Biorthogonal Greedy Algorithms in Convex Optimization

TL;DR

This paper introduces a unified analysis framework for greedy algorithms in convex optimization, demonstrating convergence, stability, and practical performance improvements in constructing sparse minimizers.

Contribution

It defines the class of Weak Biorthogonal Greedy Algorithms, analyzes their properties, and introduces a new Rescaled Weak Relaxed Greedy Algorithm for convex minimization.

Findings

Established convergence and stability of the algorithms.

Demonstrated practical efficiency through numerical experiments.

Unified analysis encompasses several known greedy algorithms.

Abstract

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.