Spectral Edge Dynamics: An Analytical-Empirical Study of Phase Transitions in Neural Network Training
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866915985710120960 |
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| author | Xu, Yongzhong |
| author_facet | Xu, Yongzhong |
| contents | We develop the spectral edge analysis: phase transitions in neural network training -- grokking, capability gains, loss plateaus -- are controlled by the spectral gap of the rolling-window Gram matrix of parameter updates. In the extreme aspect ratio regime (parameters $P \sim 10^8$, window $W \sim 10$), the classical BBP detection threshold is vacuous; the operative structure is the intra-signal gap separating dominant from subdominant modes at position $k^* = \mathrm{argmax}\, σ_j/σ_{j+1}$.
From three assumptions we derive: (i) gap dynamics governed by a Dyson-type ODE with curvature asymmetry, damping, and gradient driving; (ii) a spectral loss decomposition linking each mode's learning contribution to its Davis--Kahan stability coefficient; (iii) the Gap Maximality Principle, showing that $k^*$ is the unique dynamically privileged position -- its collapse is the only one that disrupts learning, and it sustains itself through an $α$-feedback loop requiring no assumption on the optimizer. The adiabatic parameter $\mathcal{A} = \|ΔG\|_F / (η\, g^2)$ controls circuit stability: $\mathcal{A} \ll 1$ (plateau), $\mathcal{A} \sim 1$ (phase transition), $\mathcal{A} \gg 1$ (forgetting).
Tested across six model families (150K--124M parameters): gap dynamics precede every grokking event (24/24 with weight decay, 1/24 without), the gap position is optimizer-dependent (Muon: $k^*=1$, AdamW: $k^*=2$ on the same model), and 19/20 quantitative predictions are confirmed. The framework is consistent with the edge of stability, Tensor Programs, Dyson Brownian motion, the Lottery Ticket Hypothesis, and neural scaling laws. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_28964 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Spectral Edge Dynamics: An Analytical-Empirical Study of Phase Transitions in Neural Network Training Xu, Yongzhong Machine Learning Artificial Intelligence We develop the spectral edge analysis: phase transitions in neural network training -- grokking, capability gains, loss plateaus -- are controlled by the spectral gap of the rolling-window Gram matrix of parameter updates. In the extreme aspect ratio regime (parameters $P \sim 10^8$, window $W \sim 10$), the classical BBP detection threshold is vacuous; the operative structure is the intra-signal gap separating dominant from subdominant modes at position $k^* = \mathrm{argmax}\, σ_j/σ_{j+1}$. From three assumptions we derive: (i) gap dynamics governed by a Dyson-type ODE with curvature asymmetry, damping, and gradient driving; (ii) a spectral loss decomposition linking each mode's learning contribution to its Davis--Kahan stability coefficient; (iii) the Gap Maximality Principle, showing that $k^*$ is the unique dynamically privileged position -- its collapse is the only one that disrupts learning, and it sustains itself through an $α$-feedback loop requiring no assumption on the optimizer. The adiabatic parameter $\mathcal{A} = \|ΔG\|_F / (η\, g^2)$ controls circuit stability: $\mathcal{A} \ll 1$ (plateau), $\mathcal{A} \sim 1$ (phase transition), $\mathcal{A} \gg 1$ (forgetting). Tested across six model families (150K--124M parameters): gap dynamics precede every grokking event (24/24 with weight decay, 1/24 without), the gap position is optimizer-dependent (Muon: $k^*=1$, AdamW: $k^*=2$ on the same model), and 19/20 quantitative predictions are confirmed. The framework is consistent with the edge of stability, Tensor Programs, Dyson Brownian motion, the Lottery Ticket Hypothesis, and neural scaling laws. |
| title | Spectral Edge Dynamics: An Analytical-Empirical Study of Phase Transitions in Neural Network Training |
| topic | Machine Learning Artificial Intelligence |
| url | https://arxiv.org/abs/2603.28964 |