  The weight initialization and the activation function of deep neural networks
have a crucial impact on the performance of the training procedure. An
inappropriate selection can lead to the loss of information of the input during
forward propagation and the exponential vanishing/exploding of gradients during
back-propagation. Understanding the theoretical properties of untrained random
networks is key to identifying which deep networks may be trained successfully
as recently demonstrated by Schoenholz et al. (2017) who showed that for deep
feedforward neural networks only a specific choice of hyperparameters known as
the `edge of chaos' can lead to good performance. We complete this analysis by
providing quantitative results showing that, for a class of ReLU-like
activation functions, the information propagates indeed deeper for an
initialization at the edge of chaos. By further extending this analysis, we
identify a class of activation functions that improve the information
propagation over ReLU-like functions. This class includes the Swish activation,
$\phi_{swish}(x) = x \cdot \text{sigmoid}(x)$, used in Hendrycks & Gimpel
(2016), Elfwing et al. (2017) and Ramachandran et al. (2017). This provides a
theoretical grounding for the excellent empirical performance of $\phi_{swish}$
observed in these contributions. We complement those previous results by
illustrating the benefit of using a random initialization on the edge of chaos
in this context.
