Minimum width for universal approximation using ReLU networks on compact domain

摘要

It has been shown that deep neural networks of a large enough width are universal approximators but they are not if the width is too small.There were several attempts to characterize the minimum width $w_{\min}$ enabling the universal approximation property; however, only a few of them found the exact values.In this work, we show that the minimum width for $L^p$ approximation of $L^p$ functions from $[0,1]^{d_x}$ to $\mathbb R^{d_y}$ is exactly $\max\\{d_x,d_y,2\\}$ if an activation function is ReLU-Like (e.g., ReLU, GELU, Softplus).Compared to the known result for ReLU networks, $w_{\min}=\max\\{d_x+1,d_y\\}$ when the domain is ${\mathbb R^{d_x}}$, our result first shows that approximation on a compact domain requires smaller width than on ${\mathbb R^{d_x}}$.We next prove a lower bound on $w_{\min}$ for uniform approximation using general activation functions including ReLU: $w_{\min}\ge d_y+1$ if $d_x<d_y\le2d_x$. Together with our first result, this shows a dichotomy between $L^p$ and uniform approximations for general activation functions and input/output dimensions.