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Documents authored by Mahoney, Michael W.

Document
DOI: 10.4230/LIPIcs.ICALP.2016.50
Unified Acceleration Method for Packing and Covering Problems via Diameter Reduction

Authors: Di Wang, Satish Rao, and Michael W. Mahoney

Published in: LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

Abstract
In a series of recent breakthroughs, Allen-Zhu and Orecchia [Allen-Zhu/Orecchia, STOC 2015; Allen-Zhu/Orecchia, SODA 2015] leveraged insights from the linear coupling method [Allen-Zhu/Oreccia, arXiv 2014], which is a first-order optimization scheme, to provide improved algorithms for packing and covering linear programs. The result in [Allen-Zhu/Orecchia, STOC 2015] is particularly interesting, as the algorithm for packing LP achieves both width-independence and Nesterov-like acceleration, which was not known to be possible before. Somewhat surprisingly, however, while the dependence of the convergence rate on the error parameter epsilon for packing problems was improved to O(1/epsilon), which corresponds to what accelerated gradient methods are designed to achieve, the dependence for covering problems was only improved to O(1/epsilon^{1.5}), and even that required a different more complicated algorithm, rather than from Nesterov-like acceleration. Given the primal-dual connection between packing and covering problems and since previous algorithms for these very related problems have led to the same epsilon dependence, this discrepancy is surprising, and it leaves open the question of the exact role that the linear coupling is playing in coordinating the complementary gradient and mirror descent step of the algorithm. In this paper, we clarify these issues, illustrating that the linear coupling method can lead to improved O(1/epsilon) dependence for both packing and covering problems in a unified manner, i.e., with the same algorithm and almost identical analysis. Our main technical result is a novel dimension lifting method that reduces the coordinate-wise diameters of the feasible region for covering LPs, which is the key structural property to enable the same Nesterov-like acceleration as in the case of packing LPs. The technique is of independent interest and that may be useful in applying the accelerated linear coupling method to other combinatorial problems.
Cite as

Di Wang, Satish Rao, and Michael W. Mahoney. Unified Acceleration Method for Packing and Covering Problems via Diameter Reduction. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 50:1-50:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{wang_et_al:LIPIcs.ICALP.2016.50,
  author =	{Wang, Di and Rao, Satish and Mahoney, Michael W.},
  title =	{{Unified Acceleration Method for Packing and Covering Problems via Diameter Reduction}},
  booktitle =	{43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
  pages =	{50:1--50:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-013-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{55},
  editor =	{Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.50},
  URN =		{urn:nbn:de:0030-drops-63308},
  doi =		{10.4230/LIPIcs.ICALP.2016.50},
  annote =	{Keywords: Convex optimization, Accelerated gradient descent, Linear program, Approximation algorithm, Packing and covering}
}
Document
DOI: 10.4230/LIPIcs.ICALP.2016.52
Approximating the Solution to Mixed Packing and Covering LPs in Parallel O˜(epsilon^{-3}) Time

Authors: Michael W. Mahoney, Satish Rao, Di Wang, and Peng Zhang

Published in: LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

Abstract
We study the problem of approximately solving positive linear programs (LPs). This class of LPs models a wide range of fundamental problems in combinatorial optimization and operations research, such as many resource allocation problems, solving non-negative linear systems, computing tomography, single/multi commodity flows on graphs, etc. For the special cases of pure packing or pure covering LPs, recent result by Allen-Zhu and Orecchia [Allen/Zhu/Orecchia, SODA'15] gives O˜(1/(epsilon^3))-time parallel algorithm, which breaks the longstanding O˜(1/(epsilon^4)) running time bound by the seminal work of Luby and Nisan [Luby/Nisan, STOC'93]. We present new parallel algorithm with running time O˜(1/(epsilon^3)) for the more general mixed packing and covering LPs, which improves upon the O˜(1/(epsilon^4))-time algorithm of Young [Young, FOCS'01; Young, arXiv 2014]. Our work leverages the ideas from both the optimization oriented approach [Allen/Zhu/Orecchia, SODA'15; Wang/Mahoney/Mohan/Rao, arXiv 2015], as well as the more combinatorial approach with phases [Young, FOCS'01; Young, arXiv 2014]. In addition, our algorithm, when directly applied to pure packing or pure covering LPs, gives a improved running time of O˜(1/(epsilon^2)).
Cite as

