ROOT-SGD: Sharp Nonasymptotics and Near-Optimal Asymptotics in a Single Algorithm

TL;DR

ROOT-SGD is a new stochastic optimization algorithm that achieves both sharp finite-sample bounds and near-optimal asymptotic convergence, effectively combining nonasymptotic and asymptotic analysis.

Contribution

The paper introduces ROOT-SGD, a recursive averaging method that attains state-of-the-art finite-sample risk bounds and asymptotic normality with optimal covariance in strongly convex optimization.

Findings

Achieves optimal statistical risk bounds with a sharp rate of O(n^{-3/2})

Converges asymptotically to a Gaussian distribution with Cramér-Rao optimal covariance

Performs well both in finite-sample and asymptotic regimes

Abstract

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 Recursive One-Over-T SGD (ROOT-SGD), 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 non-asymptotic side, we prove risk bounds on the last iterate of ROOT-SGD 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) ROOT-SGD converges asymptotically to a Gaussian limit with the Cram\'er-Rao optimal asymptotic covariance, for a broad range of step-size choices.