  Stochastic regularisation is an important weapon in the arsenal of a deep
learning practitioner. However, despite recent theoretical advances, our
understanding of how noise influences signal propagation in deep neural
networks remains limited. By extending recent work based on mean field theory,
we develop a new framework for signal propagation in stochastic regularised
neural networks. Our noisy signal propagation theory can incorporate several
common noise distributions, including additive and multiplicative Gaussian
noise as well as dropout. We use this framework to investigate initialisation
strategies for noisy ReLU networks. We show that no critical initialisation
strategy exists using additive noise, with signal propagation exploding
regardless of the selected noise distribution. For multiplicative noise (e.g.
dropout), we identify alternative critical initialisation strategies that
depend on the second moment of the noise distribution. Simulations and
experiments on real-world data confirm that our proposed initialisation is able
to stably propagate signals in deep networks, while using an initialisation
disregarding noise fails to do so. Furthermore, we analyse correlation dynamics
between inputs. Stronger noise regularisation is shown to reduce the depth to
which discriminatory information about the inputs to a noisy ReLU network is
able to propagate, even when initialised at criticality. We support our
theoretical predictions for these trainable depths with simulations, as well as
with experiments on MNIST and CIFAR-10
