This paper investigates the phenomenon of benign overfitting in binary classification problems with heavy-tailed input distributions, extending the analysis of maximum margin classifiers to $\alpha$ sub-exponential distributions ($\alpha \in (0, 2]$). This generalizes previous work focused on sub-gaussian inputs. We provide generalization error bounds for linear classifiers trained using gradient descent on unregularized logistic loss in this heavy-tailed setting. Our results show that, under certain conditions on the dimensionality $p$ and the distance between the centers of the distributions, the misclassification error of the maximum margin classifier asymptotically approaches the noise level, the theoretical optimal value. Moreover, we derive an upper bound on the learning rate $\beta$ for benign overfitting to occur and show that as the tail heaviness of the input distribution $\alpha$ increases, the upper bound on the learning rate decreases. These results demonstrate that benign overfitting persists even in settings with heavier-tailed inputs than previously studied, contributing to a deeper understanding of the phenomenon in more realistic data environments.

翻译：本文研究了具有重尾输入分布的二元分类问题中的良性过拟合现象，将最大间隔分类器的分析推广到α次指数分布（α ∈ (0, 2]）。这推广了先前专注于次高斯输入的工作。我们在此重尾设定下，为使用梯度下降法在无正则化逻辑损失上训练的线性分类器提供了泛化误差界。我们的结果表明，在维度p和分布中心间距离的某些条件下，最大间隔分类器的误分类误差渐近地逼近噪声水平，即理论最优值。此外，我们推导了良性过拟合发生时学习率β的一个上界，并表明随着输入分布尾部厚重程度α的增加，学习率的上界会减小。这些结果证明，即使在比先前研究更重尾的输入设定中，良性过拟合依然存在，这有助于在更现实的数据环境中更深入地理解该现象。