More Than a Toy: Random Matrix Models Predict How Real-World Neural Representations Generalize

• URL: http://arxiv.org/abs/2203.06176v1
• Date: Fri, 11 Mar 2022 18:59:01 GMT
• Title: More Than a Toy: Random Matrix Models Predict How Real-World Neural Representations Generalize
• Authors: Alexander Wei and Wei Hu and Jacob Steinhardt
• Abstract summary: We find that most theoretical analyses fall short of capturing qualitative phenomena even for kernel regression. We prove that the classical GCV estimator converges to the generalization risk whenever a local random matrix law holds. Our findings suggest that random matrix theory may be central to understanding the properties of neural representations in practice.
• Score: 94.70343385404203
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Of theories for why large-scale machine learning models generalize despite being vastly overparameterized, which of their assumptions are needed to capture the qualitative phenomena of generalization in the real world? On one hand, we find that most theoretical analyses fall short of capturing these qualitative phenomena even for kernel regression, when applied to kernels derived from large-scale neural networks (e.g., ResNet-50) and real data (e.g., CIFAR-100). On the other hand, we find that the classical GCV estimator (Craven and Wahba, 1978) accurately predicts generalization risk even in such overparameterized settings. To bolster this empirical finding, we prove that the GCV estimator converges to the generalization risk whenever a local random matrix law holds. Finally, we apply this random matrix theory lens to explain why pretrained representations generalize better as well as what factors govern scaling laws for kernel regression. Our findings suggest that random matrix theory, rather than just being a toy model, may be central to understanding the properties of neural representations in practice.

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