Online multiserver convex chasing and optimization

Sébastien Bubeck*, Yuval Rabani, Mark Sellke

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

2 Scopus citations


We introduce the problem of k-chasing of convex functions, a simultaneous generalization of both the famous k-server problem in Rd, and of the problem of chasing convex bodies and functions. Aside from fundamental interest in this general form, it has natural applications to online k-clustering problems with objectives such as k-median or k-means. We show that this problem exhibits a rich landscape of behavior. In general, if both k > 1 and d > 1 there does not exist any online algorithm with bounded competitiveness. By contrast, we exhibit a class of nicely behaved functions (which include in particular the above-mentioned clustering problems), for which we show that competitive online algorithms exist, and moreover with dimension-free competitive ratio. We also introduce a parallel question of top-k action regret minimization in the realm of online convex optimization. There, too, a much rougher landscape emerges for k > 1. While it is possible to achieve vanishing regret, unlike the top-one action case the rate of vanishing does not speed up for strongly convex functions. Moreover, vanishing regret necessitates both intractable computations and randomness. Finally we leave open whether almost dimension-free regret is achievable for k > 1 and general convex losses. As evidence that it might be possible, we prove dimension-free regret for linear losses via an information-theoretic argument.

Original languageAmerican English
Title of host publicationACM-SIAM Symposium on Discrete Algorithms, SODA 2021
EditorsDaniel Marx
PublisherAssociation for Computing Machinery
Number of pages12
ISBN (Electronic)9781611976465
StatePublished - 2021
Event32nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2021 - Alexandria, Virtual, United States
Duration: 10 Jan 202113 Jan 2021

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms


Conference32nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2021
Country/TerritoryUnited States
CityAlexandria, Virtual

Bibliographical note

Publisher Copyright:
Copyright © 2021 by SIAM


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