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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Bibliographic Details
Published in:Mathematical programming Vol. 198; no. 1; pp. 937 - 996
Main Authors: Garrigos, Guillaume, Rosasco, Lorenzo, Villa, Silvia
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2023
Springer
Springer Verlag
Subjects:
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
Online Access:Get full text
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Summary: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.
ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-022-01809-4