Scalable subspace methods for derivative-free nonlinear least-squares optimization
We introduce a general framework for large-scale model-based derivative-free optimization based on iterative minimization within random subspaces. We present a probabilistic worst-case complexity analysis for our method, where in particular we prove high-probability bounds on the number of iteration...
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| Vydané v: | Mathematical programming Ročník 199; číslo 1-2; s. 461 - 524 |
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| Hlavní autori: | , |
| Médium: | Journal Article |
| Jazyk: | English |
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Berlin/Heidelberg
Springer Berlin Heidelberg
01.05.2023
Springer |
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| ISSN: | 0025-5610, 1436-4646 |
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| Abstract | We introduce a general framework for large-scale model-based derivative-free optimization based on iterative minimization within random subspaces. We present a probabilistic worst-case complexity analysis for our method, where in particular we prove high-probability bounds on the number of iterations before a given optimality is achieved. This framework is specialized to nonlinear least-squares problems, with a model-based framework based on the Gauss–Newton method. This method achieves scalability by constructing local linear interpolation models to approximate the Jacobian, and computes new steps at each iteration in a subspace with user-determined dimension. We then describe a practical implementation of this framework, which we call DFBGN. We outline efficient techniques for selecting the interpolation points and search subspace, yielding an implementation that has a low per-iteration linear algebra cost (linear in the problem dimension) while also achieving fast objective decrease as measured by evaluations. Extensive numerical results demonstrate that DFBGN has improved scalability, yielding strong performance on large-scale nonlinear least-squares problems. |
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| AbstractList | We introduce a general framework for large-scale model-based derivative-free optimization based on iterative minimization within random subspaces. We present a probabilistic worst-case complexity analysis for our method, where in particular we prove high-probability bounds on the number of iterations before a given optimality is achieved. This framework is specialized to nonlinear least-squares problems, with a model-based framework based on the Gauss–Newton method. This method achieves scalability by constructing local linear interpolation models to approximate the Jacobian, and computes new steps at each iteration in a subspace with user-determined dimension. We then describe a practical implementation of this framework, which we call DFBGN. We outline efficient techniques for selecting the interpolation points and search subspace, yielding an implementation that has a low per-iteration linear algebra cost (linear in the problem dimension) while also achieving fast objective decrease as measured by evaluations. Extensive numerical results demonstrate that DFBGN has improved scalability, yielding strong performance on large-scale nonlinear least-squares problems. |
| Audience | Academic |
| Author | Cartis, Coralia Roberts, Lindon |
| Author_xml | – sequence: 1 givenname: Coralia surname: Cartis fullname: Cartis, Coralia organization: Mathematical Institute, University of Oxford – sequence: 2 givenname: Lindon orcidid: 0000-0001-6438-9703 surname: Roberts fullname: Roberts, Lindon email: lindon.roberts@anu.edu.au organization: Mathematical Sciences Institute, Building 145, Science Road, Australian National University |
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| Keywords | 65K05 90C30 Worst case complexity 90C56 Large-scale optimization Nonlinear least-squares Derivative-free optimization |
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| Title | Scalable subspace methods for derivative-free nonlinear least-squares optimization |
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