Collaborative Research: RI: Medium: Principles for Optimization, Generalization, and Transferability via Deep Neural Collapse

Deep learning has demonstrated unprecedented performance across various domains in engineering and science. However, the theoretical understanding of their success has remained elusive. Very recently, researchers discovered and characterized an elegant mathematical structure within the learned features and classifiers called Neural Collapse. This phenomenon persists across a variety of different network architectures, datasets, and data domains. This project will leverage the symmetry of Neural Collapse to develop a rigorous mathematical theory to explain when and why it happens and how it can be used to quantify generalization performance and provide guidelines to understand and improve transferability. By advancing the mathematical foundations of deep learning, this project is expected to influence not only the machine learning community, but also related areas such as optimization, signal and image processing, and natural language processing. The project also involves an integrated outreach and education plan, including promoting accessibility and awareness of computing and STEM concepts for K-12 students. This project will expand our understanding of the principles behind non-convex optimization of training deep learning models, and provide new mathematical insights on their generalization and transferability properties, leading to practical implications. In particular, the project is focused on the following three overarching research thrusts: (i) provide a unified framework to analyze convergence guarantees for training deep and overparametrized models through general loss functions to states of neural collapse, first for simplified cases and then for more general deep models that exhibit progressive neural collapse, with multi-labels and data imbalance; (ii) harness the structure of neural collapse to provide tighter generalization bounds for deep models, by characterizing the structure of the resulting classifiers and their mild dependence on the training data, as well as by making natural distributional assumptions; (iii) leverage the generalization of progressive neural collapse to new environments to understand transferability of deep models to new domains and tasks, and develop principled approaches for improving transferability and efficient fine-tuning. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.