  Stochastic version of alternating direction method of multiplier (ADMM) and
its variants (linearized ADMM, gradient-based ADMM) plays a key role for modern
large scale machine learning problems. One example is the regularized empirical
risk minimization problem. In this work, we put different variants of
stochastic ADMM into a unified form, which includes standard, linearized and
gradient-based ADMM with relaxation, and study their dynamics via a
continuous-time model approach. We adapt the mathematical framework of
stochastic modified equation (SME), and show that the dynamics of stochastic
ADMM is approximated by a class of stochastic differential equations with small
noise parameters in the sense of weak approximation. The continuous-time
analysis would uncover important analytical insights into the behaviors of the
discrete-time algorithm, which are non-trivial to gain otherwise. For example,
we could characterize the fluctuation of the solution paths precisely, and
decide optimal stopping time to minimize the variance of solution paths.
