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