This paper investigates the phenomenon of benign overfitting in binary classification problems with heavy-tailed input distributions. We extend the analysis of maximum margin classifiers to $\alpha$ sub-exponential distributions, where $\alpha \in (0,2]$, generalizing previous work that focused on sub-gaussian inputs. Our main result provides generalization error bounds for linear classifiers trained using gradient descent on unregularized logistic loss in this heavy-tailed setting. We prove that under certain conditions on the dimensionality $p$ and feature vector magnitude $\|\mu\|$, the misclassification error of the maximum margin classifier asymptotically approaches the noise level. This work contributes to the understanding of benign overfitting in more robust distribution settings and demonstrates that the phenomenon persists even with heavier-tailed inputs than previously studied.

本文探讨了重尾输入分布下二分类问题中的良性过拟合现象，填补了在此领域的研究空白。我们扩展了最大边际分类器的分析，提出了对使用梯度下降训练线性分类器的泛化误差界限的理解，发现误分类错误会收敛到噪声水平。这项工作有助于在更稳健的分布设置中理解良性过拟合现象，并证明了即使在比以往研究中更重尾的输入条件下，该现象依然存在。

α子指数输入下的良性过拟合