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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 servicecosts. This matches the competitive ratio achieved by Bubeck, Cohen, Lee, andLee (2019) and the refined competitive ratios obtained by Coester and Lee(2019). Those algorithms work by first randomly embedding the metric space intoan ultrametric and then solving MTS there. In contrast, our algorithm is castas regularized gradient descent where the regularizer is a multiscale metricentropy defined directly on the metric space. This answers an open question ofBubeck (Highlights of Algorithms, 2019).

Multiscale entropic regularization for MTS on general metric spaces