FISTA is an automatic geometrically optimized algorithm for strongly convex functions

In this work, we are interested in the famous FISTA algorithm. We show that FISTA is an automatic geometrically optimized algorithm for functions satisfying a quadratic growth assumption. This explains why FISTA works better than the standard Forward-Backward algorithm (FB) in such a case, although...

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Bibliographic Details
Published in:Mathematical programming 2024-03, Vol.204 (1-2), p.449-491
Main Authors: Aujol, J.-F., Dossal, Ch, Rondepierre, A.
Format: Article
Language:English
Subjects:
Algorithms
Asymptotic properties
Calculus of Variations and Optimal Control
Optimization
Combinatorics
Full Length Paper
Mathematical analysis
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Polynomials
Theoretical
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Summary:In this work, we are interested in the famous FISTA algorithm. We show that FISTA is an automatic geometrically optimized algorithm for functions satisfying a quadratic growth assumption. This explains why FISTA works better than the standard Forward-Backward algorithm (FB) in such a case, although FISTA is known to have a polynomial asymptotic convergence rate while FB is exponential. We provide a simple rule to tune the α parameter within the FISTA algorithm to reach an ε -solution with an optimal number of iterations. These new results highlight the efficiency of FISTA algorithms, and they rely on new non asymptotic bounds for FISTA.
ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-023-01960-6