Benign Overfitting in Linear Regression
This addresses the theoretical understanding of overfitting in machine learning, particularly for linear models, but is incremental as it builds on prior work on benign overfitting in deep learning.
The paper tackles the problem of when linear regression can perfectly fit noisy training data yet still predict accurately, known as benign overfitting, by characterizing conditions based on the effective rank of data covariance. It shows that overparameterization is essential, with the number of unimportant parameter directions needing to exceed sample size, and finds that finite-dimensional data leads to better accuracy over a narrower range of properties compared to infinite-dimensional cases.
The phenomenon of benign overfitting is one of the key mysteries uncovered by deep learning methodology: deep neural networks seem to predict well, even with a perfect fit to noisy training data. Motivated by this phenomenon, we consider when a perfect fit to training data in linear regression is compatible with accurate prediction. We give a characterization of linear regression problems for which the minimum norm interpolating prediction rule has near-optimal prediction accuracy. The characterization is in terms of two notions of the effective rank of the data covariance. It shows that overparameterization is essential for benign overfitting in this setting: the number of directions in parameter space that are unimportant for prediction must significantly exceed the sample size. By studying examples of data covariance properties that this characterization shows are required for benign overfitting, we find an important role for finite-dimensional data: the accuracy of the minimum norm interpolating prediction rule approaches the best possible accuracy for a much narrower range of properties of the data distribution when the data lies in an infinite dimensional space versus when the data lies in a finite dimensional space whose dimension grows faster than the sample size.