Improving "Fast Iterative Shrinkage-Thresholding Algorithm": Faster, Smarter and Greedier

TL;DR

This paper introduces modifications to FISTA that ensure convergence, enable faster 'lazy-start' strategies, and explore adaptive approaches, significantly improving practical performance in inverse problems, machine learning, and signal processing.

Contribution

It proposes a simple modification to FISTA that guarantees convergence and introduces adaptive greedy strategies for enhanced efficiency.

Findings

The 'lazy-start' strategy can be up to an order faster than original FISTA.

Proposed methods demonstrate improved performance in inverse problems, machine learning, and signal/image processing.

The modifications ensure convergence of the FISTA sequence.

Abstract

The "fast iterative shrinkage-thresholding algorithm", a.k.a. FISTA, is one of the most well-known first-order optimisation scheme in the literature, as it achieves the worst-case $O(1/k^2)$ optimal convergence rate in terms of objective function value. However, despite such an optimal theoretical convergence rate, in practice the (local) oscillatory behaviour of FISTA often damps its efficiency. Over the past years, various efforts are made in the literature to improve the practical performance of FISTA, such as monotone FISTA, restarting FISTA and backtracking strategies. In this paper, we propose a simple yet effective modification to the original FISTA scheme which has two advantages: it allows us to 1) prove the convergence of generated sequence; 2) design a so-called "lazy-start" strategy which can up to an order faster than the original scheme. Moreover, by exploring the properties of FISTA scheme, we propose novel adaptive and greedy strategies which probes the limit of the algorithm. The advantages of the proposed schemes are tested through problems arising from inverse problem, machine learning and signal/image processing.