Calibration of ϵ−insensitive loss in support vector machines regression

Support vector machines regression (SVMR) is an important tool in many machine learning applications. In this paper, we focus on the theoretical understanding of SVMR based on the ϵ−insensitive loss. For fixed ϵ ≥ 0 and general data generating distributions, we show that the minimizer of the expected risk for ϵ−insensitive loss used in SVMR is a set-valued function called conditional ϵ−median. We then establish a calibration inequality of ϵ−insensitive loss under a noise condition on the conditional distributions. This inequality also ensures us to present a nontrivial variance-expectation bound for ϵ−insensitive loss, and which is known to be important in statistical analysis of the regularized learning algorithms. With the help of the calibration inequality and variance-expectation bound, we finally derive an explicit learning rate for SVMR in some L ^r −space.