Local and global convergence of a general inertial proximal splitting scheme for minimizing composite functions

This paper is concerned with convex composite minimization problems in a Hilbert space. In these problems, the objective is the sum of two closed, proper, and convex functions where one is smooth and the other admits a computationally inexpensive proximal operator. We analyze a family of generalized...

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Vydané v:	Computational optimization and applications Ročník 67; číslo 2; s. 259 - 292
Hlavní autori:	Johnstone, Patrick R., Moulin, Pierre
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	New York Springer US 01.06.2017 Springer Nature B.V
Predmet:	Algorithms Convergence Convex analysis Convex and Discrete Geometry Hilbert space Iterative methods Management Science Manifolds Mathematical analysis Mathematical models Mathematics Mathematics and Statistics Methods Operations Research Operations Research/Decision Theory Optimization Problem solving Splitting Statistics Studies Theorems FISTA Inertial forward-backward splitting Inertial proximal gradient Local linear convergence Momentum methods Lasso
ISSN:	0926-6003, 1573-2894
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Abstract	This paper is concerned with convex composite minimization problems in a Hilbert space. In these problems, the objective is the sum of two closed, proper, and convex functions where one is smooth and the other admits a computationally inexpensive proximal operator. We analyze a family of generalized inertial proximal splitting algorithms (GIPSA) for solving such problems. We establish weak convergence of the generated sequence when the minimum is attained. Our analysis unifies and extends several previous results. We then focus on ℓ 1 -regularized optimization, which is the ubiquitous special case where the nonsmooth term is the ℓ 1 -norm. For certain parameter choices, GIPSA is amenable to a local analysis for this problem. For these choices we show that GIPSA achieves finite “active manifold identification”, i.e. convergence in a finite number of iterations to the optimal support and sign, after which GIPSA reduces to minimizing a local smooth function. We prove local linear convergence under either restricted strong convexity or a strict complementarity condition. We determine the rate in terms of the inertia, stepsize, and local curvature. Our local analysis is applicable to certain recent variants of the Fast Iterative Shrinkage–Thresholding Algorithm (FISTA), for which we establish active manifold identification and local linear convergence. Based on our analysis we propose a momentum restart scheme in these FISTA variants to obtain the optimal local linear convergence rate while maintaining desirable global properties.
AbstractList	This paper is concerned with convex composite minimization problems in a Hilbert space. In these problems, the objective is the sum of two closed, proper, and convex functions where one is smooth and the other admits a computationally inexpensive proximal operator. We analyze a family of generalized inertial proximal splitting algorithms (GIPSA) for solving such problems. We establish weak convergence of the generated sequence when the minimum is attained. Our analysis unifies and extends several previous results. We then focus on ℓ 1 -regularized optimization, which is the ubiquitous special case where the nonsmooth term is the ℓ 1 -norm. For certain parameter choices, GIPSA is amenable to a local analysis for this problem. For these choices we show that GIPSA achieves finite “active manifold identification”, i.e. convergence in a finite number of iterations to the optimal support and sign, after which GIPSA reduces to minimizing a local smooth function. We prove local linear convergence under either restricted strong convexity or a strict complementarity condition. We determine the rate in terms of the inertia, stepsize, and local curvature. Our local analysis is applicable to certain recent variants of the Fast Iterative Shrinkage–Thresholding Algorithm (FISTA), for which we establish active manifold identification and local linear convergence. Based on our analysis we propose a momentum restart scheme in these FISTA variants to obtain the optimal local linear convergence rate while maintaining desirable global properties. (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image).This paper is concerned with convex composite minimization problems in a Hilbert space. In these problems, the objective is the sum of two closed, proper, and convex functions where one is smooth and the other admits a computationally inexpensive proximal operator. We analyze a family of generalized inertial proximal splitting algorithms (GIPSA) for solving such problems. We establish weak convergence of the generated sequence when the minimum is attained. Our analysis unifies and extends several previous results. We then focus on ...-regularized optimization, which is the ubiquitous special case where the nonsmooth term is the ...-norm. For certain parameter choices, GIPSA is amenable to a local analysis for this problem. For these choices we show that GIPSA achieves finite "active manifold identification", i.e. convergence in a finite number of iterations to the optimal support and sign, after which GIPSA reduces to minimizing a local smooth function. We prove local linear convergence under either restricted strong convexity or a strict complementarity condition. We determine the rate in terms of the inertia, stepsize, and local curvature. Our local analysis is applicable to certain recent variants of the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA), for which we establish active manifold identification and local linear convergence. Based on our analysis we propose a momentum restart scheme in these FISTA variants to obtain the optimal local linear convergence rate while maintaining desirable global properties. This paper is concerned with convex composite minimization problems in a Hilbert space. In these problems, the objective is the sum of two closed, proper, and convex functions where one is smooth and the other admits a computationally inexpensive proximal operator. We analyze a family of generalized inertial proximal splitting algorithms (GIPSA) for solving such problems. We establish weak convergence of the generated sequence when the minimum is attained. Our analysis unies and extends several previous results. We then focus on 1-regularized optimization, which is the ubiquitous special case where the nonsmooth term is the 1-norm. For certain parameter choices, GIPSA is amenable to a local analysis for this problem. For these choices we show that GIPSA achieves nite active manifold identication,i.e. convergence in a nite number of iterations to the optimal support and sign, after which GIPSA reduces to minimizing a local smooth function. We prove local linear convergence under either restricted strong convexity or a strict complementarity condition. We determine the rate in terms of the inertia, stepsize, and local curvature. Our local analysis is applicable to certain recent variants of the Fast Iterative ShrinkageThresholding Algorithm (FISTA), for which we establish active manifold identication and local linear convergence. Based on our analysis we propose a momentum restart scheme in these FISTA variants to obtain the optimal local linear convergence rate while maintaining desirable global properties.
Author	Moulin, Pierre Johnstone, Patrick R.
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Keywords	FISTA Inertial forward-backward splitting Inertial proximal gradient Local linear convergence Momentum methods Lasso
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Snippet	This paper is concerned with convex composite minimization problems in a Hilbert space. In these problems, the objective is the sum of two closed, proper, and... (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image).This paper is concerned with convex composite minimization problems in a Hilbert...
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SubjectTerms	Algorithms Convergence Convex analysis Convex and Discrete Geometry Hilbert space Iterative methods Management Science Manifolds Mathematical analysis Mathematical models Mathematics Mathematics and Statistics Methods Operations Research Operations Research/Decision Theory Optimization Problem solving Splitting Statistics Studies Theorems
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Title	Local and global convergence of a general inertial proximal splitting scheme for minimizing composite functions
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