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In neural network training, RMSProp and ADAM remain widely favouredoptimization algorithms. One of the keys to their performance lies in selectingthe correct step size, which can significantly influence their effectiveness.It is worth noting that these algorithms performance can vary considerably,depending on the chosen step sizes. Additionally, questions about theirtheoretical convergence properties continue to be a subject of interest. Inthis paper, we theoretically analyze a constant stepsize version of ADAM in thenon-convex setting. We show sufficient conditions for the stepsize to achievealmost sure asymptotic convergence of the gradients to zero with minimalassumptions. We also provide runtime bounds for deterministic ADAM to reachapproximate criticality when working with smooth, non-convex functions.

Convergence of ADAM with Constant Step Size in Non-Convex Settings: A Simple Proof