A Full Adam Theorem for Spectral Heavy-Tail Onset
Abstract
We prove a full Adam theorem for spectral heavy-tail onset in a closed Gaussian Stein-Hermite teacher-student state-evolution model. The theorem begins with the actual full-batch Adam recurrences, derives the population gradient by Stein-Hermite calculus, proves finite-width covariance concentration, converts multi-step Adam momentum into an exact non-centered Gaussian sign kernel, controls the diagonal Adam denominator by a basis-homogenization theorem, derives a regularly varying projected update response from a Hermite edge-transfer theorem, pushes the response through the exact Gram update, and proves approximate-target KL contraction with matching upper and lower hitting bounds. The final law is (τ_\varepsilon=Θ(Δ_1^{-γ}d^ρ\log(Ψ_0/\varepsilon))), where (Δ_1) is the first spike-bulk spectral gap. The result is full in the following precise sense: every step from Adam's momentum and denominator to the spectral hitting law is formalized inside the closed state-evolution model. We also prove that a stronger arbitrary-gradient Adam theorem is impossible, and that exact two-step linear-network loss dynamics do not identify factor spectra or heavy-tail hitting times.
Related
- Local limit Theorem for Directed polymers in a random environment in the full weak disorder regime
- Spectral aspects of random heavy-tailed tensors
- THE OPTIMAL EIGENVALUE COUNT IN AUBRY'S SPECTRAL SPHERE THEOREM FOR DIMENSIONS AT LEAST FOUR
- A general theorem in spectral extremal graph theory
- Gradient estimates for the Green kernel under spectral Ricci bounds, and the stable Bernstein theorem in ℝ⁴
BibTeX
@misc{liu2026full,
title = {{A Full Adam Theorem for Spectral Heavy-Tail Onset}},
author = {Zongmin Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12996},
eprint = {2609.12996},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12996},
}
RIS
TY - UNPB
TI - A Full Adam Theorem for Spectral Heavy-Tail Onset
AU - Zongmin Liu
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.12996
UR - https://arxiv.org/abs/2609.12996
AB - We prove a full Adam theorem for spectral heavy-tail onset in a closed Gaussian Stein-Hermite teacher-student state-evolution model. The theorem begins with the actual full-batch Adam recurrences, derives the population gradient by Stein-Hermite calculus, proves finite-width covariance concentration, converts multi-step Adam momentum into an exact non-centered Gaussian sign kernel, controls the diagonal Adam denominator by a basis-homogenization theorem, derives a regularly varying projected update response from a Hermite edge-transfer theorem, pushes the response through the exact Gram update, and proves approximate-target KL contraction with matching upper and lower hitting bounds. The final law is (τ_\varepsilon=Θ(Δ_1^{-γ}d^ρ\log(Ψ_0/\varepsilon))), where (Δ_1) is the first spike-bulk spectral gap. The result is full in the following precise sense: every step from Adam's momentum and denominator to the spectral hitting law is formalized inside the closed state-evolution model. We also prove that a stronger arbitrary-gradient Adam theorem is impossible, and that exact two-step linear-network loss dynamics do not identify factor spectra or heavy-tail hitting times.
ER -