标题： 风险相变在尖峰回归中的表现：对齐驱动的良性过拟合和灾难性过拟合 显示英文标题

标题： Risk Phase Transitions in Spiked Regression: Alignment Driven Benign and Catastrophic Overfitting

摘要： 本文分析了在尖峰协方差数据模型下最小范数插值解的泛化误差。 本文描述了尖峰强度和目标-尖峰对齐的变化如何影响风险，尤其是在过参数化设置中。 该研究给出了泛化误差的精确表达式，根据尖峰强度、纵横比$c=d/n$（特别是$c \to \infty$）和目标对齐情况，对良性、温和和灾难性过拟合区域进行了全面分类。 值得注意的是，在正确指定的对齐问题中，增加尖峰强度可能会在达到良性过拟合之前引发灾难性过拟合。 本文还揭示了目标-尖峰对齐并不总是有利的，指出了其有益或有害的具体有时甚至出人意料的条件。 与尖峰对齐不利的情况在非线性模型中被实证证明仍然存在。 显示英文摘要

摘要： This paper analyzes the generalization error of minimum-norm interpolating solutions in linear regression using spiked covariance data models. The paper characterizes how varying spike strengths and target-spike alignments can affect risk, especially in overparameterized settings. The study presents an exact expression for the generalization error, leading to a comprehensive classification of benign, tempered, and catastrophic overfitting regimes based on spike strength, the aspect ratio $c=d/n$ (particularly as $c \to \infty$), and target alignment. Notably, in well-specified aligned problems, increasing spike strength can surprisingly induce catastrophic overfitting before achieving benign overfitting. The paper also reveals that target-spike alignment is not always advantageous, identifying specific, sometimes counterintuitive, conditions for its benefit or detriment. Alignment with the spike being detrimental is empirically demonstrated to persist in nonlinear models.