An \(\mathcal {O}(n \log n)\) Cutting Plane Algorithm for Structured Output Ranking

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8753)


In this work, we consider ranking as a training strategy for structured output prediction. Recent work has begun to explore structured output prediction in the ranking setting, but has mostly focused on the special case of bipartite preference graphs. The bipartite special case is computationally efficient as there exists a linear time cutting plane training strategy for hinge loss bounded regularized risk, but it is unclear how to feasibly extend the approach to complete preference graphs. We develop here a highly parallelizable \(\mathcal {O}(n \log n)\) algorithm for cutting plane training with complete preference graphs that is scalable to millions of samples on a single core. We explore theoretically and empirically the relationship between slack rescaling and margin rescaling variants of the hinge loss bound to structured losses, showing that the slack rescaling variant has better stability properties and empirical performance with no additional computational cost per cutting plane iteration. We further show generalization bounds based on uniform convergence. Finally, we demonstrate the effectiveness of the proposed family of approaches on the problem of object detection in computer vision.


Bipartite Graph Structure Output Pairwise Constraint Hinge Loss Preference Graph 
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.



This work is partially funded by ERC Grant 259112, and FP7-MC-CIG 334380.


  1. 1.
    Agarwal, S., Niyogi, P.: Generalization bounds for ranking algorithms via algorithmic stability. J. Mach. Learn. Res. 10, 441–474 (2009)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Alexe, B., Deselaers, T., Ferrari, V.: What is an object? In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, June 2010Google Scholar
  3. 3.
    Bakır, G.H., Hofmann, T., Schölkopf, B., Smola, A.J., Taskar, B., Vishwanathan, S.V.N.: Predicting Structured Data. MIT Press, Cambridge (2007)Google Scholar
  4. 4.
    Blaschko, M.B., Lampert, C.H.: Learning to localize objects with structured output regression. In: Forsyth, D., Torr, P., Zisserman, A. (eds.) ECCV 2008, Part I. LNCS, vol. 5302, pp. 2–15. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  5. 5.
    Blaschko, M.B., Vedaldi, A., Zisserman, A.: Simultaneous object detection and ranking with weak supervision. In: Advances in Neural Information Processing Systems (2010)Google Scholar
  6. 6.
    Bousquet, O., Elisseeff, A.: Stability and generalization. J. Mach. Learn. Res. 2, 499–526 (2002)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Everingham, M., Van Gool, L., Williams, C.K.I., Winn, J., Zisserman, A.: The Pascal visual object classes (VOC) challenge. Int. J. Comput. Vis. 88(2), 303–338 (2010)CrossRefGoogle Scholar
  8. 8.
    Felzenszwalb, P., Huttenlocher, D.: Efficient graph-based image segmentation. Int. J. Comput. Vis. 59(2), 167–181 (2004)CrossRefGoogle Scholar
  9. 9.
    Herbrich, R., Graepel, T., Obermayer, K.: Large margin rank boundaries for ordinal regression. In: Smola, A., Bartlett, P., Schölkopf, B., Schuurmans, D. (eds.) Advances in Large Margin Classifiers, pp. 115–132. MIT Press, Cambridge (2000)Google Scholar
  10. 10.
    Joachims, T.: Training linear SVMs in linear time. In: ACM SIGKDD Conference on Knowledge Discovery and Data Mining, pp. 217–226 (2006)Google Scholar
  11. 11.
    Joachims, T., Finley, T., Yu, C.N.: Cutting-plane training of structural SVMs. Mach. Learn. 77(1), 27–59 (2009)CrossRefzbMATHGoogle Scholar
  12. 12.
    Lafferty, J., Zhu, X., Liu, Y.: Kernel conditional random fields: representation and clique selection. In: Proceedings of the International Conference on Machine Learning (2004)Google Scholar
  13. 13.
    Manning, C.D., Raghavan, P., Schütze, H.: Introduction to Information Retrieval. Cambridge University Press, New York (2008)CrossRefzbMATHGoogle Scholar
  14. 14.
    McDiarmid, C.: On the method of bounded differences. In: Siemons, J. (ed.) Surveys in Combinatorics, pp. 148–188. Cambridge University Press, Cambridge (1989)Google Scholar
  15. 15.
    Mittal, A., Blaschko, M.B., Zisserman, A., Torr, P.H.S.: Taxonomic multi-class prediction and person layout using efficient structured ranking. In: Fitzgibbon, A., Lazebnik, S., Perona, P., Sato, Y., Schmid, C. (eds.) ECCV 2012, Part II. LNCS, vol. 7573, pp. 245–258. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  16. 16.
    Rahtu, E., Kannala, J., Blaschko, M.B.: Learning a category independent object detection cascade. In: Proceedings of the International Conference on Computer Vision (2011)Google Scholar
  17. 17.
    Taskar, B., Guestrin, C., Koller, D.: Max-margin Markov networks. In: Advances in Neural Information Processing Systems (2004)Google Scholar
  18. 18.
    Tsochantaridis, I., Hofmann, T., Joachims, T., Altun, Y.: Support vector machine learning for interdependent and structured output spaces. In: Proceedings of the International Conference on Machine Learning (2004)Google Scholar
  19. 19.
    Vedaldi, A., Gulshan, V., Varma, M., Zisserman, A.: Multiple kernels for object detection. In: Proceedings of the International Conference on Computer Vision (2009)Google Scholar
  20. 20.
    Vedaldi, A., Blaschko, M.B., Zisserman, A.: Learning equivariant structured output SVM regressors. In: Proceedings of the International Conference on Computer Vision, pp. 959–966 (2011)Google Scholar
  21. 21.
    Viola, P., Jones, M.: Robust real-time object detection. Int. J. Comput. Vis. 57(2), 137–154 (2002)CrossRefGoogle Scholar
  22. 22.
    Zhang, Z., Warrell, J., Torr, P.: Proposal generation for object detection using cascaded ranking SVMs. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (2011)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Inria and École Centrale ParisParisFrance
  2. 2.Department of Engineering ScienceUniversity of OxfordOxfordUK
  3. 3.Center for Machine Vision ResearchUniversity of OuluOuluFinland

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