Many popular learning-rate schedules for deep neural networks combine a decaying trend with local perturbations that attempt to escape saddle points and bad local minima. We derive convergence guarantees for bandwidth-based step-sizes, a general class of learning-rates that are allowed to vary in a banded region. This framework includes cyclic and non-monotonic step-sizes for which no theoretical guarantees were previously known. We provide worst-case guarantees for SGD on smooth non-convex problems under several bandwidth-based step sizes, including stagewise $1/sqrt{t}$ and the popular step-decay (constant and then drop by a constant), which is also shown to be optimal. Moreover, we show that its momentum variant (SGDM) converges as fast as SGD with the bandwidth-based step-decay step-size. Finally, we propose some novel step-size schemes in the bandwidth-based family and verify their efficiency on several deep neural network training tasks.
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