One of the major paradigms for supervised learning in neural network community is back-propagation learning. The standard implementations of back-propagation learning are optimal under the assumptions of identical and independent Gaussian noise. In this paper, for regression function estimation, we introduce $varepsilon-insensitive$ back-propagation learning algorithm, which corresponds to minimizing the least absolute error. We compare this algorithm with support vector machine(SVM), which is another $varepsilon-insensitive$ supervised learning algorithm and has been very successful in pattern recognition and function estimation problems. For comparison, we consider a more realistic model would allow the noise variance itself to depend on the input variables.