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

Weak Convergence of Approximate reflection coupling and its Application to Non-convex Optimization