  We present an $O((\log n)^2)$-competitive algorithm for metrical task systems
(MTS) on any $n$-point metric space that is also $1$-competitive for service
costs. This matches the competitive ratio achieved by Bubeck, Cohen, Lee, and
Lee (2019) and the refined competitive ratios obtained by Coester and Lee
(2019). Those algorithms work by first randomly embedding the metric space into
an ultrametric and then solving MTS there. In contrast, our algorithm is cast
as regularized gradient descent where the regularizer is a multiscale metric
entropy defined directly on the metric space. This answers an open question of
Bubeck (Highlights of Algorithms, 2019).
