# The double descent and Runge phenomena in overparametrized polynomial interpolation

arXiv: https://arxiv.org/abs/2609.37657 · HTML: https://mlsanity.com/paper/2609.37657/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 24 references that are in the map.
- First posted on arXiv 29 Sep 2026.

## Builds on (most cited first, 11 of 11)

- [Reconciling modern machine learning practice and the bias-variance trade-off](https://mlsanity.com/paper/1812.11118/) (2018)
- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [Two models of double descent for weak features](https://mlsanity.com/paper/1903.07571/) (2019)
- [Spurious Local Minima are Common in Two-Layer ReLU Neural Networks](https://mlsanity.com/paper/1712.08968/) (2017)
- [Towards a Mathematical Understanding of Neural Network-Based Machine Learning: what we know and what we don't](https://mlsanity.com/paper/2009.10713/) (2020)
- [Representation formulas and pointwise properties for Barron functions](https://mlsanity.com/paper/2006.05982/) (2020)
- [A Comparative Analysis of the Optimization and Generalization Property of Two-layer Neural Network and Random Feature Models Under Gradient Descent Dynamics](https://mlsanity.com/paper/1904.04326/) (2019)
- [A Brief Prehistory of Double Descent](https://mlsanity.com/paper/2004.04328/) (2020)
- [The Slow Deterioration of the Generalization Error of the Random Feature Model](https://mlsanity.com/paper/2008.05621/) (2020)
- [Stable Minima of ReLU Neural Networks Suffer from the Curse of Dimensionality: The Neural Shattering Phenomenon](https://mlsanity.com/paper/2506.20779/) (2025)
- [Minimum norm interpolation by perceptra: Explicit regularization and implicit bias](https://mlsanity.com/paper/2311.06138/) (2023)

## Abstract (arXiv)

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 ℓ2-norm (and, for the monomial basis, also those minimal in the ℓ1-norm). We present our results primarily for equidistant and Chebyshev data points, but many results are independent of the exact form of sampling.