  We consider in this work a system of two stochastic differential equations
named the perturbed compositional gradient flow. By introducing a separation of
fast and slow scales of the two equations, we show that the limit of the slow
motion is given by an averaged ordinary differential equation. We then
demonstrate that the deviation of the slow motion from the averaged equation,
after proper rescaling, converges to a stochastic process with Gaussian inputs.
This indicates that the slow motion can be approximated in the weak sense by a
standard perturbed gradient flow or the continuous-time stochastic gradient
descent algorithm that solves the optimization problem for a composition of two
functions. As an application, the perturbed compositional gradient flow
corresponds to the diffusion limit of the Stochastic Composite Gradient Descent
(SCGD) algorithm for minimizing a composition of two expected-value functions
in the optimization literatures. For the strongly convex case, such an analysis
implies that the SCGD algorithm has the same convergence time asymptotic as the
classical stochastic gradient descent algorithm. Thus it validates, at the
level of continuous approximation, the effectiveness of using the SCGD
algorithm in the strongly convex case.
