  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" (Bartlett et al. (2020)) 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.
