Calculus of the Exponent of Kurdyka–Łojasiewicz Inequality and Its Applications to Linear Convergence of First-Order Methods

In this paper, we study the Kurdyka–Łojasiewicz (KL) exponent, an important quantity for analyzing the convergence rate of first-order methods. Specifically, we develop various calculus rules to deduce the KL exponent of new (possibly nonconvex and nonsmooth) functions formed from functions with kno...

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Vydáno v:	Foundations of computational mathematics Ročník 18; číslo 5; s. 1199 - 1232
Hlavní autoři:	Li, Guoyin, Pong, Ting Kei
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	New York Springer US 01.10.2018 Springer Nature B.V
Témata:	Algorithms Applications of Mathematics Calculus Computer Science Convergence Economics Error analysis Exponents Least squares method Linear and Multilinear Algebras Math Applications in Computer Science Mathematical models Mathematics Mathematics and Statistics Matrix Theory Minimax technique Nonlinear programming Numerical Analysis Optimization Recovery Regression analysis Regularization Sparse optimization Linear convergence Luo–Tseng error bound 90C25 Convergence rate 90C26 90C05 First-order methods Kurdyka–Łojasiewicz inequality
ISSN:	1615-3375, 1615-3383
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Abstract	In this paper, we study the Kurdyka–Łojasiewicz (KL) exponent, an important quantity for analyzing the convergence rate of first-order methods. Specifically, we develop various calculus rules to deduce the KL exponent of new (possibly nonconvex and nonsmooth) functions formed from functions with known KL exponents. In addition, we show that the well-studied Luo–Tseng error bound together with a mild assumption on the separation of stationary values implies that the KL exponent is 1 2 . The Luo–Tseng error bound is known to hold for a large class of concrete structured optimization problems, and thus we deduce the KL exponent of a large class of functions whose exponents were previously unknown. Building upon this and the calculus rules, we are then able to show that for many convex or nonconvex optimization models for applications such as sparse recovery, their objective function’s KL exponent is 1 2 . This includes the least squares problem with smoothly clipped absolute deviation regularization or minimax concave penalty regularization and the logistic regression problem with ℓ 1 regularization. Since many existing local convergence rate analysis for first-order methods in the nonconvex scenario relies on the KL exponent, our results enable us to obtain explicit convergence rate for various first-order methods when they are applied to a large variety of practical optimization models. Finally, we further illustrate how our results can be applied to establishing local linear convergence of the proximal gradient algorithm and the inertial proximal algorithm with constant step sizes for some specific models that arise in sparse recovery.
AbstractList	In this paper, we study the Kurdyka–Łojasiewicz (KL) exponent, an important quantity for analyzing the convergence rate of first-order methods. Specifically, we develop various calculus rules to deduce the KL exponent of new (possibly nonconvex and nonsmooth) functions formed from functions with known KL exponents. In addition, we show that the well-studied Luo–Tseng error bound together with a mild assumption on the separation of stationary values implies that the KL exponent is 1 2 . The Luo–Tseng error bound is known to hold for a large class of concrete structured optimization problems, and thus we deduce the KL exponent of a large class of functions whose exponents were previously unknown. Building upon this and the calculus rules, we are then able to show that for many convex or nonconvex optimization models for applications such as sparse recovery, their objective function’s KL exponent is 1 2 . This includes the least squares problem with smoothly clipped absolute deviation regularization or minimax concave penalty regularization and the logistic regression problem with ℓ 1 regularization. Since many existing local convergence rate analysis for first-order methods in the nonconvex scenario relies on the KL exponent, our results enable us to obtain explicit convergence rate for various first-order methods when they are applied to a large variety of practical optimization models. Finally, we further illustrate how our results can be applied to establishing local linear convergence of the proximal gradient algorithm and the inertial proximal algorithm with constant step sizes for some specific models that arise in sparse recovery. In this paper, we study the Kurdyka–Łojasiewicz (KL) exponent, an important quantity for analyzing the convergence rate of first-order methods. Specifically, we develop various calculus rules to deduce the KL exponent of new (possibly nonconvex and nonsmooth) functions formed from functions with known KL exponents. In addition, we show that the well-studied Luo–Tseng error bound together with a mild assumption on the separation of stationary values implies that the KL exponent is 12. The Luo–Tseng error bound is known to hold for a large class of concrete structured optimization problems, and thus we deduce the KL exponent of a large class of functions whose exponents were previously unknown. Building upon this and the calculus rules, we are then able to show that for many convex or nonconvex optimization models for applications such as sparse recovery, their objective function’s KL exponent is 12. This includes the least squares problem with smoothly clipped absolute deviation regularization or minimax concave penalty regularization and the logistic regression problem with ℓ1 regularization. Since many existing local convergence rate analysis for first-order methods in the nonconvex scenario relies on the KL exponent, our results enable us to obtain explicit convergence rate for various first-order methods when they are applied to a large variety of practical optimization models. Finally, we further illustrate how our results can be applied to establishing local linear convergence of the proximal gradient algorithm and the inertial proximal algorithm with constant step sizes for some specific models that arise in sparse recovery.
Author	Li, Guoyin Pong, Ting Kei
Author_xml	– sequence: 1 givenname: Guoyin surname: Li fullname: Li, Guoyin organization: Department of Applied Mathematics, University of New South Wales – sequence: 2 givenname: Ting Kei surname: Pong fullname: Pong, Ting Kei email: tk.pong@polyu.edu.hk organization: Department of Applied Mathematics, The Hong Kong Polytechnic University
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Snippet	In this paper, we study the Kurdyka–Łojasiewicz (KL) exponent, an important quantity for analyzing the convergence rate of first-order methods. Specifically,...
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SubjectTerms	Algorithms Applications of Mathematics Calculus Computer Science Convergence Economics Error analysis Exponents Least squares method Linear and Multilinear Algebras Math Applications in Computer Science Mathematical models Mathematics Mathematics and Statistics Matrix Theory Minimax technique Nonlinear programming Numerical Analysis Optimization Recovery Regression analysis Regularization
Title	Calculus of the Exponent of Kurdyka–Łojasiewicz Inequality and Its Applications to Linear Convergence of First-Order Methods
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