  Asynchronous stochastic gradient descent (ASGD) is a popular parallel
optimization algorithm in machine learning. Most theoretical analysis on ASGD
take a discrete view and prove upper bounds for their convergence rates.
However, the discrete view has its intrinsic limitations: there is no
characterization of the optimization path and the proof techniques are
induction-based and thus usually complicated. Inspired by the recent successful
adoptions of stochastic differential equations (SDE) to the theoretical
analysis of SGD, in this paper, we study the continuous approximation of ASGD
by using stochastic differential delay equations (SDDE). We introduce the
approximation method and study the approximation error. Then we conduct
theoretical analysis on the convergence rates of ASGD algorithm based on the
continuous approximation. There are two methods: moment estimation and energy
function minimization can be used to analyze the convergence rates. Moment
estimation depends on the specific form of the loss function, while energy
function minimization only leverages the convex property of the loss function,
and does not depend on its specific form. In addition to the convergence
analysis, the continuous view also helps us derive better convergence rates.
All of this clearly shows the advantage of taking the continuous view in
gradient descent algorithms.
