  The strong lottery ticket hypothesis has highlighted the potential for
training deep neural networks by pruning, which has inspired interesting
practical and theoretical insights into how neural networks can represent
functions. For networks with ReLU activation functions, it has been proven that
a target network with depth $L$ can be approximated by the subnetwork of a
randomly initialized neural network that has double the target's depth $2L$ and
is wider by a logarithmic factor. We show that a depth $L+1$ network is
sufficient. This result indicates that we can expect to find lottery tickets at
realistic, commonly used depths while only requiring logarithmic
overparametrization. Our novel construction approach applies to a large class
of activation functions and is not limited to ReLUs.
