Calibrating the Learning Rate for Adaptive Gradient Methods to Improve Generalization Performance
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Although adaptive gradient methods (AGMs) have fast speed in training deep neural networks, it is known to generalize worse than the stochastic gradient descent (SGD) or SGD with momentum (S-Momentum). Many works have attempted to modify AGMs so to close the gap in generalization performance between AGMs and S-Momentum, but they do not answer why there is such a gap. We identify that the anisotropic scale of the adaptive learning rate (A-LR) used by AGMs contributes to the generalization performance gap, and all existing modified AGMs actually represent efforts in revising the A-LR. Because the A-LR varies significantly across the dimensions of the problem over the optimization epochs (i.e., anisotropic scale), we propose a new AGM by calibrating the A-LR with a softplus function, resulting in the Sadam and SAMSGrad methods[Code is available at https://github.com/neilliang90/Sadam.git.]. These methods have better chance to not trap at sharp local minimizers, which helps them resume the dips in the generalization error curve observed with SGD and S-Momentum. We further provide a new way to analyze the convergence of AGMs (e.g., Adam, Sadam, and SAMSGrad) under the nonconvex, non-strongly convex, and Polyak-Łojasiewicz conditions. We prove that the convergence rate of ADAM also depends on its hyper-parameter epsilon, which has been overlooked in prior convergence analysis. Empirical studies support our observation of the anisotropic A-LR and show that the proposed methods outperform existing AGMs and generalize even better than S-Momentum in multiple deep learning tasks.
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