  The activation function deployed in a deep neural network has great influence
on the performance of the network at initialisation, which in turn has
implications for training. In this paper we study how to avoid two problems at
initialisation identified in prior works: rapid convergence of pairwise input
correlations, and vanishing and exploding gradients. We prove that both these
problems can be avoided by choosing an activation function possessing a
sufficiently large linear region around the origin, relative to the bias
variance $\sigma_b^2$ of the network's random initialisation. We demonstrate
empirically that using such activation functions leads to tangible benefits in
practice, both in terms test and training accuracy as well as training time.
Furthermore, we observe that the shape of the nonlinear activation outside the
linear region appears to have a relatively limited impact on training. Finally,
our results also allow us to train networks in a new hyperparameter regime,
with a much larger bias variance than has previously been possible.
