We consider the geometric ergodicity of the Stochastic Gradient Langevin Dynamics (SGLD) algorithm under nonconvexity settings. Via the technique of reflection coupling, we prove the Wasserstein contraction of SGLD when the target distribution is log-concave only outside some compact set. The time discretization and the minibatch in SGLD introduce several difficulties when applying the reflection coupling, which are addressed by a series of careful estimates of conditional expectations. As a direct corollary, the SGLD with constant step size has an invariant distribution and we are able to obtain its geometric ergodicity in terms of $W_1$ distance. The generalization to non-gradient drifts is also included.

翻译：本文研究了随机梯度朗之万动力学（SGLD）算法在非凸设置下的几何遍历性。通过反射耦合技术，我们证明了当目标分布仅在某个紧集外满足对数凹性时，SGLD具有Wasserstein收缩性。SGLD中的时间离散化与小批量采样机制为反射耦合的应用带来了若干困难，我们通过一系列条件期望的精细估计解决了这些问题。作为直接推论，恒定步长的SGLD存在不变分布，并且我们能够以$W_1$距离度量其几何遍历性。本文结果也推广至非梯度漂移项的情形。