On Asymptotic Linear Convergence of Projected Gradient Descent for Constrained Least Squares

TL;DR

This paper provides a unified local convergence analysis framework for projected gradient descent applied to constrained least squares problems, offering sharper asymptotic convergence bounds and applicability to various signal processing tasks.

Contribution

It introduces a unified approach for analyzing local convergence of projected gradient descent in constrained least squares, applicable to multiple fundamental problems.

Findings

Conditions for linear convergence identified

Region of convergence characterized

Exact asymptotic convergence rate derived

Abstract

Many recent problems in signal processing and machine learning such as compressed sensing, image restoration, matrix/tensor recovery, and non-negative matrix factorization can be cast as constrained optimization. Projected gradient descent is a simple yet efficient method for solving such constrained optimization problems. Local convergence analysis furthers our understanding of its asymptotic behavior near the solution, offering sharper bounds on the convergence rate compared to global convergence analysis. However, local guarantees often appear scattered in problem-specific areas of machine learning and signal processing. This manuscript presents a unified framework for the local convergence analysis of projected gradient descent in the context of constrained least squares. The proposed analysis offers insights into pivotal local convergence properties such as the conditions for linear convergence, the region of convergence, the exact asymptotic rate of convergence, and the bound on the number of iterations needed to reach a certain level of accuracy. To demonstrate the applicability of the proposed approach, we present a recipe for the convergence analysis of projected gradient descent and demonstrate it via a beginning-to-end application of the recipe on four fundamental problems, namely, linear equality-constrained least squares, sparse recovery, least squares with the unit norm constraint, and matrix completion.