Convergence Analysis for Over-Parameterized Deep Learning (Communications in Computational Physics, 2024)
The success of deep learning in various applications has generated a growing interest in understanding its theoretical foundations. This paper presents a theoretical framework that explains why over-parameterized neural networks can performwell. Our analysis begins from the perspective of approximation theory and arguesthat over-parameterized deep neural networks with bounded norms can effectivelyapproximate the target. Additionally, we demonstrate that the metric entropy of suchnetworks is independent of the number of network parameters. We utilize these findings to derive consistency results for over-parameterized deep regression and the deepRitz method, respectively. Furthermore, we prove convergence rates when the targethas higher regularity, which, to our knowledge, represents the first convergence ratefor over-parameterized deep learning.
Keywords: Over-parameterization; convergence rate; approximation; generalization.