  Both accelerated and adaptive gradient methods are among state of the art
algorithms to train neural networks. The tuning of hyperparameters is needed to
make them work efficiently. For classical gradient descent, a general and
efficient way to adapt hyperparameters is the Armijo backtracking. The goal of
this work is to generalize the Armijo linesearch to Momentum and RMSProp, two
popular optimizers of this family, by means of stability theory of dynamical
systems. We establish convergence results, under the Lojasiewicz assumption,
for these strategies. As a direct result, we obtain the first guarantee on the
convergence of the iterates for RMSProp, in the non-convex setting without the
classical bounded assumptions.
