  Adam is one of the most influential adaptive stochastic algorithms for
training deep neural networks, which has been pointed out to be divergent even
in the simple convex setting via a few simple counterexamples. Many attempts,
such as decreasing an adaptive learning rate, adopting a big batch size,
incorporating a temporal decorrelation technique, seeking an analogous
surrogate, \textit{etc.}, have been tried to promote Adam-type algorithms to
converge. In contrast with existing approaches, we introduce an alternative
easy-to-check sufficient condition, which merely depends on the parameters of
the base learning rate and combinations of historical second-order moments, to
guarantee the global convergence of generic Adam for solving large-scale
non-convex stochastic optimization. This observation coupled with this
sufficient condition gives much deeper interpretations on the divergence of
Adam. On the other hand, in practice, mini-Adam and distributed-Adam are widely
used without theoretical guarantee, we further give an analysis on how will the
batch size or the number of nodes in the distributed system will affect the
convergence of Adam, which theoretically shows that mini-batch and distributed
Adam can be linearly accelerated by using a larger mini-batch size or more
number of nodes. At last, we apply the generic Adam and mini-batch Adam with a
sufficient condition for solving the counterexample and training several
different neural networks on various real-world datasets. Experimental results
are exactly in accord with our theoretical analysis.
