  We develop the mathematical foundations of the stochastic modified equations
(SME) framework for analyzing the dynamics of stochastic gradient algorithms,
where the latter is approximated by a class of stochastic differential
equations with small noise parameters. We prove that this approximation can be
understood mathematically as an weak approximation, which leads to a number of
precise and useful results on the approximations of stochastic gradient descent
(SGD), momentum SGD and stochastic Nesterov's accelerated gradient method in
the general setting of stochastic objectives. We also demonstrate through
explicit calculations that this continuous-time approach can uncover important
analytical insights into the stochastic gradient algorithms under consideration
that may not be easy to obtain in a purely discrete-time setting.
