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When training overparameterized deep networks for classification tasks, it has been widely observed that the learned features exhibit a so-called “neural collapse” phenomenon. More specifically, for the output features of the penultimate layer, for each class the within-class features converge to their means, and the means of different classes exhibit a certain tight frame structure, which is also aligned with the last layer’s classifier. As feature normalization in the last layer becomes a common practice in modern representation learning, in this work we theoretically justify the neural collapse phenomenon under normalized features. Based on an un-constrained feature model, we simplify the empirical loss function in a multi-class classification task into a nonconvex optimization problem over the Riemannian manifold by constraining all features and classifiers over the sphere. In this context, we analyze the nonconvex landscape of the Riemannian optimization problem over the product of spheres, showing a benign global landscape in the sense that the only global minimizers are the neural collapse solutions while all other critical points are strict saddle points with negative curvature. Experimental results on practical deep networks corroborate our theory and demonstrate that better representations can be learned faster via feature normalization. Code for our experiments can be found at https://github.com/cjyaras/normalized-neural-collapse.
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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 toward 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 (NC), 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.
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- PAR ID:
- 10273204
- Date Published:
- Journal Name:
- Proceedings of the National Academy of Sciences
- Volume:
- 117
- Issue:
- 40
- ISSN:
- 0027-8424
- Page Range / eLocation ID:
- 24652 to 24663
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1073/pnas.2015509117
