Spectral Asymptotics of Neural Network Loss Landscapes: An Exact Decomposition of the Curvature Exponent

arXiv:2606.02596v1 Announce Type: new Abstract: The curvature exponent $\alpha$ in $h_k \propto \sigma_k^\alpha$ -- governing how Hessian eigenvalues scale with gradient singular values -- varies systematically across layer types ($\alpha \approx 2$ for convolutions, $\approx 1$ for transformer attention, $< 1$ for MLP up-projections). Why? We prove the Spectral Alignment Decomposition: $\alpha = 2 + d\log\Phi_k / d\log\sigma_k$, where $\Phi_k$ measures alignment between Kronecker factor eigenbases and gradient singular directions. This reduces "why does $\alpha$ vary?" to a geometric question we answer for LayerNorm, residual connections, and softmax heads. The decomposition implies a spectral transfer identity $s = \alpha\gamma$ linking curvature exponent, effective gradient rank-decay $\gamma$, and Hessian decay exponent $s$. The identity is algebraic; its empirical content is that $\alpha$ and $\gamma$, fit on independent data (HVPs vs. SVD), recover $s$ to ~2% median error across 93 layers, five architectures, and three datasets -- with no free parameters. A zeta-function bound on participation ratio shows curvature concentrates onto effectively one direction per layer. As a proof of concept, we derive the architecture-adaptive preconditioner $T(\sigma;\alpha)$ and show that Spectral Newton -- implementing $T$ in the gradient singular basis -- outperforms AdamW on vision benchmarks where $\alpha \approx 2$.

Abstract:The curvature exponent $\alpha$ in $h_k \propto \sigma_k^\alpha$ -- governing how Hessian eigenvalues scale with gradient singular values -- varies systematically across layer types ($\alpha \approx 2$ for convolutions, $\approx 1$ for transformer attention, $< 1$ for MLP up-projections). Why? We prove the Spectral Alignment Decomposition: $\alpha = 2 + d\log\Phi_k / d\log\sigma_k$, where $\Phi_k$ measures alignment between Kronecker factor eigenbases and gradient singular directions. This reduces "why does $\alpha$ vary?" to a geometric question we answer for LayerNorm, residual connections, and softmax heads. The decomposition implies a spectral transfer identity $s = \alpha\gamma$ linking curvature exponent, effective gradient rank-decay $\gamma$, and Hessian decay exponent $s$. The identity is algebraic; its empirical content is that $\alpha$ and $\gamma$, fit on independent data (HVPs vs. SVD), recover $s$ to ~2% median error across 93 layers, five architectures, and three datasets -- with no free parameters. A zeta-function bound on participation ratio shows curvature concentrates onto effectively one direction per layer. As a proof of concept, we derive the architecture-adaptive preconditioner $T(\sigma;\alpha)$ and show that Spectral Newton -- implementing $T$ in the gradient singular basis -- outperforms AdamW on vision benchmarks where $\alpha \approx 2$.

13 pages, 6 figures, 3 tables. Code and data: this https URL

Machine Learning (cs.LG)

arXiv:2606.02596 [cs.LG]

(or arXiv:2606.02596v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2606.02596 Focus to learn more arXiv-issued DOI via DataCite

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Spectral Asymptotics of Neural Network Loss Landscapes: An Exact Decomposition of the Curvature Exponent

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