On the Dual Formulation of Boosting Algorithms

We study boosting algorithms from a new perspective. We show that the Lagrange dual problems of ℓ 1 -norm-regularized AdaBoost, LogitBoost, and soft-margin LPBoost with generalized hinge loss are all entropy maximization problems. By looking at the dual problems of these boosting algorithms, we show...

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Vydané v:	IEEE transactions on pattern analysis and machine intelligence Ročník 32; číslo 12; s. 2216 - 2231
Hlavní autori:	Shen, Chunhua, Li, Hanxi
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Los Alamitos, CA IEEE 01.12.2010 IEEE Computer Society The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Predmet:	AdaBoost Algorithms Applied sciences Artificial intelligence Australia Boosting Classification Computer science; control theory; systems Construction Convergence Data processing. List processing. Character string processing Entropy entropy maximization Exact sciences and technology Fasteners Formulations Lagrange duality Lagrangian functions Linear programming LogitBoost LPBoost Machine learning algorithms Maximization Memory organisation. Data processing Optimization Software Support vector machine classification Support vector machines Data analysis Linear programming LPBoost Optimization Variance Aggregate model Lagrange duality AdaBoost Supervised learning Classification Convergence rate LogitBoost entropy maximization Numerical convergence
ISSN:	0162-8828, 1939-3539, 1939-3539
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Abstract	We study boosting algorithms from a new perspective. We show that the Lagrange dual problems of ℓ 1 -norm-regularized AdaBoost, LogitBoost, and soft-margin LPBoost with generalized hinge loss are all entropy maximization problems. By looking at the dual problems of these boosting algorithms, we show that the success of boosting algorithms can be understood in terms of maintaining a better margin distribution by maximizing margins and at the same time controlling the margin variance. We also theoretically prove that approximately, ℓ 1 -norm-regularized AdaBoost maximizes the average margin, instead of the minimum margin. The duality formulation also enables us to develop column-generation-based optimization algorithms, which are totally corrective. We show that they exhibit almost identical classification results to that of standard stagewise additive boosting algorithms but with much faster convergence rates. Therefore, fewer weak classifiers are needed to build the ensemble using our proposed optimization technique.
AbstractList	We study boosting algorithms from a new perspective. We show that the Lagrange dual problems of \ell_1-norm-regularized AdaBoost, LogitBoost, and soft-margin LPBoost with generalized hinge loss are all entropy maximization problems. By looking at the dual problems of these boosting algorithms, we show that the success of boosting algorithms can be understood in terms of maintaining a better margin distribution by maximizing margins and at the same time controlling the margin variance. We also theoretically prove that approximately, \ell_1-norm-regularized AdaBoost maximizes the average margin, instead of the minimum margin. The duality formulation also enables us to develop column-generation-based optimization algorithms, which are totally corrective. We show that they exhibit almost identical classification results to that of standard stagewise additive boosting algorithms but with much faster convergence rates. Therefore, fewer weak classifiers are needed to build the ensemble using our proposed optimization technique. We study boosting algorithms from a new perspective. We show that the Lagrange dual problems of l₁-norm-regularized AdaBoost, LogitBoost, and soft-margin LPBoost with generalized hinge loss are all entropy maximization problems. By looking at the dual problems of these boosting algorithms, we show that the success of boosting algorithms can be understood in terms of maintaining a better margin distribution by maximizing margins and at the same time controlling the margin variance. We also theoretically prove that approximately, l₁-norm-regularized AdaBoost maximizes the average margin, instead of the minimum margin. The duality formulation also enables us to develop column-generation-based optimization algorithms, which are totally corrective. We show that they exhibit almost identical classification results to that of standard stagewise additive boosting algorithms but with much faster convergence rates. Therefore, fewer weak classifiers are needed to build the ensemble using our proposed optimization technique. We study boosting algorithms from a new perspective. We show that the