Representation Gap: Explaining the Unreasonable Effectiveness of Neural Networks from a Geometric Perspective

TL;DR

This paper introduces the Representation Gap, a new metric related to generalization error, which is governed by the intrinsic dimension of the task and helps explain neural networks' effectiveness.

Contribution

The paper derives a precise asymptotic equivalent of the Representation Gap, linking it to the intrinsic dimension and extending its applicability to various tasks and algorithms.

Findings

The asymptotic law accurately predicts generalization behavior on synthetic datasets.

Intrinsic dimension estimation is effective and interpretable.

Results are consistent with existing literature on realistic datasets.

Abstract

Characterizing precisely the asymptotic generalization error of neural networks using parameters that can be estimated efficiently is a crucial problem in machine learning, which relies heavily on heuristics and practitioners' intuition to make key design choices. In order to mitigate this issue, we introduce the Representation Gap, a metric closely related to the generalization error, but admitting better-behaved asymptotic dynamics. Focusing on equivariant diffusion models and leveraging results from optimal quantization and point-process theory, we derive a precise asymptotic equivalent of the Representation Gap and show that it is governed by a single parameter, the \textit{intrinsic dimension} of the task, which is easy to interpret, efficient to estimate, and can be linked to the equivariances of common neural network architectures. We show that this asymptotic dynamic also extends to a broader range of tasks and training algorithms. Finally, we demonstrate empirically that our asymptotic law and intrinsic dimension estimation are accurate on a wide range of synthetic datasets, where these quantities are known, as well as on more realistic datasets, where we obtain results consistent with the related literature.