k-server via multiscale entropic regularization

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 $\Delta$.