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.

机器学习模型分析中的关键问题之一是为模型确定合适的功能空间和范数。这是一组具有一定数量的功能，可以通过特定的机器学习模型控制近似误差和估计误差。在本文中，我们针对两个代表性的神经网络模型解决了这个问题：两层网络和残差神经网络。我们定义了Barron空间，并表明它是两层神经网络模型的正确空间，这是因为对于Barron空间中的函数而言，最优的直接和逆近似定理成立。对于残差神经网络模型，我们构造了所谓的流致函数空间，并证明了该空间的直接和逆近似定理。此外，