input dimension, i.e., around the interpolation threshold where the number of trainable parameters $kd+k$ and of data points $n$ are comparable. Our analysis tackles generic weight distributions. Focusing on binary weights, 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.

Asymptotic Bayes risk of semi-supervised multitask learning on Gaussian mixture

The article considers semi-supervised multitask learning on a Gaussian mixture model (GMM). Using methods from statistical physics, we compute the asymptotic Bayes risk of each task in the regime of large datasets in high dimension, from which we analyze the role of task similarity in learning and evaluate the performance gain when tasks are learned together rather than separately. In the supervised case, we derive a simple algorithm that attains the Bayes optimal performance.