Convergence of the forward-backward algorithm: beyond the worst-case with the help of geometry

We provide a comprehensive study of the convergence of the forward-backward algorithm under suitable geometric conditions, such as conditioning or Łojasiewicz properties. These geometrical notions are usually local by nature, and may fail to describe the fine geometry of objective functions relevant...

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Veröffentlicht in:	Mathematical programming Jg. 198; H. 1; S. 937 - 996
Hauptverfasser:	Garrigos, Guillaume, Rosasco, Lorenzo, Villa, Silvia
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2023 Springer Springer Verlag
Schlagworte:	Algorithms Analysis Calculus of Variations and Optimal Control; Optimization Combinatorics Full Length Paper Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Mathematics of Computing Numerical Analysis Optimization and Control Signal processing Theoretical Inverse problems 90C25 Convergence rates Łojasiewicz property 65K10 49M27 Forward Backward algorithm 47J26 Conditioned functions Source condition
ISSN:	0025-5610, 1436-4646
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Abstract	We provide a comprehensive study of the convergence of the forward-backward algorithm under suitable geometric conditions, such as conditioning or Łojasiewicz properties. These geometrical notions are usually local by nature, and may fail to describe the fine geometry of objective functions relevant in inverse problems and signal processing, that have a nice behaviour on manifolds, or sets open with respect to a weak topology. Motivated by this observation, we revisit those geometric notions over arbitrary sets. In turn, this allows us to present several new results as well as collect in a unified view a variety of results scattered in the literature. Our contributions include the analysis of infinite dimensional convex minimization problems, showing the first Łojasiewicz inequality for a quadratic function associated to a compact operator, and the derivation of new linear rates for problems arising from inverse problems with low-complexity priors. Our approach allows to establish unexpected connections between geometry and a priori conditions in inverse problems, such as source conditions, or restricted isometry properties.
AbstractList	We provide a comprehensive study of the convergence of the forward-backward algorithm under suitable geometric conditions, such as conditioning or Łojasiewicz properties. These geometrical notions are usually local by nature, and may fail to describe the fine geometry of objective functions relevant in inverse problems and signal processing, that have a nice behaviour on manifolds, or sets open with respect to a weak topology. Motivated by this observation, we revisit those geometric notions over arbitrary sets. In turn, this allows us to present several new results as well as collect in a unified view a variety of results scattered in the literature. Our contributions include the analysis of infinite dimensional convex minimization problems, showing the first Łojasiewicz inequality for a quadratic function associated to a compact operator, and the derivation of new linear rates for problems arising from inverse problems with low-complexity priors. Our approach allows to establish unexpected connections between geometry and a priori conditions in inverse problems, such as source conditions, or restricted isometry properties. We provide a comprehensive study of the convergence of the forward-backward algorithm under suitable geometric conditions, such as conditioning or Lojasiewicz properties. These geometrical notions are usually local by nature, and may fail to describe the fine geometry of objective functions relevant in inverse problems and signal processing, that have a nice behaviour on manifolds, or sets open with respect to a weak topology. Motivated by this observation, we revisit those geometric notions over arbitrary sets. In turn, this allows us to present several new results as well as collect in a unified view a variety of results scattered in the literature. Our contributions include the analysis of infinite dimensional convex minimization problems, showing the first Lojasiewicz inequality for a quadratic function associated to a compact operator, and the derivation of new linear rates for problems arising from inverse problems with low-complexity priors. Our approach allows to establish unexpected connections between geometry and a priori conditions in inverse problems, such as source conditions, or restricted isometry properties.
Audience	Academic
Author	Garrigos, Guillaume Villa, Silvia Rosasco, Lorenzo
Author_xml	– sequence: 1 givenname: Guillaume orcidid: 0000-0002-8613-5664 surname: Garrigos fullname: Garrigos, Guillaume email: guillaume.garrigos@lpsm.paris organization: Université de Paris and Sorbonne Université, CNRS, Laboratoire de Probabilités – sequence: 2 givenname: Lorenzo surname: Rosasco fullname: Rosasco, Lorenzo organization: MaLGa Center, DIBRIS, Università degli Studi di Genova, Istituto Italiano di Tecnologia, CBMM, Massachusetts Institute of Technology – sequence: 3 givenname: Silvia surname: Villa fullname: Villa, Silvia organization: MaLGa Center, Dipartimento di Matematica, Università degli Studi di Genova
BackLink	https://hal.science/hal-03886199$$DView record in HAL
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Copyright	Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2022 COPYRIGHT 2023 Springer Distributed under a Creative Commons Attribution 4.0 International License
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Keywords	Inverse problems 90C25 Convergence rates Łojasiewicz property 65K10 49M27 Forward Backward algorithm 47J26 Conditioned functions Source condition
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Snippet	We provide a comprehensive study of the convergence of the forward-backward algorithm under suitable geometric conditions, such as conditioning or Łojasiewicz... We provide a comprehensive study of the convergence of the forward-backward algorithm under suitable geometric conditions, such as conditioning or Lojasiewicz...
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SubjectTerms	Algorithms Analysis Calculus of Variations and Optimal Control; Optimization Combinatorics Full Length Paper Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Mathematics of Computing Numerical Analysis Optimization and Control Signal processing Theoretical
Title	Convergence of the forward-backward algorithm: beyond the worst-case with the help of geometry
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