Lagrange dual problems of l₁-norm-regularized AdaBoost, LogitBoost, and soft-margin LPBoost with generalized hinge loss are all entropy maximization problems. By looking at the dual problems of these boosting algorithms, we show that the success of boosting algorithms can be understood in terms of maintaining a better margin distribution by maximizing margins and at the same time controlling the margin variance. We also theoretically prove that approximately, l₁-norm-regularized AdaBoost maximizes the average margin, instead of the minimum margin. The duality formulation also enables us to develop column-generation-based optimization algorithms, which are totally corrective. We show that they exhibit almost identical classification results to that of standard stagewise additive boosting algorithms but with much faster convergence rates. Therefore, fewer weak classifiers are needed to build the ensemble using our proposed optimization technique.We study boosting algorithms from a new perspective. We show that the Lagrange dual problems of l₁-norm-regularized AdaBoost, LogitBoost, and soft-margin LPBoost with generalized hinge loss are all entropy maximization problems. By looking at the dual problems of these boosting algorithms, we show that the success of boosting algorithms can be understood in terms of maintaining a better margin distribution by maximizing margins and at the same time controlling the margin variance. We also theoretically prove that approximately, l₁-norm-regularized AdaBoost maximizes the average margin, instead of the minimum margin. The duality formulation also enables us to develop column-generation-based optimization algorithms, which are totally corrective. We show that they exhibit almost identical classification results to that of standard stagewise additive boosting algorithms but with much faster convergence rates. Therefore, fewer weak classifiers are needed to build the ensemble using our proposed optimization technique. We study boosting algorithms from a new perspective. We show that the Lagrange dual problems of ℓ 1 -norm-regularized AdaBoost, LogitBoost, and soft-margin LPBoost with generalized hinge loss are all entropy maximization problems. By looking at the dual problems of these boosting algorithms, we show that the success of boosting algorithms can be understood in terms of maintaining a better margin distribution by maximizing margins and at the same time controlling the margin variance. We also theoretically prove that approximately, ℓ 1 -norm-regularized AdaBoost maximizes the average margin, instead of the minimum margin. The duality formulation also enables us to develop column-generation-based optimization algorithms, which are totally corrective. We show that they exhibit almost identical classification results to that of standard stagewise additive boosting algorithms but with much faster convergence rates. Therefore, fewer weak classifiers are needed to build the ensemble using our proposed optimization technique.
Author	Hanxi Li Chunhua Shen
Author_xml	– sequence: 1 givenname: Chunhua surname: Shen fullname: Shen, Chunhua email: chunhua.shen@nicta.com.au organization: NICTA, Canberra Research Laboratory, Canberra, Australia. chunhua.shen@nicta.com.au – sequence: 2 givenname: Hanxi surname: Li fullname: Li, Hanxi
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Cites_doi	10.1007/978-0-387-21579-2_9 10.1006/jcss.1997.1504 10.1145/1102351.1102476 10.1162/089976698300017197 10.1007/s10994-010-5173-z 10.1109/AFGR.2004.1301573 10.1145/307400.307424 10.7551/mitpress/1120.003.0062 10.1007/BF01016429 10.1214/009053607000000785 10.1017/CBO9780511804441 10.1023/A:1010852229904 10.1023/B:VISI.0000013087.49260.fb 10.1109/TPAMI.2002.1033211 10.7551/mitpress/1113.003.0017 10.1198/016214505000000907 10.1145/1143844.1143939 10.1214/aos/1016120463 10.1023/A:1012470815092 10.1007/3-540-36434-X_4 10.1287/opre.1050.0234 10.1145/1143844.1143970 10.1214/aos/1015362183 10.1006/game.1999.0738 10.1023/A:1013912006537 10.1162/089976699300016106
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SubjectTerms	AdaBoost Algorithms Applied sciences Artificial intelligence Australia Boosting Classification Computer science; control theory; systems Construction Convergence Data processing. List processing. Character string processing Entropy entropy maximization Exact sciences and technology Fasteners Formulations Lagrange duality Lagrangian functions Linear programming LogitBoost LPBoost Machine learning algorithms Maximization Memory organisation. Data processing Optimization Software Support vector machine classification Support vector machines
Title	On the Dual Formulation of Boosting Algorithms
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