Minimum Width of Leaky-ReLU Neural Networks for Uniform Universal Approximation

• URL: http://arxiv.org/abs/2305.18460v3
• Date: Thu, 1 Feb 2024 03:36:36 GMT
• Title: Minimum Width of Leaky-ReLU Neural Networks for Uniform Universal Approximation
• Authors: Li'ang Li, Yifei Duan, Guanghua Ji, Yongqiang Cai
• Abstract summary: This paper examines a uniform UAP for the function class $C(K,mathbbRd_y)$. It gives the exact minimum width of the leaky-ReLU NN as $w_min=max(d_x,d_y)+Delta (d_x, d_y)$.
• Score: 10.249623880822055
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The study of universal approximation properties (UAP) for neural networks (NN) has a long history. When the network width is unlimited, only a single hidden layer is sufficient for UAP. In contrast, when the depth is unlimited, the width for UAP needs to be not less than the critical width $w^*_{\min}=\max(d_x,d_y)$, where $d_x$ and $d_y$ are the dimensions of the input and output, respectively. Recently, \cite{cai2022achieve} shows that a leaky-ReLU NN with this critical width can achieve UAP for $L^p$ functions on a compact domain ${K}$, \emph{i.e.,} the UAP for $L^p({K},\mathbb{R}^{d_y})$. This paper examines a uniform UAP for the function class $C({K},\mathbb{R}^{d_y})$ and gives the exact minimum width of the leaky-ReLU NN as $w_{\min}=\max(d_x,d_y)+\Delta (d_x, d_y)$, where $\Delta (d_x, d_y)$ is the additional dimensions for approximating continuous functions with diffeomorphisms via embedding. To obtain this result, we propose a novel lift-flow-discretization approach that shows that the uniform UAP has a deep connection with topological theory.

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