We investigate kernel-based quantile regression based on the pinball loss and support vector regression based on the $\e$-insensitive loss. Conditions are given which quarantee that the set of exact minimizers contains only one function. Some results about oracle inequalities and learning rates of these methods are presented. We show that the stopping criteria used in many support vector machine (SVM) algorithms working on the dual can be interpreted as primal optimality bounds which in turn are known to be important for the statistical analysis of SVMs. To this end we revisit the duality theory underlying the derivation of the dual and show that in many interesting cases primal optimality bounds are the same as known dual optimality bounds.
I. Steinwart and A. Christmann, How SVMs can Estimate Quantiles and the Median. In Neural Information Processing Systems 20, pp. 305-312, 2008. Los Alamos National Laboratory Technical Report LA-UR-07-6041, 2007. [ Abstract | PDF (211 KB) ]
