The double descent and Runge phenomena in overparametrized polynomial interpolation

2026-09-29

AI Summary

This study investigates the intrinsic relationship between the Runge phenomenon and the double descent phenomenon in overparameterized polynomial interpolation. Methodologically, it systematically compares overparameterized interpolation behaviors under equispaced and Chebyshev sampling strategies by employing minimum l2- and l1-norm optimization across three bases: monomial, Chebyshev, and Legendre. The analysis demonstrates that the effects of different sampling strategies on interpolation behavior are independent of one another. The core contribution of this work lies in revealing a deep theoretical connection between the classical Runge phenomenon in numerical analysis and the modern double descent effect in machine learning. By establishing this link, the paper provides a novel perspective for understanding the generalization mechanisms of overparameterized models.

Abstract

The Runge phenomenon in polynomial interpolation is often considered a classical analogue of the double descent phenomenon in machine learning. In this note, we explore overparameterized polynomial interpolation in three popular polynomial bases: Monomial, Chebyshev and Legendre basis with coefficients that are minimal in the $\ell^2$-norm (and, for the monomial basis, also those minimal in the $\ell^1$-norm). We present our results primarily for equidistant and Chebyshev data points, but many results are independent of the exact form of sampling.

Problem

• double descent
• Runge phenomenon
• overparameterized polynomial interpolation
• polynomial bases

Innovation

• double descent
• Runge phenomenon
• overparameterized polynomial interpolation
• minimal norm coefficients
• polynomial bases