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Article Dans Une Revue ESAIM: Control, Optimisation and Calculus of Variations Année : 2019

Thresholding gradient methods in Hilbert spaces: support identification and linear convergence

Lorenzo Rosasco
  • Fonction : Auteur
Silvia Villa
  • Fonction : Auteur

Résumé

We study the ℓ 1 regularized least squares optimization problem in a separable Hilbert space. We show that the iterative soft-thresholding algorithm (ISTA) converges linearly, without making any assumption on the linear operator into play or on the problem. The result is obtained combining two key concepts: the notion of extended support , a finite set containing the support, and the notion of conditioning over finite-dimensional sets . We prove that ISTA identifies the solution extended support after a finite number of iterations, and we derive linear convergence from the conditioning property, which is always satisfied for ℓ 1 regularized least squares problems. Our analysis extends to the entire class of thresholding gradient algorithms, for which we provide a conceptually new proof of strong convergence, as well as convergence rates.
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Dates et versions

hal-03886153 , version 1 (06-12-2022)

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Guillaume Garrigos, Lorenzo Rosasco, Silvia Villa. Thresholding gradient methods in Hilbert spaces: support identification and linear convergence. ESAIM: Control, Optimisation and Calculus of Variations, 2019, 26, ⟨10.1051/cocv/2019011⟩. ⟨hal-03886153⟩
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