All types of part-of-speech (POS) tagging errors have been equally treated by existing taggers. However, the errors are not equally important, since some errors affect the performance of subsequent natural language processing seriously while others do not. This paper aims to minimize these serious errors while retaining the overall performance of POS tagging. Two gradient loss functions are proposed to reflect the different types of errors. They are designed to assign a larger cost for serious errors and a smaller cost for minor errors. Through a series of experiments, it is shown that the classifier trained with the proposed loss functions not only reduces serious errors but also achieves slightly higher accuracy than ordinary classifiers.