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One of the most popular training algorithms for deep neural networks is theAdaptive Moment Estimation (Adam) introduced by Kingma and Ba. Despite itssuccess in many applications there is no satisfactory convergence analysis:only local convergence can be shown for batch mode under some restrictions onthe hyperparameters, counterexamples exist for incremental mode. Recent resultsshow that for simple quadratic objective functions limit cycles of period 2exist in batch mode, but only for atypical hyperparameters, and only for thealgorithm without bias correction. %More general there are several moreadaptive gradient methods which try to estimate a fitting learning rate and /or search direction from the training data to improve the learning processcompared to pure gradient descent with fixed learningrate. We extend theconvergence analysis for Adam in the batch mode with bias correction and showthat even for quadratic objective functions as the simplest case of convexfunctions 2-limit-cycles exist, for all choices of the hyperparameters. Weanalyze the stability of these limit cycles and relate our analysis to otherresults where approximate convergence was shown, but under the additionalassumption of bounded gradients which does not apply to quadratic functions.The investigation heavily relies on the use of computer algebra due to thecomplexity of the equations.

Non-Convergence and Limit Cycles in the Adam optimizer