Variational formulations of ODE-Net as a mean-field optimal control problem and existence results

• URL: http://arxiv.org/abs/2303.05924v4
• Date: Sun, 20 Oct 2024 15:13:21 GMT
• Title: Variational formulations of ODE-Net as a mean-field optimal control problem and existence results
• Authors: Noboru Isobe, Mizuho Okumura,
• Abstract summary: 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.
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• 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 its 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, which describes 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. The proofs of these existence results use variational methods, differential equations, and mean-field optimal control theory. They will stand for a new analytic way to investigate the learning process of deep neural networks.

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