In this work we consider a model problem of deep neural learning, namely the learning of a given function when it is assumed that we have access to its point values on a finite set of points. The deep neural network interpolant is the the resulting approximation of f, which is obtained by a typical machine learning algorithm involving a given DNN architecture and an optimisation step, which is assumed to be solved exactly. These are among the simplest regression algorithms based on neural networks. In this work we introduce a new approach to the estimation of the (generalisation) error and to convergence. Our results include (i) estimates of the error without any structural assumption on the neural networks and under mild regularity assumptions on the learning function f (ii) convergence of the approximations to the target function f by only requiring that the neural network spaces have appropriate approximation capability.

翻译：本文研究深度神经学习的一个模型问题，即假设能够获取给定函数在有限点集上的点值时，对该函数的学习过程。深度神经网络插值是通过典型机器学习算法（包含特定深度神经网络架构和假设精确求解的优化步骤）获得的函数f的近似结果，这类算法属于基于神经网络的最简回归方法。本文提出一种评估（泛化）误差与收敛性的新方法，主要成果包括：(i) 无需对神经网络做任何结构假设，仅要求学习函数f满足温和正则性条件即可获得误差估计；(ii) 仅需神经网络空间具备适当的逼近能力，即可保证逼近结果收敛于目标函数f。