Fast Distributed Gradient Methods
We study distributed optimization problems when N nodes minimize the sum of their individual costs subject to a common vector variable. The costs are convex, have Lipschitz continuous gradient (with constant L), and bounded gradient. We propose two fast distributed gradient algorithms based on the c...
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| Vydáno v: | IEEE transactions on automatic control Ročník 59; číslo 5; s. 1131 - 1146 |
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| Hlavní autoři: | , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
New York
IEEE
01.05.2014
The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
| Témata: | |
| ISSN: | 0018-9286, 1558-2523 |
| On-line přístup: | Získat plný text |
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| Abstract | We study distributed optimization problems when N nodes minimize the sum of their individual costs subject to a common vector variable. The costs are convex, have Lipschitz continuous gradient (with constant L), and bounded gradient. We propose two fast distributed gradient algorithms based on the centralized Nesterov gradient algorithm and establish their convergence rates in terms of the per-node communications K and the per-node gradient evaluations k. Our first method, Distributed Nesterov Gradient, achieves rates O( logK/K) and O(logk/k). Our second method, Distributed Nesterov gradient with Consensus iterations, assumes at all nodes knowledge of L and μ(W) - the second largest singular value of the N ×N doubly stochastic weight matrix W. It achieves rates O( 1/ K 2-ξ ) and O( 1/k 2 ) ( ξ > 0 arbitrarily small). Further, we give for both methods explicit dependence of the convergence constants on N and W. Simulation examples illustrate our findings. |
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| AbstractList | We study distributed optimization problems when N nodes minimize the sum of their individual costs subject to a common vector variable. The costs are convex, have Lipschitz continuous gradient (with constant L), and bounded gradient. We propose two fast distributed gradient algorithms based on the centralized Nesterov gradient algorithm and establish their convergence rates in terms of the per-node communications K and the per-node gradient evaluations k. Our first method, Distributed Nesterov Gradient, achieves rates O( logK/K) and O(logk/k). Our second method, Distributed Nesterov gradient with Consensus iterations, assumes at all nodes knowledge of L and μ(W) - the second largest singular value of the N ×N doubly stochastic weight matrix W. It achieves rates O( 1/ K 2-ξ ) and O( 1/k 2 ) ( ξ > 0 arbitrarily small). Further, we give for both methods explicit dependence of the convergence constants on N and W. Simulation examples illustrate our findings. We study distributed optimization problems when [Formula Omitted] nodes minimize the sum of their individual costs subject to a common vector variable. The costs are convex, have Lipschitz continuous gradient (with constant [Formula Omitted]), and bounded gradient. We propose two fast distributed gradient algorithms based on the centralized Nesterov gradient algorithm and establish their convergence rates in terms of the per-node communications [Formula Omitted] and the per-node gradient evaluations [Formula Omitted]. Our first method, Distributed Nesterov Gradient, achieves rates [Formula Omitted] and [Formula Omitted]. Our second method, Distributed Nesterov gradient with Consensus iterations, assumes at all nodes knowledge of [Formula Omitted] and [Formula Omitted] - the second largest singular value of the [Formula Omitted] doubly stochastic weight matrix [Formula Omitted]. It achieves rates [Formula Omitted] and [Formula Omitted] ([Formula Omitted] arbitrarily small). Further, we give for both methods explicit dependence of the convergence constants on [Formula Omitted] and [Formula Omitted]. Simulation examples illustrate our findings. We study distributed optimization problems when N nodes minimize the sum of their individual costs subject to a common vector variable. The costs are convex, have Lipschitz continuous gradient (with constant L ), and bounded gradient. We propose two fast distributed gradient algorithms based on the centralized Nesterov gradient algorithm and establish their convergence rates in terms of the per-node communications calK and the per-node gradient evaluations k . Our first method, Distributed Nesterov Gradient, achieves rates syntax error at token { and syntax error at token { . Our second method, Distributed Nesterov gradient with Consensus iterations, assumes at all nodes knowledge of L and mu openbracket W [ closebracket ] - the second largest singular value of the N timesN doubly stochastic weight matrix W . It achieves rates syntax error at token { and syntax error at token { ( xi > 0 arbitrarily small). Further, we give for both methods explicit dependence of the convergence constants on N and W . Simulation examples illustrate our findings. |
| Author | Xavier, Joao Moura, Jose M. F. Jakovetic, Dusan |
| Author_xml | – sequence: 1 givenname: Dusan surname: Jakovetic fullname: Jakovetic, Dusan email: djakovet@uns.ac.rs organization: Inst. for Syst. & Robot., Univ. of Lisbon, Lisbon, Portugal – sequence: 2 givenname: Joao surname: Xavier fullname: Xavier, Joao email: jxavier@isr.ist.utl.pt organization: Inst. de Sist. e Robot. (ISR), Univ. of Lisbon, Lisbon, Portugal – sequence: 3 givenname: Jose M. F. surname: Moura fullname: Moura, Jose M. F. email: moura@ece.cmu.edu organization: Dept. of Electr. & Comput. Eng., Carnegie Mellon Univ., Pittsburgh, PA, USA |
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| Copyright | Copyright The Institute of Electrical and Electronics Engineers, Inc. (IEEE) May 2014 |
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| Snippet | We study distributed optimization problems when N nodes minimize the sum of their individual costs subject to a common vector variable. The costs are convex,... We study distributed optimization problems when [Formula Omitted] nodes minimize the sum of their individual costs subject to a common vector variable. The... |
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| SubjectTerms | Acceleration Algorithm design and analysis Algorithms Consensus Constants Convergence convergence rate Costs distributed optimization Educational institutions Errors Gradient methods Mathematical analysis Nesterov gradient Optimization Syntax Vectors |
| Title | Fast Distributed Gradient Methods |
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