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

• 成果类型: Article
• 署名作者: Garrigos, Guillaume; Rosasco, Lorenzo; Villa, Silvia
• 署名单位: Centre National de la Recherche Scientifique (CNRS); Universite Paris Cite; University of Genoa; Istituto Italiano di Tecnologia - IIT; University of Genoa
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-022-01809-4
• 发表日期: 2023
• 页码: 937-996
• 关键词: error-bounds; descent methods; linear convergence; thresholding algorithm; metric regularity; convex; systems; optimization; STABILITY; nonconvex

摘要

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.