Year: 2024
Author: Noboru Isobe, Mizuho Okumura
Journal of Machine Learning, Vol. 3 (2024), Iss. 4 : pp. 413–444
Abstract
This paper presents a mathematical analysis of ODE-Net, a continuum model of deep neural networks (DNNs). In recent years, machine learning researchers have introduced ideas of replacing the deep structure of DNNs with ODEs as a continuum limit. These studies regard the “learning” of ODE-Net as the minimization of a “loss” constrained by a parametric ODE. Although the existence of a minimizer for this minimization problem needs to be assumed, only a few studies have investigated the existence analytically in detail. In the present paper, the existence of a minimizer is discussed based on a formulation of ODE-Net as a measure-theoretic mean-field optimal control problem. The existence result is proved when a neural network describing a vector field of ODE-Net is linear with respect to learnable parameters. The proof employs the measure-theoretic formulation combined with the direct method of calculus of variations. Secondly, an idealized minimization problem is proposed to remove the above linearity assumption. Such a problem is inspired by a kinetic regularization associated with the Benamou-Brenier formula and universal approximation theorems for neural networks.
Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jml.231210
Journal of Machine Learning, Vol. 3 (2024), Iss. 4 : pp. 413–444
Published online: 2024-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 32
Keywords: Deep learning ResNet ODE-Net Benamou-Brenier formula Mean-field game.