Advertisement

Online Prediction under Submodular Constraints

  • Daiki Suehiro
  • Kohei Hatano
  • Shuji Kijima
  • Eiji Takimoto
  • Kiyohito Nagano
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7568)

Abstract

We consider an online prediction problem of combinatorial concepts where each combinatorial concept is represented as a vertex of a polyhedron described by a submodular function (base polyhedron). In general, there are exponentially many vertices in the base polyhedron. We propose polynomial time algorithms with regret bounds. In particular, for cardinality-based submodular functions, we give O(n 2)-time algorithms.

Keywords

Extreme Point Convex Combination Submodular Function Combinatorial Concept Procedure Projection 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press (2004)Google Scholar
  2. 2.
    Cesa-Bianchi, N., Lugosi, G.: Combinatorial Bandits. In: Proceedings of the 22nd Conference on Learning Theory (COLT 2009) (2009)Google Scholar
  3. 3.
    Chopra, S.: On the spanning tree polyhedron. Operations Research Letters 8(1), 25–29 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Edmonds, J.: Submodular functions, matroids, and certain polyhedra. In: Combinatorial Structures and Their Applications, pp. 69–87 (1970)Google Scholar
  5. 5.
    Edmonds, J.: Matroids and the greedy algorithm. Mathematical Programming 1(1), 127–136 (1971)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    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(1), 119–139 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Fujishige, S.: Submodular functions and optimization, 2nd edn. Elsevier Science (2005)Google Scholar
  8. 8.
    Hazan, E.: The convex optimization approach to regret minimization. In: Sra, S., Nowozin, S., Wright, S.J. (eds.) Optimization for Machine Learning, ch. 10, pp. 287–304. MIT Press (2011)Google Scholar
  9. 9.
    Helmbold, D.P., Warmuth, M.K.: Learning Permutations with Exponential Weights. Journal of Machine Learning Research 10, 1705–1736 (2009)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Iwata, S.: Submodular function minimization. Mathematical Programming, Ser. B 112, 45–64 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Kakade, S., Kalai, A.T., Ligett, L.: Playing games with approximation algorithms. SIAM Journal on Computing 39(3), 1018–1106 (2009)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kalai, A., Vempala, S.: Efficient algorithms for online decision problems. Journal of Computer and System Sciences 71(3), 291–307 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Koolen, W.M., Warmuth, M.K., Kivinen, J.: Hedging Structured Concepts. In: Proceedings of the 23rd Conference on Learning Theory (COLT 2010), pp. 93–105 (2010)Google Scholar
  14. 14.
    Nagano, K.: A faster parametric submodular function minimization algorithm and applications. Technical Report METR 2007–43, Department of Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo (2007)Google Scholar
  15. 15.
    Orlin, J.B.: A Faster Strongly Polynomial Time Algorithm for Submodular Function Minimization. In: Fischetti, M., Williamson, D.P. (eds.) IPCO 2007. LNCS, vol. 4513, pp. 240–251. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  16. 16.
    Warmuth, M.K., Kuzmin, D.: Randomized Online PCA Algorithms with Regret Bounds that are Logarithmic in the Dimension. Journal of Machine Learning Research 9, 2287–2320 (2008)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Yasutake, S., Hatano, K., Kijima, S., Takimoto, E., Takeda, M.: Online Linear Optimization over Permutations. In: Asano, T., Nakano, S.-I., Okamoto, Y., Watanabe, O. (eds.) ISAAC 2011. LNCS, vol. 7074, pp. 534–543. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  18. 18.
    Ziegler, G.M.: Lectures on Polytopes. Graduate Texts in Mathematics, vol. 152. Springer (1995)Google Scholar
  19. 19.
    Zinkevich, M.: Online convex programming and generalized infinitesimal gradient ascent. In: Proceedings of the Twentieth International Conference on Machine Learning (ICML 2003), pp. 928–936 (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Daiki Suehiro
    • 1
  • Kohei Hatano
    • 1
  • Shuji Kijima
    • 1
  • Eiji Takimoto
    • 1
  • Kiyohito Nagano
    • 2
  1. 1.Department of InformaticsKyushu UniversityJapan
  2. 2.University of TokyoJapan

Personalised recommendations