One of the key issues in the analysis of machine learning models is to
identify the appropriate function space and norm for the model. This is the set
of functions endowed with a quantity which can control the approximation and
estimation errors by a particular machine learning model. In this paper, we
address this issue for two representative neural network models: the two-layer
networks and the residual neural networks. We define the Barron space and show
that it is the right space for two-layer neural network models in the sense
that optimal direct and inverse approximation theorems hold for functions in
the Barron space. For residual neural network models, we construct the
so-called flow-induced function space, and prove direct and inverse
approximation theorems for this space. In addition, we show that the Rademacher
complexity for bounded sets under these norms has the optimal upper bounds.

本文研究了机器学习模型分析中的关键问题，即如何确定适当的函数空间和模型的范数，针对两种典型的神经网络模型（即双层网络和残差神经网络），提出了 Barron 空间和流引导函数空间，并证明了这些空间下的函数具有最优的直接和反向逼近定理，同时表明这些范数下的有界集的 Rademacher 复杂度具有最优的上界。