A Flatness-Generalization Relation in the Teacher-Student Tree-Committee Machine
Abstract
The flatness of the loss landscape at a minimizer is a widely used heuristic for reasoning about neural-network generalization, yet evidence for this relation is mostly empirical and controversial. We study this relation in a teacher-student tree committee machine, where both the ERM estimator and the Hessian spectrum are analytically tractable in the proportional high-dimensional limit. First, we use a zero-temperature Gibbs formulation to obtain predictions for the observables of the typical minimizers of the empirical loss. Secondly, we use Edwards-Jones formalism to derive the limiting Hessian resolvent around these typical minimizers. All predictions agree with finite-size gradient-descent simulations. Finally, we study three measures of flatness, namely the left and right edges and the spectral mean, and check if a decrease in generalization error as the dataset size is increased corresponds to an increase in flatness. We find that the answer strongly depends on the learning task and on the ratio of the number of parameters to the number of data points. In regression, the spectral mean and right edge correlate with the generalization error, while the left edge does so only in the overparametrized regime. In classification this correlation reliably holds only in the highly overparametrized phase, while for underparametrized networks it can even reverse.
Related
BibTeX
@misc{annesi2026flatnessgeneralization,
title = {{A Flatness-Generalization Relation in the Teacher-Student Tree-Committee Machine}},
author = {Brandon Livio Annesi and Davide Straziota and Enrico Maria Malatesta},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31101},
eprint = {2609.31101},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31101},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Flatness-Generalization Relation in the Teacher-Student Tree-Committee Machine
AU - Brandon Livio Annesi
AU - Davide Straziota
AU - Enrico Maria Malatesta
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.31101
UR - https://arxiv.org/abs/2609.31101
AB - The flatness of the loss landscape at a minimizer is a widely used heuristic for reasoning about neural-network generalization, yet evidence for this relation is mostly empirical and controversial. We study this relation in a teacher-student tree committee machine, where both the ERM estimator and the Hessian spectrum are analytically tractable in the proportional high-dimensional limit. First, we use a zero-temperature Gibbs formulation to obtain predictions for the observables of the typical minimizers of the empirical loss. Secondly, we use Edwards-Jones formalism to derive the limiting Hessian resolvent around these typical minimizers. All predictions agree with finite-size gradient-descent simulations. Finally, we study three measures of flatness, namely the left and right edges and the spectral mean, and check if a decrease in generalization error as the dataset size is increased corresponds to an increase in flatness. We find that the answer strongly depends on the learning task and on the ratio of the number of parameters to the number of data points. In regression, the spectral mean and right edge correlate with the generalization error, while the left edge does so only in the overparametrized regime. In classification this correlation reliably holds only in the highly overparametrized phase, while for underparametrized networks it can even reverse.
ER -