  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.
