Prevalence of neural collapse during the terminal phase of deep learning training

By Vardan Papyan, Xuemei Han, David L. Donoho. 2020

962 citations recorded by the source corpus.

Significance Modern deep neural networks for image classification have achieved superhuman performance. Yet, the complex details of trained networks have forced most practitioners and researchers to regard them as black boxes with little that could be understood. This paper considers in detail a now-standard training methodology: driving the cross-entropy loss to zero, continuing long after the classification error is already zero. Applying this methodology to an authoritative collection of standard deepnets and datasets, we observe the emergence of a simple and highly symmetric geometry of the deepnet features and of the deepnet classifier, and we document important benefits that the geometry conveys—thereby helping us understand an important component of the modern deep learning training paradigm. 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.

Subfields: Artificial Intelligence

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Citations in the corpus

Continual Learning by Modeling Intra-Class Variation (2022, background): “Second, the representation space is more regularized, since it converges to the simplex equiangular tight frame (Papyan et al., 2020) which is also equivalent to a hyperspherically uniform space (Liu et al., 2018a; Lin et al., 2020; Liu et al., 2021c;b).”
Incremental Learning via Rate Reduction (2020, background): “Recent work [15] shows that this direct label fitting leads to a phenomena called neural collapse, where within-class variability and structural information are completely suppressed.”
Why Do Better Loss Functions Lead to Less Transferable Features? (2020, background): “[50] describe a phenomenon they call “neural collapse,” which occurs after training error vanishes and is associated with the collapse of the class means, classifier, and activations to the vertices of an equiangular tight frame, implying maximal class separation.”
Dissecting Supervised Constrastive Learning (2021, result): “This is in line with the special case of equinorm representations and weights, presented in (Wang et al., 2017), and complements a recent optimality result by Papyan et al. (2020) (cf. Remark 2 for details).”
Pre-trained Vision and Language Transformers are Few-Shot Incremental Learners (2024, background): “Yang et al. [39] proposed the framework inspired by neural collapse (Papyan et al. [22]) which aims to align between the feature and corresponding weight of the classifier.”
Continual Learning with Foundation Models: An Empirical Study of Latent Replay (2022, background): “This can be attributed to the phenomenon discussed by Papyan et al. (2020) and Galanti et al. (2021), who point out that supervised pre-training can lead to features collapsing to the mean feature vector of the same classes facilitating the applicability of linear and metric based classiﬁers.”
Incremental Learning via Rate Reduction (2021, background): “Recent work [15] shows that this direct label fitting leads to a phenomena called neural collapse in deep networks, where within-class variability and structural information are completely suppressed.”
Neural Collapse Inspired Feature-Classifier Alignment for Few-Shot Class Incremental Learning (2023, background): “…the terminal phase of training (after 0 training error rate), the last-layer features of the same class will collapse into a single vertex, and the vertices of all classes will be aligned with their classiﬁer prototypes and be formed as a simplex equiangular tight frame (ETF) (Papyan et al., 2020).”
Learning to Give Checkable Answers with Prover-Verifier Games (2021, neutral): “We link this to the Neural Collapse phenomenon elaborated by Papyan et al. [26] and leave a more detailed examination to future works.”
The Platonic Representation Hypothesis (2024, neutral): “Nonetheless, similar arguments have been made theoretically and empirically that cooccurrence relations are learned by practical contrastive (Wang & Isola, 2020; Zimmermann et al., 2021) and predictive learners (Papyan et al., 2020; Roeder et al., 2021).”
On the Representation Collapse of Sparse Mixture of Experts (2022, background): “Several studies observe that the within-class variation of the representations in classiﬁcation networks becomes negligible at the terminal phase of training (Papyan et al., 2020; Zhu et al., 2021; Tirer and Bruna, 2022).”
Sparse Prototype Code Underlies Classification and Prediction Across Modalities (2026, unknown): “…geometry —manifold capacity [Chung et al., 2018, Cohen et al., 2020, Stephenson et al., 2021, Wakhloo et al., 2023] and neural collapse [Papyan et al., 2020, Han et al., 2021, Galanti et al., 2021] — have made important progress in connecting geometry to performance within their…”