Minimum Width for Universal Approximation Using Squashable Activation Functions

Abstract

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.