Michael W. Mahoney, Satish Rao, Di Wang, and Peng Zhang. Approximating the Solution to Mixed Packing and Covering LPs in Parallel O˜(epsilon^{-3}) Time. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 52:1-52:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{mahoney_et_al:LIPIcs.ICALP.2016.52,
  author =	{Mahoney, Michael W. and Rao, Satish and Wang, Di and Zhang, Peng},
  title =	{{Approximating the Solution to Mixed Packing and Covering LPs in Parallel O˜(epsilon^\{-3\}) Time}},
  booktitle =	{43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
  pages =	{52:1--52:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-013-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{55},
  editor =	{Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.52},
  URN =		{urn:nbn:de:0030-drops-63335},
  doi =		{10.4230/LIPIcs.ICALP.2016.52},
  annote =	{Keywords: Mixed packing and covering, Linear program, Approximation algorithm, Parallel algorithm}
}
Document
DOI: 10.4230/DagSemProc.07071.1
07071 Abstracts Collection – Web Information Retrieval and Linear Algebra Algorithms

Authors: Andreas Frommer, Michael W. Mahoney, and Daniel B. Szyld

Published in: Dagstuhl Seminar Proceedings, Volume 7071, Web Information Retrieval and Linear Algebra Algorithms (2007)

Abstract
From 12th to 16th February 2007, the Dagstuhl Seminar 07071 ``Web Information Retrieval and Linear Algebra Algorithms'' was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available.
Cite as

Andreas Frommer, Michael W. Mahoney, and Daniel B. Szyld. 07071 Abstracts Collection – Web Information Retrieval and Linear Algebra Algorithms. In Web Information Retrieval and Linear Algebra Algorithms. Dagstuhl Seminar Proceedings, Volume 7071, pp. 1-20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2007)

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@InProceedings{frommer_et_al:DagSemProc.07071.1,
  author =	{Frommer, Andreas and Mahoney, Michael W. and Szyld, Daniel B.},
  title =	{{07071 Abstracts Collection – Web Information Retrieval and Linear Algebra Algorithms}},
  booktitle =	{Web Information Retrieval and Linear Algebra Algorithms},
  pages =	{1--20},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2007},
  volume =	{7071},
  editor =	{Andreas Frommer and Michael W. Mahoney and Daniel B. Szyld},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.07071.1},
  URN =		{urn:nbn:de:0030-drops-10743},
  doi =		{10.4230/DagSemProc.07071.1},
  annote =	{Keywords: Information retrieval, Markov chains, PageRank, numerical linear algebra, low rank approximations, sparsity, ranking}
}
Document
DOI: 10.4230/DagSemProc.07071.2
07071 Report on Dagstuhl Seminar – Web Information Retrieval and Linear Algebra Algorithms

Authors: Andreas Frommer, Michael W. Mahoney, and Daniel B. Szyld

Published in: Dagstuhl Seminar Proceedings, Volume 7071, Web Information Retrieval and Linear Algebra Algorithms (2007)

Abstract
A seminar concentrating on the intersection of the fields of information retrieval and other web-related aspects with numerical and applied linear algebra techniques was held with the attendance of scientists from industry and academia.
Cite as

Andreas Frommer, Michael W. Mahoney, and Daniel B. Szyld. 07071 Report on Dagstuhl Seminar – Web Information Retrieval and Linear Algebra Algorithms. In Web Information Retrieval and Linear Algebra Algorithms. Dagstuhl Seminar Proceedings, Volume 7071, pp. 1-3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2007)

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@InProceedings{frommer_et_al:DagSemProc.07071.2,
  author =	{Frommer, Andreas and Mahoney, Michael W. and Szyld, Daniel B.},
  title =	{{07071 Report on Dagstuhl Seminar – Web Information Retrieval and Linear Algebra Algorithms}},
  booktitle =	{Web Information Retrieval and Linear Algebra Algorithms},
  pages =	{1--3},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2007},
  volume =	{7071},
  editor =	{Andreas Frommer and Michael W. Mahoney and Daniel B. Szyld},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.07071.2},
  URN =		{urn:nbn:de:0030-drops-10709},
  doi =		{10.4230/DagSemProc.07071.2},
  annote =	{Keywords: Information retrieval, Markov chains, PageRank, numerical linear algebra, low rank approximations, sparsity , ranking}
}
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