Revised: October 8, 2020
Published: December 29, 2022
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Abstract: [Plain Text Version]
We show that on every $n$-point HST metric, there is a randomized online algorithm for metrical task systems (MTS) that is $1$-competitive for service costs and $O(\log n)$-competitive for movement costs. In general, these refined guarantees are optimal up to the implicit constant. While an $O(\log n)$-competitive algorithm for MTS on HST metrics was developed by Bubeck et al. (SODA'19), that approach could only establish an $O((\log n)^2)$-competitive ratio when the service costs are required to be $O(1)$-competitive. Our algorithm can be viewed as an instantiation of online mirror descent with the regularizer derived from a multiscale conditional entropy.
In fact, our algorithm satisfies a set of even more refined guarantees;
we are able to exploit this
property to combine it with known random embedding theorems and obtain,
An extended abstract of this paper appeared in the Proceedings of the 32nd Ann. Conference on Learning Theory (COLT 2019).