Stochastic gradient descent (SGD) optimization schemes are the methods of choice for the optimization of deep neural networks (DNNs) in artificial intelligence (AI) systems. Often not the standard SGD method is used but instead suitable accelerated, adaptive, and/or normalized variants of standard SGD such as Adam, AdamW, and MUON are employed to train large scale AI systems in practically relevant settings. The acceleration (higher order convergence speed) in all these popular optimizers relies on the momentum SGD optimizer. In this work we provide a rigorous error analysis for the momentum SGD optimizer. In particular, we establish convergence rates for the momentum optimizer in terms of the size of the learning rate (step size), the size of the mini-batch, and the size of the one-point convexity constant.
Gradient based optimization methods are nowadays the methods of choice for training deep neural networks (DNNs) in artificial intelligence (AI) systems. In practically relevant DNN training problems, one does usually not apply the standard gradient descent (GD) optimization method but instead one employs suitable sophisticated GD optimization methods, which incorporate adaptivity and/or acceleration techniques, such as the famous Adam optimizer. It is a key contribution of this work to provide a general unified convergence analysis for GD optimization methods in the training of DNNs with analytic activations such as the softplus and the popular Gaussian error linear unit (GeLU) activation. Our general unified convergence result applies to a large class of gradient based optimization methods such as the standard GD, the momentum, the Nesterov accelerated gradient (NAG), the RMSprop, the Adam, the Adamax, the Nadam, the Nadamax, the Adan, the AdaBelief, the AMSGrad, and the Yogi optimizers. Our analysis employs the theory of Kurdyka-{\L}ojasiewicz (KL) inequalities to establish convergence to critical points in the training of DNNs. To the best of our knowledge, the generality of our convergence analysis is also just in the special situation of the Adam optimizer a new contribution to the literature on the analysis of AI optimization algorithms.