  Gaussian noise injections (GNIs) are a family of simple and widely-used
regularisation methods for training neural networks, where one injects additive
or multiplicative Gaussian noise to the network activations at every iteration
of the optimisation algorithm, which is typically chosen as stochastic gradient
descent (SGD). In this paper we focus on the so-called `implicit effect' of
GNIs, which is the effect of the injected noise on the dynamics of SGD. We show
that this effect induces an asymmetric heavy-tailed noise on SGD gradient
updates. In order to model this modified dynamics, we first develop a
Langevin-like stochastic differential equation that is driven by a general
family of asymmetric heavy-tailed noise. Using this model we then formally
prove that GNIs induce an `implicit bias', which varies depending on the
heaviness of the tails and the level of asymmetry. Our empirical results
confirm that different types of neural networks trained with GNIs are
well-modelled by the proposed dynamics and that the implicit effect of these
injections induces a bias that degrades the performance of networks.
