Namjun Kim

Namjun Kim, Chanho Min, Sejun Park

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 $\smash{\mathbb R^{d_x}}$, our result first shows that approximation on a compact domain requires smaller width than on $\smash{\mathbb R^{d_x}}$. 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.

Jonghyun Shin, Namjun Kim, Geonho Hwang et al.

The exact minimum width that allows for universal approximation of unbounded-depth networks is known only for ReLU and its variants. In this work, we study the minimum width of networks using general activation functions. Specifically, we focus on squashable functions that can approximate the identity function and binary step function by alternatively composing with affine transformations. We show that for networks using a squashable activation function to universally approximate $L^p$ functions from $[0,1]^{d_x}$ to $\mathbb R^{d_y}$, the minimum width is $\max\{d_x,d_y,2\}$ unless $d_x=d_y=1$; the same bound holds for $d_x=d_y=1$ if the activation function is monotone. We then provide sufficient conditions for squashability and show that all non-affine analytic functions and a class of piecewise functions are squashable, i.e., our minimum width result holds for those general classes of activation functions.