Support vector machines with adaptive Lq penalty

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 [Bradley, P., Mangasarian, O., 1998. Feature selection via concave minimization and support vector machines. In: Shavlik, J. (Ed.), ICML'98. Morgan Kaufmann, Los Altos, CA; Zhu, J., Hastie, T., Rosset, S., Tibshirani, R., 2003. 1-norm support vector machines. Neural Inform. Process. Systems 16]. 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 L"q 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.