A Globally Convergent Algorithm for Nonconvex Optimization Based on Block Coordinate Update
Nonconvex optimization arises in many areas of computational science and engineering. However, most nonconvex optimization algorithms are only known to have local convergence or subsequence convergence properties. In this paper, we propose an algorithm for nonconvex optimization and establish its gl...
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| Veröffentlicht in: | Journal of scientific computing Jg. 72; H. 2; S. 700 - 734 |
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| Format: | Journal Article |
| Sprache: | Englisch |
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Springer US
01.08.2017
Springer Nature B.V |
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| ISSN: | 0885-7474, 1573-7691 |
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| Abstract | Nonconvex optimization arises in many areas of computational science and engineering. However, most nonconvex optimization algorithms are only known to have local convergence or subsequence convergence properties. In this paper, we propose an algorithm for nonconvex optimization and establish its global convergence (of the whole sequence) to a critical point. In addition, we give its asymptotic convergence rate and numerically demonstrate its efficiency. In our algorithm, the variables of the underlying problem are either treated as one block or multiple disjoint blocks. It is assumed that each non-differentiable component of the objective function, or each constraint, applies only to one block of variables. The differentiable components of the objective function, however, can involve multiple blocks of variables together. Our algorithm updates one block of variables at a time by minimizing a certain prox-linear surrogate, along with an extrapolation to accelerate its convergence. The order of update can be either deterministically cyclic or randomly shuffled for each cycle. In fact, our convergence analysis only needs that each block be updated at least once in every fixed number of iterations. We show its global convergence (of the whole sequence) to a critical point under fairly loose conditions including, in particular, the Kurdyka–Łojasiewicz condition, which is satisfied by a broad class of nonconvex/nonsmooth applications. These results, of course, remain valid when the underlying problem is convex. We apply our convergence results to the coordinate descent iteration for non-convex regularized linear regression, as well as a modified rank-one residue iteration for nonnegative matrix factorization. We show that both applications have global convergence. Numerically, we tested our algorithm on nonnegative matrix and tensor factorization problems, where random shuffling clearly improves the chance to avoid low-quality local solutions. |
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| AbstractList | Nonconvex optimization arises in many areas of computational science and engineering. However, most nonconvex optimization algorithms are only known to have local convergence or subsequence convergence properties. In this paper, we propose an algorithm for nonconvex optimization and establish its global convergence (of the whole sequence) to a critical point. In addition, we give its asymptotic convergence rate and numerically demonstrate its efficiency. In our algorithm, the variables of the underlying problem are either treated as one block or multiple disjoint blocks. It is assumed that each non-differentiable component of the objective function, or each constraint, applies only to one block of variables. The differentiable components of the objective function, however, can involve multiple blocks of variables together. Our algorithm updates one block of variables at a time by minimizing a certain prox-linear surrogate, along with an extrapolation to accelerate its convergence. The order of update can be either deterministically cyclic or randomly shuffled for each cycle. In fact, our convergence analysis only needs that each block be updated at least once in every fixed number of iterations. We show its global convergence (of the whole sequence) to a critical point under fairly loose conditions including, in particular, the Kurdyka–Łojasiewicz condition, which is satisfied by a broad class of nonconvex/nonsmooth applications. These results, of course, remain valid when the underlying problem is convex. We apply our convergence results to the coordinate descent iteration for non-convex regularized linear regression, as well as a modified rank-one residue iteration for nonnegative matrix factorization. We show that both applications have global convergence. Numerically, we tested our algorithm on nonnegative matrix and tensor factorization problems, where random shuffling clearly improves the chance to avoid low-quality local solutions. |
| Author | Xu, Yangyang Yin, Wotao |
| Author_xml | – sequence: 1 givenname: Yangyang orcidid: 0000-0002-4163-3723 surname: Xu fullname: Xu, Yangyang email: yangyang.xu@ua.edu organization: Department of Mathematics, University of Alabama – sequence: 2 givenname: Wotao orcidid: 0000-0001-6697-9731 surname: Yin fullname: Yin, Wotao organization: Department of Mathematics, UCLA |
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| Keywords | Nonsmooth optimization Nonconvex optimization Prox-linear Block coordinate descent Whole sequence convergence Kurdyka–Łojasiewicz inequality |
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| SubjectTerms | Algorithms Computational Mathematics and Numerical Analysis Convergence Critical point Factorization Iterative methods Mathematical and Computational Engineering Mathematical and Computational Physics Mathematics Mathematics and Statistics Methods Optimization Sparsity Tensors Theoretical Variables |
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| Title | A Globally Convergent Algorithm for Nonconvex Optimization Based on Block Coordinate Update |
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