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We investigate the stochastic gradient descent (SGD) method where the stepsize lies within a banded region instead of being given by a fixed formula. Theoptimal convergence rate under mild conditions and large initial step size isproved. Our analysis provides comparable theoretical error bounds for SGDassociated with a variety of step sizes. In addition, the convergence rates forsome existing step size strategies, e.g., triangular policy and cosine-wave,can be revealed by our analytical framework under the boundary constraints. Thebandwidth-based step size provides efficient and flexible step size selectionin optimization. We also propose a $1/t$ up-down policy and give severalnon-monotonic step sizes. Numerical experiments demonstrate the efficiency andsignificant potential of the bandwidth-based step-size in many applications.

On the Convergence of Stochastic Gradient Descent with Bandwidth-based Step Size