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

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