Approximating Parameterized Convex Optimization Problems

  • Joachim Giesen
  • Martin Jaggi
  • Sören Laue
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6346)

Abstract

We extend Clarkson’s framework by considering parameterized convex optimization problems over the unit simplex, that depend on one parameter. We provide a simple and efficient scheme for maintaining an ε-approximate solution (and a corresponding ε-coreset) along the entire parameter path. We prove correctness and optimality of the method. Practically relevant instances of the abstract parameterized optimization problem are for example regularization paths of support vector machines, multiple kernel learning, and minimum enclosing balls of moving points.

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References

  1. 1.
    Agarwal, P., Har-Peled, S., Varadarajan, K.: Approximating extent measures of points. Journal of the ACM 51(4), 606–635 (2004)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Agarwal, P., Har-Peled, S., Yu, H.: Embeddings of surfaces, curves, and moving points in euclidean space. In: SCG 2007: Proceedings of the Twenty-third Annual Symposium on Computational Geometry (2007)Google Scholar
  3. 3.
    Bach, F., Lanckriet, G., Jordan, M.: Multiple kernel learning, conic duality, and the smo algorithm. In: ICML 2004: Proceedings of the Twenty-first International Conference on Machine Learning (2004)Google Scholar
  4. 4.
    Bădoiu, M., Clarkson, K.L.: Optimal core-sets for balls. Computational Geometry: Theory and Applications 40(1), 14–22 (2007)Google Scholar
  5. 5.
    Clarkson, K.L.: Coresets, sparse greedy approximation, and the frank-wolfe algorithm. In: SODA 2008: Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms (2008)Google Scholar
  6. 6.
    Friedman, J., Hastie, T., Höfling, H., Tibshirani, R.: Pathwise coordinate optimization. The Annals of Applied Statistics 1(2), 302–332 (2007)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Gärtner, B., Giesen, J., Jaggi, M.: An exponential lower bound on the complexity of regularization paths. arXiv, cs.LG (2009)Google Scholar
  8. 8.
    Gärtner, B., Giesen, J., Jaggi, M., Welsch, T.: A combinatorial algorithm to compute regularization paths. arXiv, cs.LG (2009)Google Scholar
  9. 9.
    Gärtner, B., Jaggi, M.: Coresets for polytope distance. In: SCG 2009: Proceedings of the 25th Annual Symposium on Computational Geometry (2009)Google Scholar
  10. 10.
    Hastie, T., Rosset, S., Tibshirani, R., Zhu, J.: The entire regularization path for the support vector machine. The Journal of Machine Learning Research 5, 1391–1415 (2004)MathSciNetGoogle Scholar
  11. 11.
    Matousek, J., Gärtner, B.: Understanding and Using Linear Programming (Universitext). Springer, New York (2006)Google Scholar
  12. 12.
    Rosset, S., Zhu, J.: Piecewise linear regularized solution paths. Annals of Statistics 35(3), 1012–1030 (2007)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Wu, Z., Zhang, A., Li, C., Sudjianto, A.: Trace solution paths for svms via parametric quadratic programming. In: KDD 2008 DMMT Workshop (2008)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Joachim Giesen
    • 1
  • Martin Jaggi
    • 2
  • Sören Laue
    • 1
  1. 1.Friedrich-Schiller-Universität JenaGermany
  2. 2.ETH ZürichSwitzerland

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