Algebraic and Statistical Properties of the Ordinary Least Squares Interpolator

TL;DR

This paper explores the algebraic and statistical properties of the minimum $\\ell_2$-norm OLS interpolator in high-dimensional, overparameterized settings, providing new theoretical insights and simulations to understand its generalization and causal inference capabilities.

Contribution

It offers the first algebraic equivalents of classical statistical formulas in the overparameterized regime and extends key statistical theorems to this setting.

Findings

Algebraic equivalents of residual formulas and theorems in overparameterized models.

Extended Gauss-Markov theorem for high-dimensional OLS.

Simulation results illustrating stochastic properties of the OLS interpolator.

Abstract

Deep learning research has uncovered the phenomenon of benign overfitting for overparameterized statistical models, which has drawn significant theoretical interest in recent years. Given its simplicity and practicality, the ordinary least squares (OLS) interpolator has become essential to gain foundational insights into this phenomenon. While properties of OLS are well established in classical, underparameterized settings, its behavior in high-dimensional, overparameterized regimes is less explored (unlike for ridge or lasso regression) though significant progress has been made of late. We contribute to this growing literature by providing fundamental algebraic and statistical results for the minimum $\ell_2$-norm OLS interpolator. In particular, we provide algebraic equivalents of (i) the leave-$k$-out residual formula, (ii) Cochran's formula, and (iii) the Frisch-Waugh-Lovell theorem in the overparameterized regime. These results aid in understanding the OLS interpolator's ability to generalize and have substantive implications for causal inference. Under the Gauss-Markov model, we present statistical results such as an extension of the Gauss-Markov theorem and an analysis of variance estimation under homoskedastic errors for the overparameterized regime. To substantiate our theoretical contributions, we conduct simulations that further explore the stochastic properties of the OLS interpolator.