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

Jiping Li; Rishi Sonthalia

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arXiv · 2510.01414

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

Jiping Li·Rishi Sonthalia

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Abstract

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.

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Jiping Li, Rishi Sonthalia. 2025-10-01. Risk Phase Transitions in Spiked Regression: Alignment Driven Benign and Catastrophic Overfitting. https://arxiv.org/abs/2510.01414

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