Biorthogonal Greedy Algorithms in convex optimization

The study of greedy approximation in the context of convex optimization is becoming a promising research direction as greedy algorithms are actively being employed to construct sparse minimizers for convex functions with respect to given sets of elements. In this paper we propose a unified way of an...

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Bibliographic Details
Published in:Applied and computational harmonic analysis Vol. 60; pp. 489 - 511
Main Authors: Dereventsov, A.V., Temlyakov, V.N.
Format: Journal Article
Language:English
Published: Elsevier Inc 01.09.2022
Subjects:
Banach space
Biorthogonal Greedy Algorithm
Convex optimization
Greedy algorithm
Sparsity
Greedy algorithm
Sparsity
Biorthogonal Greedy Algorithm
Banach space
Convex optimization
ISSN:1063-5203, 1096-603X
Online Access:Get full text
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Summary:The study of greedy approximation in the context of convex optimization is becoming a promising research direction as greedy algorithms are actively being employed to construct sparse minimizers for convex functions with respect to given sets of elements. In this paper we propose a unified way of analyzing a certain kind of greedy-type algorithms for the minimization of convex functions on Banach spaces. Specifically, we define the class of Weak Biorthogonal Greedy Algorithms for convex optimization that contains a wide range of greedy algorithms. We analyze the introduced class of algorithms and establish the properties of convergence, rate of convergence, and numerical stability, which is understood in the sense that the steps of the algorithm are allowed to be performed not precisely but with controlled computational inaccuracies. We show that the following well-known algorithms for convex optimization — the Weak Chebyshev Greedy Algorithm (co) and the Weak Greedy Algorithm with Free Relaxation (co) — belong to this class, and introduce a new algorithm — the Rescaled Weak Relaxed Greedy Algorithm (co). Presented numerical experiments demonstrate the practical performance of the aforementioned greedy algorithms in the setting of convex minimization as compared to optimization with regularization, which is the conventional approach of constructing sparse minimizers.
ISSN:1063-5203
1096-603X
DOI:10.1016/j.acha.2022.05.001