Nearly Optimal VC-Dimension and Pseudo-Dimension Bounds for Deep Neural Network Derivatives

Published: 21 Sept 2023, Last Modified: 02 Nov 2023NeurIPS 2023 posterEveryoneRevisionsBibTeX
Keywords: VC-dimension, pseudo-dimension, Sobolev space, generalization error, nearly optimal approximation
Abstract: This paper addresses the problem of nearly optimal Vapnik--Chervonenkis dimension (VC-dimension) and pseudo-dimension estimations of the derivative functions of deep neural networks (DNNs). Two important applications of these estimations include: 1) Establishing a nearly tight approximation result of DNNs in the Sobolev space; 2) Characterizing the generalization error of machine learning methods with loss functions involving function derivatives. This theoretical investigation fills the gap of learning error estimations for a wide range of physics-informed machine learning models and applications including generative models, solving partial differential equations, operator learning, network compression, distillation, regularization, etc.
Supplementary Material: pdf
Submission Number: 10386