MINIMUM WIDTH FOR UNIVERSAL APPROXIMATION USING RELU NETWORKS ON COMPACT DOMAIN - Kim, Namjun; Min, Chanho; Park, Sejun (12th International Conference on Learning Representations, ICLR 2024, 2024)
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 wmin 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 Lp approximation of Lp functions from [0, 1]dx to ℝdy is exactly max{dx, dy, 2} if an activation function is RELU-LIKE (e.g., RELU, GELU, SOFTPLUS). Compared to the known result for RELU networks, wmin = max{dx + 1, dy} when the domain is ℝdx, our result first shows that approximation on a compact domain requires smaller width than on Rdx. We next prove a lower bound on wmin for uniform approximation using general activation functions including RELU: wmin ≥ dy + 1 if dx < dy ≤ 2dx. Together with our first result, this shows a dichotomy between Lp and uniform approximations for general activation functions and input/output dimensions.
Keywords: Activation functions; Input-output; Low bound; Min-max; p-Approximation; P-function; Uniform approximation; Universal approximation; Universal approximation properties; Universal approximators