  Stochastic optimization methods have been hugely successful in making
large-scale optimization problems feasible when computing the full gradient is
computationally prohibitive. Using the theory of modified equations for
numerical integrators, we propose a class of stochastic differential equations
that approximate the dynamics of general stochastic optimization methods more
closely than the original gradient flow. Analyzing a modified stochastic
differential equation can reveal qualitative insights about the associated
optimization method. Here, we study mean-square stability of the modified
equation in the case of stochastic coordinate descent.
