# Mathematics of Deep Learning Seminar: David Donoho

Date & Time

Speaker: David Donoho
Department of Statistics, Stanford University

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

Abstract: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 classifier’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 confers important benefits, including better generalization performance, better robustness, and better interpretability.

This is joint work with Vardan Papyan, U Toronto and XY Han, Cornell

Reference papers:

Prevalence of neural collapse during the terminal phase of deep learning training
Papyan, V., Han, X.Y. and Donoho, D.L., 2020.
Proceedings of the National Academy of Sciences, 117(40), pp.24652-24663
arXiv:2008.08186

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