Multiscale entropic regularization for MTS on general metric spaces

Abstract We present an $((log =)2)-competitive algorithm for metrical task systems (MTS) on any =-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).

[1]  Yair Bartal,et al.  Probabilistic approximation of metric spaces and its algorithmic applications , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

[2]  Yuval Rabani,et al.  A Decomposition Theorem for Task Systems and Bounds for Randomized Server Problems , 2000, SIAM J. Comput..

[3]  Amos Fiat,et al.  Better algorithms for unfair metrical task systems and applications , 2000, STOC '00.

[4]  Satish Rao,et al.  A tight bound on approximating arbitrary metrics by tree metrics , 2003, STOC '03.

[5]  Yuval Rabani,et al.  Approximation algorithms for the 0-extension problem , 2001, SODA '01.

[6]  Allan Borodin,et al.  An optimal on-line algorithm for metrical task system , 1992, JACM.

[7]  Nathan Linial,et al.  On metric ramsey-type phenomena , 2003, STOC '03.

[8]  Steven S. Seiden,et al.  Unfair Problems and Randomized Algorithms for Metrical Task Systems , 1999, Inf. Comput..

[9]  Andrew Tomkins,et al.  A polylog(n)-competitive algorithm for metrical task systems , 1997, STOC '97.

[10]  Béla Bollobás,et al.  Ramsey-type theorems for metric spaces with applications to online problems , 2004, J. Comput. Syst. Sci..

[11]  Joseph Naor,et al.  Competitive Analysis via Regularization , 2014, SODA.

[12]  Christian Coester,et al.  Pure entropic regularization for metrical task systems , 2019, COLT.

[13]  James R. Lee,et al.  Metrical task systems on trees via mirror descent and unfair gluing , 2018, SODA.

[14]  Peter L. Bartlett,et al.  A Regularization Approach to Metrical Task Systems , 2010, ALT.