Nonclosedness of Sets of Neural Networks in Sobolev Spaces

Abstract: We examine the closedness of sets of realized neural networks of a fixed architecture in Sobolev spaces. For an exactly $m$-times differentiable activation function $\rho$, we construct a sequence of neural networks $(\Phi_n)_{n \in \mathbb{N}}$ whose realizations converge in order-$(m-1)$ Sobolev norm to a function that cannot be realized exactly by a neural network. Thus, sets of realized neural networks are not closed in order-$(m-1)$ Sobolev spaces $W<sup>{m-1,p}$ for $p \in [1,\infty]$. We further show that these sets are not closed in $W<sup>{m,p}$ under slightly stronger conditions on the $m$-th derivative of $\rho$. For a real analytic activation function, we show that sets of realized neural networks are not closed in $W<sup>{k,p}$ for any $k \in \mathbb{N}$. The nonclosedness allows for approximation of non-network target functions with unbounded parameter growth. We partially characterize the rate of parameter growth for most activation functions by showing that a specific sequence of realized neural networks can approximate the activation function's derivative with weights increasing inversely proportional to the $L<sup>p$ approximation error. Finally, we present experimental results showing that networks are capable of closely approximating non-network target functions with increasing parameters via training.