Overparameterization and generalization error: weighted trigonometric interpolation
This provides theoretical insight into why overparameterized models like deep neural networks generalize well, addressing a fundamental problem in machine learning theory.
The paper analyzes how smoothness affects generalization error in overparameterized linear learning by studying a random Fourier series model, showing that weighted trigonometric interpolation reduces generalization error in overparameterized regimes compared to underparameterized ones.
Motivated by surprisingly good generalization properties of learned deep neural networks in overparameterized scenarios and by the related double descent phenomenon, this paper analyzes the relation between smoothness and low generalization error in an overparameterized linear learning problem. We study a random Fourier series model, where the task is to estimate the unknown Fourier coefficients from equidistant samples. We derive exact expressions for the generalization error of both plain and weighted least squares estimators. We show precisely how a bias towards smooth interpolants, in the form of weighted trigonometric interpolation, can lead to smaller generalization error in the overparameterized regime compared to the underparameterized regime. This provides insight into the power of overparameterization, which is common in modern machine learning.