Stochastic momentum methods have been widely adopted in training deep neural networks. However, their theoretical analysis of convergence of the training objective and the generalization error for prediction is still under-explored. This paper aims to bridge the gap between practice and theory by analyzing the stochastic gradient (SG) method, and the stochastic momentum methods including two famous variants, i.e., the stochastic heavy-ball (SHB) method and the stochastic variant of Nesterov's accelerated gradient (SNAG) method. We propose a framework that unifies the three variants. We then derive the convergence rates of the norm of gradient for the non-convex optimization problem, and analyze the generalization performance through the uniform stability approach. Particularly, the convergence analysis of the training objective exhibits that SHB and SNAG have no advantage over SG. However, the stability analysis shows that the momentum term can improve the stability of the learned model and hence improve the generalization performance. These theoretical insights verify the common wisdom and are also corroborated by our empirical analysis on deep learning.

本文研究随机动量方法，包含随机梯度法（SG），随机重球方法（SHB）和随机Nesterov's加速梯度方法（SNAG）。我们提出了一个框架，统一了这三种方法，并通过一致稳定性方法推导了梯度范数的收敛速率和推导了非凸优化问题。同时，我们也分别分析了这三个方法的收敛率和泛化性能。研究结果表明，动量项可以提高学习模型的稳定性和泛化性能。

深度学习随机动量方法的统一分析