Risk Bounds for Over-parameterized Maximum Margin Classification on Sub-Gaussian Mixtures
This work addresses the theoretical understanding of benign overfitting in machine learning, which is incremental as it builds on existing studies of over-parameterized models.
The paper tackles the problem of understanding when over-parameterized maximum margin classifiers achieve small test errors despite fitting noisy training data, by providing a tight risk bound for linear classification on sub-Gaussian mixtures. The results characterize conditions for benign overfitting and improve on prior work, with implications for logistic regression.
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.