  We study the effect of imperfect training data labels on the performance of
classification methods. In a general setting, where the probability that an
observation in the training dataset is mislabelled may depend on both the
feature vector and the true label, we bound the excess risk of an arbitrary
classifier trained with imperfect labels in terms of its excess risk for
predicting a noisy label. This reveals conditions under which a classifier
trained with imperfect labels remains consistent for classifying uncorrupted
test data points. Furthermore, under stronger conditions, we derive detailed
asymptotic properties for the popular $k$-nearest neighbour ($k$nn), support
vector machine (SVM) and linear discriminant analysis (LDA) classifiers. One
consequence of these results is that the knn and SVM classifiers are robust to
imperfect training labels, in the sense that the rate of convergence of the
excess risks of these classifiers remains unchanged; in fact, our theoretical
and empirical results even show that in some cases, imperfect labels may
improve the performance of these methods. On the other hand, the LDA classifier
is shown to be typically inconsistent in the presence of label noise unless the
prior probabilities of each class are equal. Our theoretical results are
supported by a simulation study.
