Speed and Sparsity of Regularized Boosting

Yongxin Xi, Zhen Xiang, Peter Ramadge, Robert Schapire
Proceedings of the Twelfth International Conference on Artificial Intelligence and Statistics, PMLR 5:615-622, 2009.

Abstract

Boosting algorithms with $l_1$ regularization are of interest because $l_1$ regularization leads to sparser composite classifiers. Moreover, Rosset et al. have shown that for separable data, standard $l_p$ regularized loss minimization results in a margin maximizing classifier in the limit as regularization is relaxed. For the case $p=1$, we extend these results by obtaining explicit convergence bounds on the regularization required to yield a margin within prescribed accuracy of the maximum achievable margin. We derive similar rates of convergence for the $\epsilon$-AdaBoost algorithm, in the process providing a new proof that $\epsilon$-AdaBoost is margin maximizing as $\epsilon$ converges to $0$. Because both of these known algorithms are computationally expensive, we introduce a new hybrid algorithm, AdaBoost+$L_1$, that combines the virtues of AdaBoost with the sparsity of $l_1$ regularization in a computationally efficient fashion. We prove that the algorithm is margin maximizing and empirically examine its performance on five datasets.

Cite this Paper

BibTeX
@InProceedings{pmlr-v5-xi09a, title = {Speed and Sparsity of Regularized Boosting}, author = {Xi, Yongxin and Xiang, Zhen and Ramadge, Peter and Schapire, Robert}, booktitle = {Proceedings of the Twelfth International Conference on Artificial Intelligence and Statistics}, pages = {615--622}, year = {2009}, editor = {van Dyk, David and Welling, Max}, volume = {5}, series = {Proceedings of Machine Learning Research}, address = {Hilton Clearwater Beach Resort, Clearwater Beach, Florida USA}, month = {16--18 Apr}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v5/xi09a/xi09a.pdf}, url = {https://proceedings.mlr.press/v5/xi09a.html}, abstract = {Boosting algorithms with $l_1$ regularization are of interest because $l_1$ regularization leads to sparser composite classifiers. Moreover, Rosset et al. have shown that for separable data, standard $l_p$ regularized loss minimization results in a margin maximizing classifier in the limit as regularization is relaxed. For the case $p=1$, we extend these results by obtaining explicit convergence bounds on the regularization required to yield a margin within prescribed accuracy of the maximum achievable margin. We derive similar rates of convergence for the $\epsilon$-AdaBoost algorithm, in the process providing a new proof that $\epsilon$-AdaBoost is margin maximizing as $\epsilon$ converges to $0$. Because both of these known algorithms are computationally expensive, we introduce a new hybrid algorithm, AdaBoost+$L_1$, that combines the virtues of AdaBoost with the sparsity of $l_1$ regularization in a computationally efficient fashion. We prove that the algorithm is margin maximizing and empirically examine its performance on five datasets.} }
Endnote
%0 Conference Paper %T Speed and Sparsity of Regularized Boosting %A Yongxin Xi %A Zhen Xiang %A Peter Ramadge %A Robert Schapire %B Proceedings of the Twelfth International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2009 %E David van Dyk %E Max Welling %F pmlr-v5-xi09a %I PMLR %P 615--622 %U https://proceedings.mlr.press/v5/xi09a.html %V 5 %X Boosting algorithms with $l_1$ regularization are of interest because $l_1$ regularization leads to sparser composite classifiers. Moreover, Rosset et al. have shown that for separable data, standard $l_p$ regularized loss minimization results in a margin maximizing classifier in the limit as regularization is relaxed. For the case $p=1$, we extend these results by obtaining explicit convergence bounds on the regularization required to yield a margin within prescribed accuracy of the maximum achievable margin. We derive similar rates of convergence for the $\epsilon$-AdaBoost algorithm, in the process providing a new proof that $\epsilon$-AdaBoost is margin maximizing as $\epsilon$ converges to $0$. Because both of these known algorithms are computationally expensive, we introduce a new hybrid algorithm, AdaBoost+$L_1$, that combines the virtues of AdaBoost with the sparsity of $l_1$ regularization in a computationally efficient fashion. We prove that the algorithm is margin maximizing and empirically examine its performance on five datasets.
RIS
TY - CPAPER TI - Speed and Sparsity of Regularized Boosting AU - Yongxin Xi AU - Zhen Xiang AU - Peter Ramadge AU - Robert Schapire BT - Proceedings of the Twelfth International Conference on Artificial Intelligence and Statistics DA - 2009/04/15 ED - David van Dyk ED - Max Welling ID - pmlr-v5-xi09a PB - PMLR DP - Proceedings of Machine Learning Research VL - 5 SP - 615 EP - 622 L1 - http://proceedings.mlr.press/v5/xi09a/xi09a.pdf UR - https://proceedings.mlr.press/v5/xi09a.html AB - Boosting algorithms with $l_1$ regularization are of interest because $l_1$ regularization leads to sparser composite classifiers. Moreover, Rosset et al. have shown that for separable data, standard $l_p$ regularized loss minimization results in a margin maximizing classifier in the limit as regularization is relaxed. For the case $p=1$, we extend these results by obtaining explicit convergence bounds on the regularization required to yield a margin within prescribed accuracy of the maximum achievable margin. We derive similar rates of convergence for the $\epsilon$-AdaBoost algorithm, in the process providing a new proof that $\epsilon$-AdaBoost is margin maximizing as $\epsilon$ converges to $0$. Because both of these known algorithms are computationally expensive, we introduce a new hybrid algorithm, AdaBoost+$L_1$, that combines the virtues of AdaBoost with the sparsity of $l_1$ regularization in a computationally efficient fashion. We prove that the algorithm is margin maximizing and empirically examine its performance on five datasets. ER -
APA
Xi, Y., Xiang, Z., Ramadge, P. & Schapire, R.. (2009). Speed and Sparsity of Regularized Boosting. Proceedings of the Twelfth International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 5:615-622 Available from https://proceedings.mlr.press/v5/xi09a.html.

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