Unified Optimal Analysis of the (Stochastic) Gradient Method

07/09/2019

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by   Sebastian U. Stich, et al.

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In this note we give a simple proof for the convergence of stochastic gradient (SGD) methods on μ-strongly convex functions under a (milder than standard) L-smoothness assumption. We show that SGD converges after T iterations as O( L x_0-x^^2 [-μ/4LT ] + σ^2/μ T) where σ^2 measures the variance. For deterministic gradient descent (GD) and SGD in the interpolation setting we have σ^2 =0 and we recover the exponential convergence rate. The bound matches with the best known iteration complexity of GD and SGD, up to constants.