The restricted strong convexity revisited: Analysis of equivalence to error bound and quadratic growth

TL;DR

This paper revisits the concept of restricted strong convexity, demonstrating its equivalence with error bounds and quadratic growth, and extends these notions to structured convex optimization to establish new convergence results.

Contribution

It proves the equivalence of restricted strong convexity, error bounds, and quadratic growth, and extends these concepts to structured convex problems with convergence guarantees.

Findings

Proves equivalence of three convexity notions using Ekeland's principle.

Establishes asymptotically linear convergence of the proximal gradient method.

Strengthens properties of restricted strong convexity for affine compositions over polyhedral sets.

Abstract

The restricted strong convexity is an effective tool for deriving globally linear convergence rates of descent methods in convex minimization. Recently, the global error bound and quadratic growth properties appeared as new competitors. In this paper, with the help of Ekeland's variational principle, we show the equivalence between these three notions. To deal with convex minimization over a closed convex set and structured convex optimization, we propose a group of modified versions and a group of extended versions of these three notions by using gradient mapping and proximal gradient mapping separately, and prove that the equivalence for the modified and extended versions still holds. Based on these equivalence notions, we establish new asymptotically linear convergence results for the proximal gradient method. Finally, we revisit the problem of minimizing the composition of an affine mapping with a strongly convex differentiable function over a polyhedral set, and obtain a strengthened property of the restricted strong convex type under mild assumptions.