License: 
Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ESA.2022.13
URN: urn:nbn:de:0030-drops-169515
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/16951/
Bansal, Nikhil ;
Coester, Christian
Online Metric Allocation and Time-Varying Regularization
Abstract
We introduce a general online allocation problem that connects several of the most fundamental problems in online optimization. Let M be an n-point metric space. Consider a resource that can be allocated in arbitrary fractions to the points of M. At each time t, a convex monotone cost function c_t: [0,1] → ℝ_+ appears at some point r_t ∈ M. In response, an algorithm may change the allocation of the resource, paying movement cost as determined by the metric and service cost c_t(x_{r_t}), where x_{r_t} is the fraction of the resource at r_t at the end of time t. For example, when the cost functions are c_t(x) = α x, this is equivalent to randomized MTS, and when the cost functions are c_t(x) = ∞⋅1_{x < 1/k}, this is equivalent to fractional k-server.
Because of an inherent scale-freeness property of the problem, existing techniques for MTS and k-server fail to achieve similar guarantees for metric allocation. To handle this, we consider a generalization of the online multiplicative update method where we decouple the rate at which a variable is updated from its value, resulting in interesting new dynamics. We use this to give an O(log n)-competitive algorithm for weighted star metrics. We then show how this corresponds to an extension of the online mirror descent framework to a setting where the regularizer is time-varying. Using this perspective, we further refine the guarantees of our algorithm.
We also consider the case of non-convex cost functions. Using a simple ?₂²-regularizer, we give tight bounds of Θ(n) on tree metrics, which imply deterministic and randomized competitive ratios of O(n²) and O(nlog n) respectively on arbitrary metrics.
BibTeX - Entry
@InProceedings{bansal_et_al:LIPIcs.ESA.2022.13,
author = {Bansal, Nikhil and Coester, Christian},
title = {{Online Metric Allocation and Time-Varying Regularization}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {13:1--13:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/16951},
URN = {urn:nbn:de:0030-drops-169515},
doi = {10.4230/LIPIcs.ESA.2022.13},
annote = {Keywords: Online algorithms, competitive analysis, k-server, metrical task systems, mirror descent, regularization}
}
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Keywords: |
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Online algorithms, competitive analysis, k-server, metrical task systems, mirror descent, regularization |
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Collection: |
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30th Annual European Symposium on Algorithms (ESA 2022) |
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Issue Date: |
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2022 |
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Date of publication: |
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01.09.2022 |
