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

• Published in: arXiv.org 2024-09
• Main Authors: Li, Chris Junchi, Mou, Wenlong, Wainwright, Martin J, Jordan, Michael I
• Format: Article
• Language: English
• Subjects: Algorithms; Approximation; Asymptotic methods; Asymptotic properties; Convergence; Covariance; Normal distribution; Optimization; Statistical analysis; Tightness; Variance
• ISSN: 2331-8422

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 Cramér-Rao optimal asymptotic covariance, for a broad range of step-size choices.