  Stochastic Gradient Descent (SGD) and its variants are the most used
algorithms in machine learning applications. In particular, SGD with adaptive
learning rates and momentum is the industry standard to train deep networks.
Despite the enormous success of these methods, our theoretical understanding of
these variants in the nonconvex setting is not complete, with most of the
results only proving convergence in expectation and with strong assumptions on
the stochastic gradients. In this paper, we present a high probability analysis
for adaptive and momentum algorithms, under weak assumptions on the function,
stochastic gradients, and learning rates. We use it to prove for the first time
the convergence of the gradients to zero in high probability in the smooth
nonconvex setting for Delayed AdaGrad with momentum.
