  We study the limiting dynamics of a large class of noisy gradient descent
systems in the overparameterized regime. In this regime the set of global
minimizers of the loss is large, and when initialized in a neighbourhood of
this zero-loss set a noisy gradient descent algorithm slowly evolves along this
set. In some cases this slow evolution has been related to better
generalisation properties. We characterize this evolution for the broad class
of noisy gradient descent systems in the limit of small step size. Our results
show that the structure of the noise affects not just the form of the limiting
process, but also the time scale at which the evolution takes place. We apply
the theory to Dropout, label noise and classical SGD (minibatching) noise, and
show that these evolve on different two time scales. Classical SGD even yields
a trivial evolution on both time scales, implying that additional noise is
required for regularization. The results are inspired by the training of neural
networks, but the theorems apply to noisy gradient descent of any loss that has
a non-trivial zero-loss set.
