  Modern machine learning systems such as deep neural networks are often highly
over-parameterized so that they can fit the noisy training data exactly, yet
they can still achieve small test errors in practice. In this paper, we study
this "benign overfitting" phenomenon of the maximum margin classifier for
linear classification problems. Specifically, we consider data generated from
sub-Gaussian mixtures, and provide a tight risk bound for the maximum margin
linear classifier in the over-parameterized setting. Our results precisely
characterize the condition under which benign overfitting can occur in linear
classification problems, and improve on previous work. They also have direct
implications for over-parameterized logistic regression.
