On Projected Stochastic Gradient Descent Algorithm with Weighted Averaging for Least Squares Regression

TL;DR

This paper analyzes a stochastic gradient descent algorithm with weighted averaging for least squares regression, providing convergence rates, bounds on asymptotic ratio to the ERM, and demonstrating strong empirical performance.

Contribution

It introduces a single-pass weighted averaging SGD algorithm for least squares regression with explicit convergence bounds and asymptotic ratio analysis, improving understanding of its efficiency.

Findings

Convergence rate is $O(1/k)$ with variance dominating error.

Asymptotic ratio $ ho$ between proposed method and ERM is at most 4, improved to 4/3 for $d=1$.

Simulation results confirm strong practical performance and theoretical bounds.

Abstract

The problem of least squares regression of a $d$-dimensional unknown parameter is considered. A stochastic gradient descent based algorithm with weighted iterate-averaging that uses a single pass over the data is studied and its convergence rate is analyzed. We first consider a bounded constraint set of the unknown parameter. Under some standard regularity assumptions, we provide an explicit $O(1/k)$ upper bound on the convergence rate, depending on the variance (due to the additive noise in the measurements) and the size of the constraint set. We show that the variance term dominates the error and decreases with rate $1/k$, while the term which is related to the size of the constraint set decreases with rate $\log k/k^2$. We then compare the asymptotic ratio $\rho$ between the convergence rate of the proposed scheme and the empirical risk minimizer (ERM) as the number of iterations approaches infinity. We show that $\rho\leq 4$ under some mild conditions for all $d\geq 1$. We further improve the upper bound by showing that $\rho\leq 4/3$ for the case of $d=1$ and unbounded parameter set. Simulation results demonstrate strong performance of the algorithm as compared to existing methods, and coincide with $\rho\leq 4/3$ even for large $d$ in practice.