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

TL;DR

This paper introduces a weak approximation of reflection coupling for SDEs, proving its convergence and demonstrating its effectiveness in non-convex optimization scenarios like stochastic gradient descent.

Contribution

It proposes a new approximate reflection coupling that avoids hitting time considerations and applies it to analyze stochastic gradient descent in non-convex settings.

Findings

ARC converges weakly to the true coupling.

Uniform evaluation bounds for SGD in non-convex optimization.

Sample size, step size, and batch size impact on convergence rates.

Abstract

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.