The expressive power of neural networks is important for understanding deep learning. Most existing works consider this problem from the view of the depth of a network. In this paper, we study how width affects the expressiveness of neural networks. Classical results state that \emph{depth-bounded} (e.g. depth-$2$) networks with suitable activation functions are universal approximators. We show a universal approximation theorem for \emph{width-bounded} ReLU networks: width-$(n+4)$ ReLU networks, where $n$ is the input dimension, are universal approximators. Moreover, except for a measure zero set, all functions cannot be approximated by width-$n$ ReLU networks, which exhibits a phase transition. Several recent works demonstrate the benefits of depth by proving the depth-efficiency of neural networks. That is, there are classes of deep networks which cannot be realized by any shallow network whose size is no more than an \emph{exponential} bound. Here we pose the dual question on the width-efficiency of ReLU networks: Are there wide networks that cannot be realized by narrow networks whose size is not substantially larger? We show that there exist classes of wide networks which cannot be realized by any narrow network whose depth is no more than a \emph{polynomial} bound. On the other hand, we demonstrate by extensive experiments that narrow networks whose size exceed the polynomial bound by a constant factor can approximate wide and shallow network with high accuracy. Our results provide more comprehensive evidence that depth is more effective than width for the expressiveness of ReLU networks.

本文研究神经网络的宽度对其表达能力的影响，证明了width-$(n+4)$ ReLU神经网络是一种通用逼近器，同时存在一些无法用宽度为$n$的神经网络进行逼近的函数，表现出相变现象，结果展示了深度对ReLU网络的表达能力比宽度更为有效。

神经网络的表达能力：基于宽度的视角