Quadratic Weak-to-Strong Generalization in Random Feature Networks via Random Matrix Theory
Abstract
Weak-to-strong generalization is the phenomenon where a strong student model trained with labels produced by a weak teacher model is able to generalize better than the teacher. In this paper, we study this phenomenon in two-layer random feature networks where the model strength is determined by its width. Using tools from random matrix theory, we derive deterministic equivalents for the population errors of an optimally trained teacher and a student trained with gradient flow. For ReLU activation and a pure spherical harmonic target, we obtain sharp asymptotics under a Gaussian universality assumption, showing a quadratic improvement: the student error scales as the square of the teacher error. These results attain the general lower bound of Medvedev at al (2025). We also analyze how the student behaves under more general stopping times and targets supported on multiple harmonic degrees, characterizing the regimes in which weak-to-strong generalization occurs and identifying the transition between quadratic, non-quadratic, and no improvement.
Related
- A Smoothed Discrepancy Principle for Random Feature Methods and Neural Networks
- Physics Informed Random Feature Neural Networks for Solving PDEs
- Characterizing Identifiability and Generalization for Inverse Receding-Horizon Linear-Quadratic Regulator Problems
- Generalization in Nonlinear Least Squares via Learned Feature Geometry
- An Active-Bottleneck Mechanism for Weak-to-Strong Generalization
BibTeX
@misc{oliveira2026quadratic,
title = {{Quadratic Weak-to-Strong Generalization in Random Feature Networks via Random Matrix Theory}},
author = {Deborah Oliveira and Elliot Paquette},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09044},
eprint = {2610.09044},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09044},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Quadratic Weak-to-Strong Generalization in Random Feature Networks via Random Matrix Theory
AU - Deborah Oliveira
AU - Elliot Paquette
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.09044
UR - https://arxiv.org/abs/2610.09044
AB - Weak-to-strong generalization is the phenomenon where a strong student model trained with labels produced by a weak teacher model is able to generalize better than the teacher. In this paper, we study this phenomenon in two-layer random feature networks where the model strength is determined by its width. Using tools from random matrix theory, we derive deterministic equivalents for the population errors of an optimally trained teacher and a student trained with gradient flow. For ReLU activation and a pure spherical harmonic target, we obtain sharp asymptotics under a Gaussian universality assumption, showing a quadratic improvement: the student error scales as the square of the teacher error. These results attain the general lower bound of Medvedev at al (2025). We also analyze how the student behaves under more general stopping times and targets supported on multiple harmonic degrees, characterizing the regimes in which weak-to-strong generalization occurs and identifying the transition between quadratic, non-quadratic, and no improvement.
ER -