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We consider metrical task systems on tree metrics, and present an$O(\mathrm{depth} \times \log n)$-competitive randomized algorithm based on themirror descent framework introduced in our prior work on the $k$-serverproblem. For the special case of hierarchically separated trees (HSTs), we usemirror descent to refine the standard approach based on gluing unfair metricaltask systems. This yields an $O(\log n)$-competitive algorithm for HSTs, thusremoving an extraneous $\log\log n$ in the bound of Fiat and Mendel (2003).Combined with well-known HST embedding theorems, this also gives an $O((\logn)^2)$-competitive randomized algorithm for every $n$-point metric space.

Metrical task systems on trees via mirror descent and unfair gluing