This paper studies a class of adaptive gradient based momentum algorithms that update the  search directions and learning rates simultaneously using past gradients. This class, which we refer to as the ''``Adam-type'', includes the popular algorithms such as Adam, AMSGrad, AdaGrad. Despite their popularity in training deep neural networks (DNNs), the convergence of these algorithms for solving  non-convex problems remains an open question. In this paper, we develop an analysis framework and a set of mild sufficient conditions that guarantee the convergence of the Adam-type methods, with a convergence rate of order   $O(\log{T}/\sqrt{T})$ for non-convex stochastic optimization. Our convergence analysis applies to a new algorithm called AdaFom (AdaGrad with First Order Momentum). We show that the conditions are essential, by identifying concrete examples in which violating the conditions makes an algorithm diverge. Besides providing one of the first comprehensive analysis for Adam-type methods in the non-convex setting, our results can also help the practitioners to easily  monitor the progress of algorithms and determine their convergence behavior.
We analyze convergence of Adam-type algorithms and provide mild sufficient conditions to guarantee their convergence, we also show  violating the conditions can makes an algorithm diverge. Presents a convergence analysis in the non-convex setting for a family of optimization algorithms. This paper investigates the convergence condition of Adam-type optimizers in the unconstrained non-convex optimization problems.