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This is an unreviewed arXiv preprint proposing a theoretical framework for why trained neural network layers often show heavy-tailed power-law weight spectra with exponent near 2. It links that observation to renormalization-group ideas, a proposed SETOL condition, and a new optimizer extension, WW-PGD. The claims are largely hypotheses, so readers should treat the α≈2 heuristic and the optimizer results as candidates needing independent empirical replication across architectures and training setups.
A Spectral Renormalization-Group View of Learning
arXiv:2611.00422v1 Announce Type: cross Abstract: We study how the layers in deep neural networks (NNs) converge, using Wilsonian Renormalization-Group (RG) ideas to organize the spectral picture. During training, these layers often develop heavy-tailed power-law (PL) spectra, and the strongest checkpoints often appear near the PL exponent $\alpha\approx 2$. This pattern raises a simple question: why should this value mark a well-trained layer? The answer is both statistical and physical. Our statistical hypothesis is that $\alpha=2$ marks the boundary between stable if not optimal learning and overfitting via non-self-averaging. The physical answer is that the same value makes the spectral energy gain scale evenly across logarithmic spectral scales, which is the proposed Wilsonian marginal condition. We then apply the ERG trace-log condition, the proposed SETOL theory, which we justify by taking volume-preserving, scale-invariant transformations to remove arbitrary layer scale along the proposed RG flow. The resulting spectral RG flow connects the HTSR and SETOL theories to optimizers such as Muon, and motivates a new optimizer extension, WW-PGD. Together, these results point toward a spectral Renormalization-Group view of learning, in which NNs may approach shared spectral structure in the settings studied here.
Abstract:We study how the layers in deep neural networks (NNs) converge, using Wilsonian Renormalization-Group (RG) ideas to organize the spectral picture. During training, these layers often develop heavy-tailed power-law (PL) spectra, and the strongest checkpoints often appear near the PL exponent $\alpha\approx 2$. This pattern raises a simple question: why should this value mark a well-trained layer? The answer is both statistical and physical. Our statistical hypothesis is that $\alpha=2$ marks the boundary between stable if not optimal learning and overfitting via non-self-averaging. The physical answer is that the same value makes the spectral energy gain scale evenly across logarithmic spectral scales, which is the proposed Wilsonian marginal condition. We then apply the ERG trace-log condition, the proposed SETOL theory, which we justify by taking volume-preserving, scale-invariant transformations to remove arbitrary layer scale along the proposed RG flow. The resulting spectral RG flow connects the HTSR and SETOL theories to optimizers such as Muon, and motivates a new optimizer extension, WW-PGD. Together, these results point toward a spectral Renormalization-Group view of learning, in which NNs may approach shared spectral structure in the settings studied here.
Machine Learning (cs.LG); Disordered Systems and Neural Networks (cond-mat.dis-nn); Artificial Intelligence (cs.AI)
arXiv:2611.00422 [cs.LG]
(or arXiv:2611.00422v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2611.00422 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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