Predictive Model Degrees of Freedom in Linear Regression

TL;DR

This paper introduces a new measure of model complexity for linear regression that explains the success of overparametrized models and reconciles the double descent phenomenon with classical theory.

Contribution

It proposes an adjusted degrees of freedom measure based on covariance, extending classical concepts to modern interpolating models.

Findings

The new measure differentiates among interpolating models.

It explains the double descent phenomenon.

It extends classical model complexity theory.

Abstract

Overparametrized interpolating models have drawn increasing attention from machine learning. Some recent studies suggest that regularized interpolating models can generalize well. This phenomenon seemingly contradicts the conventional wisdom that interpolation tends to overfit the data and performs poorly on test data. Further, it appears to defy the bias-variance trade-off. As one of the shortcomings of the existing theory, the classical notion of model degrees of freedom fails to explain the intrinsic difference among the interpolating models since it focuses on estimation of in-sample prediction error. This motivates an alternative measure of model complexity which can differentiate those interpolating models and take different test points into account. In particular, we propose a measure with a proper adjustment based on the squared covariance between the predictions and observations. Our analysis with least squares method reveals some interesting properties of the measure, which can reconcile the "double descent" phenomenon with the classical theory. This opens doors to an extended definition of model degrees of freedom in modern predictive settings.