Overparameterization and generalization error: weighted trigonometric interpolation

TL;DR

This paper investigates how overparameterization and smoothness influence generalization error in linear models, revealing that weighted trigonometric interpolation can improve generalization in overparameterized settings.

Contribution

It provides exact formulas for generalization error in a Fourier series model and demonstrates how bias towards smooth interpolants benefits overparameterized learning.

Findings

Weighted trigonometric interpolation reduces generalization error in overparameterized regimes.

Exact expressions for generalization error in Fourier coefficient estimation.

Overparameterization can be advantageous for smoothness-based generalization.

Abstract

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.