Abstract
In this paper we discuss boosting algorithms that maximize the soft margin of the produced linear combination of base hypotheses. LPBoost is the most straightforward boosting algorithm for doing this. It maximizes the soft margin by solving a linear programming problem. While it performs well on natural data, there are cases where the number of iterations is linear in the number of examples instead of logarithmic.
By simply adding a relative entropy regularization to the linear objective of LPBoost, we arrive at the Entropy Regularized LPBoost algorithm for which we prove a logarithmic iteration bound. A previous algorithm, called SoftBoost, has the same iteration bound, but the generalization error of this algorithm often decreases slowly in early iterations. Entropy Regularized LPBoost does not suffer from this problem and has a simpler, more natural motivation.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Abe, N., Takeuchi, J., Warmuth, M.K.: Polynomial learnability of stochastic rules with respect to the KL-divergence and quadratic distance. IEICE Transactions on Information and Systems E84-D(3), 299–316 (2001)
Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)
Demiriz, A., Bennett, K.P., Shawe-Taylor, J.: Linear programming boosting via column generation. Mach. Learn. 46(1-3), 225–254 (2002)
Duffy, N., Helmbold, D.: Potential boosters? In: Solla, S., Leen, T., Müller, K.-R. (eds.) Advances in Neural Information Processing Systems 12, pp. 258–264. MIT Press, Cambridge (2000)
Freund, Y.: Boosting a weak learning algorithm by majority. Inform. Comput. 121(2), 256–285 (1995); In: COLT 1990
Freund, Y.: An adaptive version of the boost by majority algorithm. In: Proceedings of the 12th annual conference on Computational learning theory, pp. 102–113. ACM Press, New York (1999)
Freund, Y., Schapire, R.: A decision-theoretic generalization of on-line learning and an application to boosting. Journal of Computer and System Sciences 55(1), 119–139 (1997)
Friedman, J., Hastie, T., Tibshirani, R.: Additive Logistic Regression: a Statistical View of Boosting. The Annals of Statistics 38(2) (2000)
Grove, A.J., Schuurmans, D.: Boosting in the limit: maximizing the margin of learned ensembles. In: AAAI 1998/IAAI 1998, Menlo Park, CA, USA, pp. 692–699 (1998)
Helmbold, D., Schapire, R.E., Singer, Y., Warmuth, M.K.: A comparison of new and old algorithms for a mixture estimation problem. Machine Learning 27(1), 97–119 (1997)
Kivinen, J., Warmuth, M.K.: Boosting as entropy projection. In: Proc. 12th Annu. Conf. on Comput. Learning Theory, pp. 134–144. ACM Press, New York (1999)
Lafferty, J.: Additive models, boosting, and inference for generalized divergences. In: Proceedings of the 12th Annual Conference on Computional Learning Theory, pp. 125–133. ACM Press, New York (1999)
Liao, J.: Totally Corrective Boosting Algorithms that Maximize the Margin. PhD thesis, University of California at Santa Cruz (December 2006)
Rätsch, G.: Robust Boosting via Convex Optimization: Theory and Applications. PhD thesis, University of Potsdam (2001)
Rätsch, G., Onoda, T., Müller, K.-R.: Soft margins for adaboost. Mach. Learn. 42(3), 287–320 (2001)
Rätsch, G., Schölkopf, B., Smola, A., Mika, S., Onoda, T., Müller, K.-R.: Robust ensemble learning. In: Smola, A., Bartlett, P., Schölkopf, B., Schuurmans, D. (eds.) Advances in Large Margin Classifiers, pp. 207–219. MIT Press, Cambridge, MA (2000)
Rätsch, G., Warmuth, M.: Efficient margin maximizing with boosting. Journal of Machine Learning Research 6, 2131–2152 (2005)
Rudin, C., Schapire, R., Daubechies, I.: Boosting based on a smooth margin. In: Proceedings of the 17th Annual Conference on Computational Learning Theory, pp. 502–517 (2004)
Rudin, C., Schapire, R., Daubechies, I.: Analysis of boosting algorithms using the smooth margin function. The Annals of Statistics 6(35), 2723–2768 (2007)
Schapire, R., Freund, Y., Bartlett, P., Lee, W.: Boosting the margin: A new explanation for the effectiveness of voting methods. The Annals of Statistics 26(5), 1651–1686 (1998)
Shalev-Shwartz, S., Singer, Y.: On the equivalence of weak learnability and linear separability: New relaxations and efficient boosting algorithms. In: Proceedings of the 21st annual conference on Computational learning theory, pp. 311–321. Omicron (2008)
Smola, A., Vishwanathan, S.V.N., Le, Q.: Bundle methods for machine learning. In: Platt, J., Koller, D., Singer, Y., Roweis, S. (eds.) Advances in Neural Information Processing Systems 20, pp. 1377–1384. MIT Press, Cambridge (2008)
Warmuth, M., Liao, J., Rätsch, G.: Totally corrective boosting algorithms that maximize the margin. In: ICML 2006, pp. 1001–1008. ACM Press, New York (2006)
Warmuth, M.K., Glocer, K., Rätsch, G.: Boosting algorithms for maximizing the soft margin. In: Platt, J., Koller, D., Singer, Y., Roweis, S. (eds.) Advances in Neural Information Processing Systems 20. MIT Press, Cambridge (2007)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Warmuth, M.K., Glocer, K.A., Vishwanathan, S.V.N. (2008). Entropy Regularized LPBoost. In: Freund, Y., Györfi, L., Turán, G., Zeugmann, T. (eds) Algorithmic Learning Theory. ALT 2008. Lecture Notes in Computer Science(), vol 5254. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-87987-9_23
Download citation
DOI: https://doi.org/10.1007/978-3-540-87987-9_23
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-87986-2
Online ISBN: 978-3-540-87987-9
eBook Packages: Computer ScienceComputer Science (R0)