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We study the problem of solving strongly convex and smooth unconstrainedoptimization problems using stochastic first-order algorithms. We devise anovel algorithm, referred to as \emph{Recursive One-Over-T SGD} (\ROOTSGD),based on an easily implementable, recursive averaging of past stochasticgradients. We prove that it simultaneously achieves state-of-the-artperformance in both a finite-sample, nonasymptotic sense and an asymptoticsense. On the nonasymptotic side, we prove risk bounds on the last iterate of\ROOTSGD with leading-order terms that match the optimal statistical risk witha unity pre-factor, along with a higher-order term that scales at the sharprate of $O(n^{-3/2})$ under the Lipschitz condition on the Hessian matrix. Onthe asymptotic side, we show that when a mild, one-point Hessian continuitycondition is imposed, the rescaled last iterate of (multi-epoch) \ROOTSGDconverges asymptotically to a Gaussian limit with the Cram\'{e}r-Rao optimalasymptotic covariance, for a broad range of step-size choices.

ROOT-SGD: Sharp Nonasymptotics and Asymptotic Efficiency in a Single Algorithm