Deep learning has exhibited remarkable results across diverse areas. To
understand its success, substantial research has been directed towards its
theoretical foundations. Nevertheless, the majority of these studies examine
how well deep neural networks can model functions with uniform regularity. In
this paper, we explore a different angle: how deep neural networks can adapt to
different regularity in functions across different locations and scales and
nonuniform data distributions. More precisely, we focus on a broad class of
functions defined by nonlinear tree-based approximation. This class encompasses
a range of function types, such as functions with uniform regularity and
discontinuous functions. We develop nonparametric approximation and estimation
theories for this function class using deep ReLU networks. Our results show
that deep neural networks are adaptive to different regularity of functions and
nonuniform data distributions at different locations and scales. We apply our
results to several function classes, and derive the corresponding approximation
and generalization errors. The validity of our results is demonstrated through
numerical experiments.

深度学习在不同领域展现了显著的成果，但为了理解其成功，我们需要研究其理论基础。本文探讨了一个不同的角度：深度神经网络如何适应不同地点、尺度和非均匀数据分布的函数的不同规则性。我们使用深层 ReLU 网络发展了非参数逼近和估计理论，并在多个函数类上应用了我们的结果，推导出相应的逼近误差和泛化误差。通过数值实验验证了我们结果的有效性。