Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik GmbH Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik GmbH scholarly article en Wang, Di; Rao, Satish; Mahoney, Michael W. http://www.dagstuhl.de/lipics License
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URN: urn:nbn:de:0030-drops-63308
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Unified Acceleration Method for Packing and Covering Problems via Diameter Reduction

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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.

BibTeX - Entry

@InProceedings{wang_et_al:LIPIcs:2016:6330,
  author =	{Di Wang and Satish Rao and Michael W. Mahoney},
  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 =	{Ioannis Chatzigiannakis and Michael Mitzenmacher and Yuval Rabani and Davide Sangiorgi},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/6330},
  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}
}

Keywords: Convex optimization, Accelerated gradient descent, Linear program, Approximation algorithm, Packing and covering
Seminar: 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)
Issue date: 2016
Date of publication: 2016


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