  Adam and RMSProp are two of the most influential adaptive stochastic
algorithms for training deep neural networks, which have been pointed out to be
divergent even in the 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, etc., have been tried to promote Adam/RMSProp-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/RMSProp for solving
large-scale non-convex stochastic optimization. Moreover, we show that the
convergences of several variants of Adam, such as AdamNC, AdaEMA, etc., can be
directly implied via the proposed sufficient condition in the non-convex
setting. In addition, we illustrate that Adam is essentially a specifically
weighted AdaGrad with exponential moving average momentum, which provides a
novel perspective for understanding Adam and RMSProp. This observation coupled
with this sufficient condition gives much deeper interpretations on their
divergences. At last, we validate the sufficient condition by applying Adam and
RMSProp to tackle a certain counterexample and train deep neural networks.
Numerical results are exactly in accord with our theoretical analysis.
