# Codes and Expansions (CodEx) Seminar

## X.Y. Han (Cornell University)

Prevalence of Neural Collapse during the terminal phase of deep learning training

Modern practice for training classification deepnets involves a Terminal Phase of Training (TPT), which begins at the epoch where training error first vanishes; During TPT, the training error stays effectively zero while training loss is pushed towards zero.
Direct measurements of TPT, for three prototypical deepnet architectures and across seven canonical classification datasets, expose a pervasive inductive bias we call Neural Collapse, involving four deeply interconnected phenomena: (NC1) Cross-example within-class variability of last-layer training activations collapses to zero, as the individual activations themselves collapse to their class-means; (NC2) The class-means collapse to the vertices of a Simplex Equiangular Tight Frame (ETF); (NC3) Up to rescaling, the last-layer classifiers collapse to the class-means, or in other words to the Simplex ETF, i.e. to a self-dual configuration; (NC4) For a given activation, the classifierâ€™s decision collapses to simply choosing whichever class has the closest train class-mean, i.e. the Nearest Class-Center (NCC) decision rule. The symmetric and very simple geometry induced by the TPT occur simultaneously with important benefits, including better generalization performance, better robustness, and better interpretability. I will discuss the Neural Collapse phenomenon as well as recent theoretical models of this behavior.

This is joint work with Vardan Papyan, U Toronto and David Donoho, Stanford.