@Article{JML-3-413,
author = {Isobe , Noboru and Okumura , Mizuho},
title = {Variational Formulations of ODE-Net as a Mean-Field Optimal Control Problem and Existence Results},
journal = {Journal of Machine Learning},
year = {2024},
volume = {3},
number = {4},
pages = {413--444},
abstract = {<p style="text-align: justify;">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.</p>},
issn = {2790-2048},
doi = {https://doi.org/10.4208/jml.231210},
url = {http://global-sci.org/intro/article_detail/jml/23501.html}
}