  We present an $O((\log k)^2)$-competitive randomized algorithm for the
$k$-server problem on hierarchically separated trees (HSTs). This is the first
$o(k)$-competitive randomized algorithm for which the competitive ratio is
independent of the size of the underlying HST. Our algorithm is designed in the
framework of online mirror descent where the mirror map is a multiscale
entropy. When combined with Bartal's static HST embedding reduction, this leads
to an $O((\log k)^2 \log n)$-competitive algorithm on any $n$-point metric
space. We give a new dynamic HST embedding that yields an $O((\log k)^3 \log
\Delta)$-competitive algorithm on any metric space where the ratio of the
largest to smallest non-zero distance is at most $\Delta$.
