We study the problem of chasing convex bodies online: given a sequence of convex bodies the algorithm must respond with points in an online fashion (i.e., is chosen before is revealed). The objective is to minimize the sum of distances between successive points in this sequence. Bubeck et al. (STOC 2019) gave a -competitive algorithm for this problem. We give an algorithm that is -competitive for any sequence of length .
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Leibniz International Proceedings in Informatics (LIPIcs):13th Innovations in Theoretical Computer Science Conference (ITCS 2022)
We present an O((log n)²)-competitive algorithm for metrical task systems (MTS) on any n-point metric space that is also 1-competitive for service costs. This matches the competitive ratio achieved by Bubeck, Cohen, Lee, and Lee (2019) and the refined competitive ratios obtained by Coester and Lee (2019). Those algorithms work by first randomly embedding the metric space into an ultrametric and then solving MTS there. In contrast, our algorithm is cast as regularized gradient descent where the regularizer is a multiscale metric entropy defined directly on the metric space. This answers an open question of Bubeck (Highlights of Algorithms, 2019).
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- Award ID(s):
- 2007079
- NSF-PAR ID:
- 10483029
- Editor(s):
- Braverman, Mark
- Publisher / Repository:
- Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- Date Published:
- Subject(s) / Keyword(s):
- Metrical task systems online algorithms metric embeddings convex optimization Theory of computation → Online algorithms Theory of computation → Mathematical optimization Theory of computation → Random projections and metric embeddings
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.4230/LIPIcs.ITCS.2022.60
