Algebraic and Statistical Properties of the Partially Regularized Ordinary Least Squares Interpolator

Letian Yang; Dennis Shen

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arXiv · 2411.06593

Algebraic and Statistical Properties of the Partially Regularized Ordinary Least Squares Interpolator

Letian Yang·Dennis Shen

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Abstract

Modern deep learning has revealed a surprising statistical phenomenon known as benign overfitting, with high-dimensional linear regression being a prominent example. This paper contributes to ongoing research on the ordinary least squares (OLS) interpolator, focusing on the partial regression setting, where only a subset of coefficients is implicitly regularized. On the algebraic front, we extend Cochran's formula and the leave-one-out residual formula for the partial regularization framework. On the stochastic front, we leverage our algebraic results to design several homoskedastic variance estimators under the Gauss-Markov model. These estimators serve as a basis for conducting statistical inference, albeit with slight conservatism in their performance. Through simulations, we study the finite-sample properties of these variance estimators across various generative models.

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Letian Yang, Dennis Shen. 2024-11-10. Algebraic and Statistical Properties of the Partially Regularized Ordinary Least Squares Interpolator. https://arxiv.org/abs/2411.06593

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