Multiscale entropic regularization for MTS on general metric spaces

TL;DR

This paper introduces a new multiscale entropic regularization approach for metrical task systems on general metric spaces, achieving competitive ratios comparable to previous embedding-based methods.

Contribution

It proposes a novel regularized gradient descent algorithm using multiscale metric entropy directly on the metric space, answering an open question in the field.

Findings

Achieves $O((\log n)^2)$-competitive ratio for MTS on any n-point metric space.

Matches the best known competitive ratios from prior embedding-based algorithms.

Provides a new direct regularization method avoiding metric embeddings.

Abstract

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).