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 npoint metric space that is also 1competitive 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
 NSFPAR ID:
 10483029
 Editor(s):
 Braverman, Mark
 Publisher / Repository:
 Schloss Dagstuhl – LeibnizZentrum 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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