  In this paper, we propose a weak approximation of the reflection coupling
(RC) for stochastic differential equations (SDEs), and prove it converges
weakly to the desired coupling. In contrast to the RC, the proposed approximate
reflection coupling (ARC) need not take the hitting time of processes to the
diagonal set into consideration and can be defined as the solution of some SDEs
on the whole time interval. Therefore, ARC can work effectively against SDEs
with different drift terms. As an application of ARC, an evaluation on the
effectiveness of the stochastic gradient descent in a non-convex setting is
also described. For the sample size $n$, the step size $\eta$, and the batch
size $B$, we derive uniform evaluations on the time with orders $n^{-1}$,
$\eta^{1/2}$, and $\sqrt{(n - B) / B (n - 1)}$, respectively.
