Dense Hebbian neural networks : a replica symmetric picture of supervised learning

We consider dense, associative neural-networks trained by a teacher (i.e., with supervision) and we investigate their computational capabilities analytically, via statistical-mechanics tools, and numerically, via Monte Carlo simulations. In particular, we obtain a phase diagram which summarizes their performance as a function of the control parameters (e.g., quality and quantity of the training dataset, network storage, noise), that is valid in the limit of large network-size and structureless datasets. We also numerically test the learning, storing and retrieval capabilities of these networks on structured datasets such as MNist and Fashion MNist. As technical remarks, on the analytic side, we extend Guerra's interpolation to tackle the non-Gaussian distributions involved in the post-synaptic potentials while, on the computational side, we insert Plefka's approximation in the Monte Carlo scheme, to speed up the evaluation of the synaptic tensors, overall obtaining a novel and broad approach to investigate supervised learning in neural networks, beyond the shallow limit.