Understanding FISTA's weak convergence: A step-by-step introduction to the 2025 milestone

TL;DR

This paper provides a clear, step-by-step introduction to the recent proof of weak convergence of FISTA iterates, a milestone achieved in 2025 that enhances understanding of this influential optimization algorithm.

Contribution

It offers an accessible, self-contained explanation of the 2025 proof of FISTA's weak convergence, clarifying a significant theoretical advancement.

Findings

Proof of weak convergence of FISTA iterates in 2025

Enhanced understanding of FISTA's theoretical properties

Clarification of the convergence behavior of FISTA

Abstract

Beck and Teboulle's FISTA for finding the minimizer of the sum of two convex functions is one of the most important algorithms of the past decades. While function value convergence of the iterates was known, the actual convergence of the iterates remained elusive until October 2025 when Jang and Ryu, as well as Bo\c{t}, Fadili, and Nguyen proved weak convergence. In this paper, we provide a gentle self-contained introduction to the proof of their remarkable result.