论文标题
近似反射耦合及其应用于非凸优化的弱收敛性
Weak Convergence of Approximate reflection coupling and its Application to Non-convex Optimization
论文作者
论文摘要
在本文中，我们提出了用于随机微分方程（SDES）的反射耦合（RC）的弱近似值，并证明其弱收敛到所需的耦合。与RC相反，提出的近似反射耦合（ARC）不必将过程的打击时间用于对角线设置，并且可以定义为在整个时间间隔中的某些SDE的解决方案。因此，ARC可以针对具有不同漂移项的SDE有效工作。作为ARC的应用，还描述了对随机梯度下降在非convex设置中的有效性的评估。对于样本量$ n $，步长$η$和批处理大小$ b $，我们在订单上得出统一的评估，分别是$ n^{ - 1} $，$η^{1/2} $和$ \ sqrt {（n -b） / b（n -1）} $。
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 $η$, and the batch size $B$, we derive uniform evaluations on the time with orders $n^{-1}$, $η^{1/2}$, and $\sqrt{(n - B) / B (n - 1)}$, respectively.
