Momentum Stochastic Gradient Descent (MSGD) algorithm has been widely applied
to many nonconvex optimization problems in machine learning, e.g., training
deep neural networks, variational Bayesian inference, and etc. Despite its
empirical success, there is still a lack of theoretical understanding of
convergence properties of MSGD. To fill this gap, we propose to analyze the
algorithmic behavior of MSGD by diffusion approximations for nonconvex
optimization problems with strict saddle points and isolated local optima. Our
study shows that the momentum helps escape from saddle points, but hurts the
convergence within the neighborhood of optima (if without the step size
annealing or momentum annealing). Our theoretical discovery partially
corroborates the empirical success of MSGD in training deep neural networks.

本文通过对非凸优化问题的扩散逼近，分析了 Momentum 随机梯度下降算法的算法行为，发现该算法对于强鞍点的逃逸具有帮助，但在优化器的周围区域内妨碍了收敛（未进行步长退火或动量退火），本文的理论发现部分验证了 MSGD 在训练深度神经网络中的实证成功。