Combining Initial Segments of Lists

  • Manfred K. Warmuth
  • Wouter M. Koolen
  • David P. Helmbold
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6925)

Abstract

We propose a new way to build a combined list from K base lists, each containing N items. A combined list consists of top segments of various sizes from each base list so that the total size of all top segments equals N. A sequence of item requests is processed and the goal is to minimize the total number of misses. That is, we seek to build a combined list that contains all the frequently requested items. We first consider the special case of disjoint base lists. There, we design an efficient algorithm that computes the best combined list for a given sequence of requests. In addition, we develop a randomized online algorithm whose expected number of misses is close to that of the best combined list chosen in hindsight. We prove lower bounds that show that the expected number of misses of our randomized algorithm is close to the optimum. In the presence of duplicate items, we show that computing the best combined list is NP-hard. We show that our algorithms still apply to a linearized notion of loss in this case. We expect that this new way of aggregating lists will find many ranking applications.

Keywords

Initial Segment Online Algorithm Deterministic Algorithm Probabilistic Algorithm Cache Strategy 
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.

References

  1. [AHR08]
    Abernethy, J., Hazan, E., Rakhlin, A.: Competing in the dark: An efficient algorithm for bandit linear optimization. In: Proceedings of the 21st Annual Conference on Learning Theory (July 2008)Google Scholar
  2. [AWY08]
    Abernethy, J., Warmuth, M.K., Yellin, J.: Optimal strategies for random walks. In: Proceedings of the 21st Annual Conference on Learning Theory (July 2008)Google Scholar
  3. [BCD+06]
    Bremner, D., Chan, T.M., Demaine, E.D., Erickson, J., Hurtado, F., Iacono, J., Langerman, S., Taslakian, P.: Necklaces, convolutions, and x + y. In: Azar, Y., Erlebach, T. (eds.) ESA 2006. LNCS, vol. 4168, pp. 160–171. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  4. [BW02]
    Bousquet, O., Warmuth, M.K.: Tracking a small set of experts by mixing past posteriors. Journal of Machine Learning Research 3, 363–396 (2002)MathSciNetMATHGoogle Scholar
  5. [CBL09]
    Cesa-Bianchi, N., Lugosi, G.: Combinatorial bandits. In: Proceedings of the 22nd Annual Conference on Learning Theory (June 2009)Google Scholar
  6. [CT65]
    Cooley, J.W., Tukey, J.W.: An algorithm for the machine calculation of complex Fourier series. Mathematics of Computation 19(90), 297–301 (1965)MathSciNetCrossRefMATHGoogle Scholar
  7. [FS97]
    Freund, Y., Schapire, R.E.: A decision-theoretic generalization of on-line learning and an application to boosting. Journal of Computer and System Sciences 55, 119–139 (1997)MathSciNetCrossRefMATHGoogle Scholar
  8. [GLL05]
    György, A., Linder, T., Lugosi, G.: Tracking the best of many experts. In: Auer, P., Meir, R. (eds.) COLT 2005. LNCS (LNAI), vol. 3559, pp. 204–216. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  9. [Gra03]
    Gramacy, R.B.: Adaptive caching by experts. PhD thesis, University of California at Santa Cruz (2003)Google Scholar
  10. [GVW10]
    Geulen, S., Vöcking, B., Winkler, M.: Regret minimization for online buffering problems using the Weighted Majority algorithm. In: Proceedings of the 23rd Annual Conference on Learning Theory (2010)Google Scholar
  11. [GWBA02]
    Gramacy, R.B., Warmuth, M.K., Brandt, S.A., Ari, I.: Adaptive caching by refetching. In: Becker, S., Thrun, S., Obermayer, K. (eds.) NIPS, pp. 1465–1472. MIT Press, Cambridge (2002)Google Scholar
  12. [HLSS00]
    Helmbold, D.P., Long, D.D.E., Sconyers, T.L., Sherrod, B.: Adaptive disk spin-down for mobile computers. In: ACM/Baltzer Mobile Networks and Applications (MONET), pp. 285–297 (2000)Google Scholar
  13. [HP05]
    Hutter, M., Poland, J.: Adaptive online prediction by following the perturbed leader. Journal of Machine Learning Research 6, 639–660 (2005)MathSciNetMATHGoogle Scholar
  14. [HW98]
    Herbster, M., Warmuth, M.K.: Tracking the best expert. Machine Learning 32, 151–178 (1998)CrossRefMATHGoogle Scholar
  15. [KM05]
    Kontkanen, P., Myllymäki, P.: A fast normalized maximum likelihood algorithm for multinomial data. In: IJCAI, pp. 1613–1615 (2005)Google Scholar
  16. [KV05]
    Kalai, A., Vempala, S.: Efficient algorithms for online decision problems. J. Comput. Syst. Sci. 71(3), 291–307 (2005)MathSciNetCrossRefMATHGoogle Scholar
  17. [KWK10]
    Koolen, W.M., Warmuth, M.K., Kivinen, J.: Hedging structured concepts. In: Proceedings of the 23rd Annual Conference on Learning Theory, pp. 93–105 (June 2010)Google Scholar
  18. [Lit88]
    Littlestone, N.: Learning quickly when irrelevant attributes abound: A new linear-threshold algorithm. Machine Learning 2(4), 285–318 (1988)Google Scholar
  19. [LW94]
    Littlestone, N., Warmuth, M.K.: The Weighted Majority algorithm. Information and Computation 108(2), 212–261 (1994)MathSciNetCrossRefMATHGoogle Scholar
  20. [MM03]
    Megiddo, N., Modha, D.S.: One up on LRU. login: The Magazine of USENIX and SAGE 28(4), 6–11 (2003)Google Scholar
  21. [MM04]
    Megiddo, N., Modha, D.S.: Outperforming LRU with an adaptive replacement cache algorithm. IEEE Computer 37(4), 58–65 (2004)CrossRefGoogle Scholar
  22. [Sca07]
    Scalisi, C.A.: An adaptive caching algorithm. Master’s thesis, University of California Santa Cruz (June 2007)Google Scholar
  23. [SS07]
    Sen, A., Scalisi, C.: Making online predictions from k-lists. Project report for CMPS 290C, Advanced Topics in Machine Learning (June 2007)Google Scholar
  24. [TW03]
    Takimoto, E., Warmuth, M.K.: Path kernels and multiplicative updates. Journal of Machine Learning Research 4, 773–818 (2003)MathSciNetMATHGoogle Scholar
  25. [Vov98]
    Vovk, V.: A game of prediction with expert advice. J. of Comput. Syst. Sci. 56(2), 153–173 (1998)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Manfred K. Warmuth
    • 1
  • Wouter M. Koolen
    • 2
    • 3
  • David P. Helmbold
    • 1
  1. 1.Department of Computer ScienceUC Santa CruzUSA
  2. 2.Department of Computer ScienceRoyal Holloway, University of LondonUK
  3. 3.Centrum Wiskunde en InformaticaAmsterdam

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