Convergence of the forward-backward algorithm: beyond the worst-case with the help of geometry
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10107_2022_1809_ReferencePDF.pdf
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Author(s)
Garrigos, Guillaume
Rosasco, Lorenzo
Villa, Silvia
Date Issued
June 21, 2022
Publisher
Springer Berlin Heidelberg
Citation
Garrigos, Guillaume, Rosasco, Lorenzo and Villa, Silvia. 2022. "Convergence of the forward-backward algorithm: beyond the worst-case with the help of geometry."
Version
Author's final manuscript
Abstract
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.
MIT Department
Center for Brains, Minds, and Machines
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1007/s10107-022-01809-4
