Review for NeurIPS paper: Generalization error in high-dimensional perceptrons: Approaching Bayes error with convex optimization

Specifically their rigorous results concern the model: for i 1,2,...,n Y_i sign( X_i,w *) for Gaussian prior on w * and Gaussian X_i where they live on d dimensions. They assume n/d \alpha (constant) and n,d grow to infinity. In [10] the Bayes optimal reconstruction error has been studied (verifying a stats physics prediction) and here they discuss about the performance of regularize ERM (potentially convex methods) to achieve it. Their first set of results is about the performance of any \ell_2 regularized convex loss and showing that their performance can be tracked using a fixed point equation. The result is based on Gordon's minimax theory and is shown then to be verifying also the replica (stats physics) prediction.