TY - JOUR T1 - Variational Formulations of ODE-Net as a Mean-Field Optimal Control Problem and Existence Results AU - Isobe , Noboru AU - Okumura , Mizuho JO - Journal of Machine Learning VL - 4 SP - 413 EP - 444 PY - 2024 DA - 2024/11 SN - 3 DO - http://doi.org/10.4208/jml.231210 UR - https://global-sci.org/intro/article_detail/jml/23501.html KW - Deep learning, ResNet, ODE-Net, Benamou-Brenier formula, Mean-field game. AB -

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.