Solution Path for Pin-SVM Classifiers With Positive and Negative τ Values

Applying the pinball loss in a support vector machine (SVM) classifier results in pin-SVM. The pinball loss is characterized by a parameter tau. Its value is related to the quantile level and different tau values are suitable for different problems. In this paper, we establish an algorithm to find the entire solution path for pin-SVM with different tau values. This algorithm is based on the fact that the optimal solution to pin-SVM is continuous and piecewise linear with respect to tau. We also show that the nonnegativity constraint on tau is not necessary, i.e., tau can be extended to negative values. First, in some applications, a negative tau leads to better accuracy. Second, tau = -1 corresponds to a simple solution that links SVM and the classical kernel rule. The solution for tau = -1 can be obtained directly and then be used as a starting point of the solution path. The proposed method efficiently traverses tau values through the solution path, and then achieves good performance by a suitable tau. In particular, tau = 0 corresponds to C-SVM, meaning that the traversal algorithm can output a result at least as good as C-SVM with respect to validation error.