Accelerated Backward Forward Method for Convex Optimization

TL;DR

This paper analyzes an accelerated backward forward method for convex optimization, establishing fast convergence rates and weak convergence of iterates, with variants for strongly convex cases.

Contribution

It provides convergence rate analysis for a novel accelerated method, differing from FISTA, including variants with linear convergence for strongly convex functions.

Findings

Achieves $ ext{O}(1/k^2)$ convergence rate for convex functions.

Proves weak convergence of the iterates.

Establishes linear convergence for strongly convex functions.

Abstract

We analyze the convergence rate of an accelerated backward forward method for solving convex composite optimization problems. The method was developed by Taylor, Hendrickx and Glineur, and is different from the FISTA algorithm in its placement of the proximal operator. When the smooth part of the objective function is convex, we establish a fast convergence rate of $\mathcal{O}\left( \frac{1}{k^2} \right)$ for the function values, and prove the weak convergence of the iterates. When the smooth part is strongly convex, we propose a variant of the method, and establish an accelerated linear convergence rate for the function values.