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Support vector machines with adaptive $$L_q$$ penalty. (English) Zbl 1446.62179
Summary: The standard support vector machine (SVM) minimizes the hinge loss function subject to the $$L_{2}$$ penalty or the roughness penalty. Recently, the $$L_{1}$$ SVM was suggested for variable selection by producing sparse solutions [P. S. Bradley and O. L. Mangasarian, “Feature selection via concave minimization and support vector machines”, in: Proceedings of the Fifteenth International Conference on Machine Learning, ICML’98. San Francisco, CA: Morgan Kaufmann (1998; doi:10.5555/645527.657467); J. Zhu et al., “1-norm support vector machines”, in: Proceedings of the 16th international conference on neural information processing systems, NIPS’03. Cambridge, MA: MIT Press. 49–56 (2003; doi:10.5555/2981345.2981352)]. These learning methods are non-adaptive since their penalty forms are pre-determined before looking at data, and they often perform well only in a certain type of situation. For instance, the $$L_{2}$$ SVM generally works well except when there are too many noise inputs, while the $$L_{1}$$ SVM is more preferred in the presence of many noise variables. In this article we propose and explore an adaptive learning procedure called the $$Lq$$ SVM, where the best $$q>0$$ is automatically chosen by data. Both two- and multi-class classification problems are considered. We show that the new adaptive approach combines the benefit of a class of non-adaptive procedures and gives the best performance of this class across a variety of situations. Moreover, we observe that the proposed $$L_q$$ penalty is more robust to noise variables than the $$L_{1}$$ and $$L_{2}$$ penalties. An iterative algorithm is suggested to solve the $$L_q$$ SVM efficiently. Simulations and real data applications support the effectiveness of the proposed procedure.

##### MSC:
 62H30 Classification and discrimination; cluster analysis (statistical aspects) 68T05 Learning and adaptive systems in artificial intelligence
PDCO
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