  Stochastic gradient descent (SGD) has been widely used in machine learning
due to its computational efficiency and favorable generalization properties.
Recently, it has been empirically demonstrated that the gradient noise in
several deep learning settings admits a non-Gaussian, heavy-tailed behavior.
This suggests that the gradient noise can be modeled by using $\alpha$-stable
distributions, a family of heavy-tailed distributions that appear in the
generalized central limit theorem. In this context, SGD can be viewed as a
discretization of a stochastic differential equation (SDE) driven by a L\'{e}vy
motion, and the metastability results for this SDE can then be used for
illuminating the behavior of SGD, especially in terms of `preferring wide
minima'. While this approach brings a new perspective for analyzing SGD, it is
limited in the sense that, due to the time discretization, SGD might admit a
significantly different behavior than its continuous-time limit. Intuitively,
the behaviors of these two systems are expected to be similar to each other
only when the discretization step is sufficiently small; however, to the best
of our knowledge, there is no theoretical understanding on how small the
step-size should be chosen in order to guarantee that the discretized system
inherits the properties of the continuous-time system. In this study, we
provide formal theoretical analysis where we derive explicit conditions for the
step-size such that the metastability behavior of the discrete-time system is
similar to its continuous-time limit. We show that the behaviors of the two
systems are indeed similar for small step-sizes and we identify how the error
depends on the algorithm and problem parameters. We illustrate our results with
simulations on a synthetic model and neural networks.
