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

Heinz H. Bauschke, Walaa M. Moursi

TL;DR

The paper proves the weak convergence of FISTA iterates for minimizing $F(x)=f(x)+g(x)$ in a real Hilbert space, a milestone achieved in 2025 by independent teams. It weaves an accessible, self-contained exposition around boundedness, cluster-point analysis, and auxiliary sequences, culminating in a structured proof that leverages the BCCH Lemma and Salzo sequences. The work connects finite-sample rate results, continuous-time perspectives, and recent developments to deliver a teaching-friendly account, including optional bonus material and a convex feasibility example with a non-unique minimizer. The findings solidify the theoretical foundations of FISTA in infinite-dimensional settings and enhance its reliability for practical, accelerated proximal-gradient optimization.

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 Boţ, Fadili, and Nguyen proved weak convergence. In this paper, we provide a gentle self-contained introduction to the proof of their remarkable result.

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

TL;DR

The paper proves the weak convergence of FISTA iterates for minimizing in a real Hilbert space, a milestone achieved in 2025 by independent teams. It weaves an accessible, self-contained exposition around boundedness, cluster-point analysis, and auxiliary sequences, culminating in a structured proof that leverages the BCCH Lemma and Salzo sequences. The work connects finite-sample rate results, continuous-time perspectives, and recent developments to deliver a teaching-friendly account, including optional bonus material and a convex feasibility example with a non-unique minimizer. The findings solidify the theoretical foundations of FISTA in infinite-dimensional settings and enhance its reliability for practical, accelerated proximal-gradient optimization.

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 Boţ, Fadili, and Nguyen proved weak convergence. In this paper, we provide a gentle self-contained introduction to the proof of their remarkable result.
Paper Structure (16 sections, 7 theorems, 57 equations, 1 figure)

This paper contains 16 sections, 7 theorems, 57 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The state of the art on FISTA in September 2025
  3. Approaching the proof of weak convergence of FISTA
  4. The FISTA sequence $(x_k)_{k\in{\mathbb N}}$ is bounded, and its weak cluster points lie in $S$
  5. A note on the parameter sequence $(t_k)_{k\in{\mathbb N}}$
  6. The Boţ-Chenchene-Csetnek-Hulett (BCCH) Lemma
  7. The BCCH Lemma: statement and proof
  8. Optional Bonus material on the BCCH Lemma
  9. Beyond Fejér and Opial: Salzo sequences
  10. Bounded Salzo sequences are weakly convergent
  11. Optional Bonus material on Salzo sequences
  12. The grand finale: FISTA converges weakly!
  13. Proof of \ref{['lem:key']}
  14. Proof of \ref{['l:Radu']} when $\ell=\pm\infty$
  15. Proof of \ref{['ex:sinh']}
  16. ...and 1 more sections

Key Result

Theorem 3.1

The FISTA sequence $(x_k)_{k\in\mathbb{N}}$ (see 30.3) satisfies the following:

Figures (1)

  • Figure 1: The first few iterates of FISTA with parameter sequence \ref{['BTpar']} and starting point $x_0=(5,0)$. Note that $\lim_{k\to\infty} x_k \approx (0.4829,0.5171)$; in contrast, PGM converges to $(1,0)$.

Theorems & Definitions (27)

  • proof
  • proof
  • proof
  • Theorem 3.1
  • proof
  • Lemma 1
  • proof
  • Lemma 2: the BCCH Lemma
  • proof
  • Example 1: converse of the BCCH Lemma fails with $(g_k)_{k\in{\mathbb N}}$ bounded
  • ...and 17 more