On the Convergence of Weighted AdaGrad with Momentum for Training Deep Neural Networks
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Adaptive stochastic gradient descent methods, such as AdaGrad, RMSProp, Adam, AMSGrad, etc., have been demonstrated efficacious in solving non-convex stochastic optimization, such as training deep neural networks. However, their convergence rates have not been touched under the non-convex stochastic circumstance except recent breakthrough results on AdaGrad, perturbed AdaGrad and AMSGrad. In this paper, we propose two new adaptive stochastic gradient methods called AdaHB and AdaNAG which integrate a novel weighted coordinate-wise AdaGrad with heavy ball momentum and Nesterov accelerated gradient momentum, respectively. The O(T/√(T)) non-asymptotic convergence rates of AdaHB and AdaNAG in non-convex stochastic setting are also jointly established by leveraging a newly developed unified formulation of these two momentum mechanisms. Moreover, comparisons have been made between AdaHB, AdaNAG, Adam and RMSProp, which, to a certain extent, explains the reasons why Adam and RMSProp are divergent. In particular, when momentum term vanishes we obtain convergence rate of coordinate-wise AdaGrad in non-convex stochastic setting as a byproduct.
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