Linear Convergence of the Proximal Gradient Method for Composite Optimization Under the Polyak-{\L}ojasiewicz Inequality and Its Variant

TL;DR

This paper establishes and improves linear convergence rates for the proximal gradient method applied to composite functions satisfying Polyak-Łojasiewicz inequalities, supported by theoretical analysis and numerical experiments.

Contribution

It provides new explicit convergence rates and enhances existing complexity bounds for the proximal gradient method under PL inequalities.

Findings

Derived explicit linear convergence rates for the proximal gradient method.

Improved complexity bounds for composite function minimization.

Numerical illustrations confirm theoretical results.

Abstract

We study the linear convergence rates of the proximal gradient method for composite functions satisfying two classes of Polyak-{\L}ojasiewicz (PL) inequality: the PL inequality, the variant of PL inequality defined by the proximal map-based residual. Using the performance estimation problem, we either provide new explicit linear convergence rates or improve existing complexity bounds for minimizing composite functions under the two classes of PL inequality. Finally, we illustrate numerically the effects of our theoretical results.