  Adaptive Moment Estimation (ADAM) is a very popular training algorithm for
deep neural networks and belongs to the family of adaptive gradient descent
optimizers. However to the best of the authors knowledge no complete
convergence analysis exists for ADAM. The contribution of this paper is a
method for the local convergence analysis in batch mode for a deterministic
fixed training set, which gives necessary conditions for the hyperparameters of
the ADAM algorithm. Due to the local nature of the arguments the objective
function can be non-convex but must be at least twice continuously
differentiable. Then we apply this procedure to other adaptive gradient descent
algorithms and show for most of them local convergence with hyperparameter
bounds.
