Strong Convergence of FISTA Iterates under H{\"o}lderian and Quadratic Growth Conditions

TL;DR

This paper proves strong convergence of FISTA iterates under H"olderian and quadratic growth conditions, extending known results without requiring a unique minimizer, and improves convergence rate understanding for these cases.

Contribution

It establishes the strong convergence of FISTA sequences under local growth conditions without the need for minimizer uniqueness, and analyzes related dynamical systems for improved convergence insights.

Findings

Strong convergence of FISTA under H"olderian and quadratic growth conditions

Enhanced convergence rates for function values in these settings

Finite length of trajectories in the associated dynamical system

Abstract

Introduced by Beck and Teboulle, FISTA (for Fast Iterative Shrinkage-Thresholding Algorithm) is a first-order method widely used in convex optimization. Adapted from Nesterov's accelerated gradient method for convex functions, the generated sequence guarantees a decay of the function values of $\mathcal{O}\left(n^{-2}\right)$ in the convex setting. We show that for coercive functions satisfying some local growth condition (namely a H\''olderian or quadratic growth condition), this sequence strongly converges to a minimizer. This property, which has never been proved without assuming the uniqueness of the minimizer, is associated with improved convergence rates for the function values. The proposed analysis is based on a preliminary study of the Asymptotic Vanishing Damping system introduced by Su et al. in to modelNesterov's accelerated gradient method in a continuous setting. Novel improved convergence results are also shown for the solutions of this dynamical system, including the finite length of the trajectory under the aforementioned geometry conditions.