Weighted Averaged Stochastic Gradient Descent: Asymptotic Normality and Optimality

TL;DR

This paper studies a general weighted averaging scheme for stochastic gradient descent, establishing its asymptotic normality, proposing an adaptive method with optimal statistical properties, and enabling valid online inference.

Contribution

It introduces a broad class of weighted averaging schemes for SGD, proves their asymptotic normality, and develops an adaptive averaging method with optimal convergence and inference capabilities.

Findings

Asymptotic normality of weighted averaged SGD solutions.

An adaptive averaging scheme with optimal statistical rate.

Insights into optimal weights for linear models in non-asymptotic MSE.

Abstract

Stochastic Gradient Descent (SGD) is one of the most popular algorithms in statistical and machine learning due to its computational and memory efficiency. Various averaging schemes have been proposed to accelerate the convergence of SGD in different settings. In this paper, we explore a general averaging scheme for SGD. Specifically, we establish the asymptotic normality of a broad range of weighted averaged SGD solutions and provide asymptotically valid online inference approaches. Furthermore, we propose an adaptive averaging scheme that exhibits both optimal statistical rate and favorable non-asymptotic convergence, drawing insights from the optimal weight for the linear model in terms of non-asymptotic mean squared error (MSE).