A Regularization Approach to Metrical Task Systems

  • Jacob Abernethy
  • Peter L. Bartlett
  • Niv Buchbinder
  • Isabelle Stanton
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6331)


We address the problem of constructing randomized online algorithms for the Metrical Task Systems (MTS) problem on a metric δ against an oblivious adversary. Restricting our attention to the class of “work-based” algorithms, we provide a framework for designing algorithms that uses the technique of regularization. For the case when δ is a uniform metric, we exhibit two algorithms that arise from this framework, and we prove a bound on the competitive ratio of each. We show that the second of these algorithms is ln n + O(loglogn) competitive, which is the current state-of-the art for the uniform MTS problem.


Competitive Ratio Online Algorithm Competitive Algorithm Cost Vector General Metrics 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Borodin, A., Linial, N., Saks, M.: An optimal on-line algorithm for metrical task system. JACM: Journal of the ACM 39(4), 745–763 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Manasse, M., McGeoch, L., Sleator, D.: Competitive algorithms for server problems. J. Algorithms 11(2), 208–230 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Freund, S.: A decision-theoretic generalization of on-line learning and an application to boosting. JCSS: Journal of Computer and System Sciences 55 (1997)Google Scholar
  4. 4.
    Schafer, G., Sivadasan, N.: Topology matters: Smoothed competitiveness of metrical task systems. TCS: Theoretical Computer Science 341 (2005)Google Scholar
  5. 5.
    Irani, S., Seiden, S.: Randomized algorithms for metrical task systems. Theoretical Computer Science 194 (1998)Google Scholar
  6. 6.
    Bartal, Y., Blum, A., Burch, C., Tomkins, A.: A polylog( n )-competitive algorithm for metrical task systems. In: Symposium on Theory Of Computing (STOC), pp. 711–719 (1997)Google Scholar
  7. 7.
    Bartal, Y.: On approximating arbitrary metrics by tree metrics. In: Symposium Theory Of Computing (STOC), pp. 161–168 (1998)Google Scholar
  8. 8.
    Fakcharoenphol, J., Rao, S., Talwar, K.: A tight bound on approximating arbitrary metrics by tree metrics. In: Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of computing (STOC), pp. 448–455 (2003)Google Scholar
  9. 9.
    Fiat, A., Mendel, M.: Better algorithms for unfair metrical task systems and applications. SIAM Journal on Computing 32 (2003)Google Scholar
  10. 10.
    Bansal, N., Buchbinder, N., Naor, S.: Metrical task systems and the k-server problem on hsts (2009) (manuscript)Google Scholar
  11. 11.
    Bartal, Y., Bollobás, B., Mendel, M.: A ramsey-type theorem for metric spaces and its applications for metrical task systems and related problems. In: IEEE Symposium on Foundations of Computer Science (FOCS), pp. 396–405 (2001)Google Scholar
  12. 12.
    Blum, A., Karloff, H., Rabani, Y., Saks, M.: A decomposition theorem and lower bounds for randomized server problems. SIAM Journal on Computing 30, 1624–1661 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Bansal, N., Buchbinder, N., Naor, J.: A primal-dual randomized algorithm for weighted paging. In: IEEE Symposium on Foundations of Computer Science, FOCS (2007)Google Scholar
  14. 14.
    Bein, W., Larmore, L., Noga, J.: Uniform metrical task systems with a limited number of states. IPL: Information Processing Letters 104 (2007)Google Scholar
  15. 15.
    Bansal, N., Buchbinder, N., Naor, S.: Towards the randomized k-server conjecture: A primal-dual approach. In: ACM-SIAM Symposium on Discrete Algorithms, SODA (2010)Google Scholar
  16. 16.
    Buchbinder, N., Naor, S.: The design of competitive online algorithms via a primal-dual approach. Foundations and Trends in Theoretical Computer Science 3(2-3), 93–263 (2009)MathSciNetGoogle Scholar
  17. 17.
    Blum, A., Burch, C.: On-line learning and the metrical task system problem. Machine Learning 39(1), 35–58 (2000)zbMATHCrossRefGoogle Scholar
  18. 18.
    Herbster, M., Warmuth, M.K.: Tracking the best expert. Machine Learning 32, 151 (1998)zbMATHCrossRefGoogle Scholar
  19. 19.
    Kivinen, J., Warmuth, M.: Exponentiated gradient versus gradient descent for linear predictors. Information and Computation (1997)Google Scholar
  20. 20.
    Gordon, G.: Regret bounds for prediction problems. In: Proceedings of the Twelfth Annual Conference on Computational Learning Theory, pp. 29–40. ACM, New York (1999)CrossRefGoogle Scholar
  21. 21.
    Cesa-Bianchi, N., Lugosi, G.: Prediction, Learning, and Games. Cambridge University Press, New York (2006)zbMATHCrossRefGoogle Scholar
  22. 22.
    Rakhlin, A.: Lecture Notes on Online Learning DRAFT (2009)Google Scholar
  23. 23.
    Guiau, S.: Weighted entropy. Reports on Mathematical Physics 2(3), 165–179 (1971)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Jacob Abernethy
    • 1
  • Peter L. Bartlett
    • 1
  • Niv Buchbinder
    • 2
  • Isabelle Stanton
    • 1
  1. 1.UC Berkeley 
  2. 2.Microsoft Research 

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