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

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 condition of 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 PGD and demonstrate it via a beginning-to-end application of the recipe on four fundamental problems, namely, linearly constrained least squares, sparse recovery, least squares with the unit norm constraint, and matrix completion.

On Local Linear Convergence of Projected Gradient Descent for Unit-Modulus Least Squares

“…The proof of this lemma is based on the following result for the projection onto the unit sphere [30]: Lemma 10. (Rephrased from Lemma 5 in [30]) Let x be a point on the unit sphere S n−1 . Then, for any δ ∈ R n , the projection onto S n−1 satisfies…”