Convergence Rate Analysis for Monotone Accelerated Proximal Gradient Method

TL;DR

This paper establishes the fastest known linear convergence rate for the monotone accelerated proximal gradient method in strongly convex settings, and proves boundedness and optimality of its iterates.

Contribution

It provides a new convergence rate analysis for the method without requiring knowledge of the strong convexity parameter.

Findings

Linear convergence rate established for strongly convex cases

Boundedness of iterates proven in convex setting

Limit points are all minimizers of the objective

Abstract

We analyze the convergence rate of the monotone accelerated proximal gradient method, which can be used to solve structured convex composite optimization problems. A linear convergence rate is established when the smooth part of the objective function is strongly convex, without knowledge of the strong convexity parameter. This is the fastest convergence rate known for this algorithm. As a byproduct, we also establish the boundedness of the iterates in the convex setting, and prove that the limit points of the iterates are all minimizers of the objective function.