Solution Path for Pin-SVM Classifiers With Positive and Negative $\tau $ Values

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