The Prevalence of Neural Collapse in Neural Multivariate Regression

TL;DR

This paper demonstrates that neural collapse phenomena also occur in multivariate regression tasks, supported by empirical evidence and theoretical modeling, suggesting neural collapse may be a universal behavior in deep learning.

Contribution

It introduces Neural Regression Collapse (NRC), extending neural collapse concepts to regression, with empirical validation and a theoretical explanation via the Unconstrained Feature Model.

Findings

NRC occurs in various datasets and architectures.

NRC involves feature vectors collapsing to target and weight subspaces.

Regularization is crucial for the emergence of NRC.

Abstract

Recently it has been observed that neural networks exhibit Neural Collapse (NC) during the final stage of training for the classification problem. We empirically show that multivariate regression, as employed in imitation learning and other applications, exhibits Neural Regression Collapse (NRC), a new form of neural collapse: (NRC1) The last-layer feature vectors collapse to the subspace spanned by the $n$ principal components of the feature vectors, where $n$ is the dimension of the targets (for univariate regression, $n=1$); (NRC2) The last-layer feature vectors also collapse to the subspace spanned by the last-layer weight vectors; (NRC3) The Gram matrix for the weight vectors converges to a specific functional form that depends on the covariance matrix of the targets. After empirically establishing the prevalence of (NRC1)-(NRC3) for a variety of datasets and network architectures, we provide an explanation of these phenomena by modeling the regression task in the context of the Unconstrained Feature Model (UFM), in which the last layer feature vectors are treated as free variables when minimizing the loss function. We show that when the regularization parameters in the UFM model are strictly positive, then (NRC1)-(NRC3) also emerge as solutions in the UFM optimization problem. We also show that if the regularization parameters are equal to zero, then there is no collapse. To our knowledge, this is the first empirical and theoretical study of neural collapse in the context of regression. This extension is significant not only because it broadens the applicability of neural collapse to a new category of problems but also because it suggests that the phenomena of neural collapse could be a universal behavior in deep learning.