# When Does Learning Renormalize? Sufficient Conditions for Power Law Spectral Dynamics

arXiv: https://arxiv.org/abs/2512.18209 · HTML: https://mlsanity.com/paper/2512.18209/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 29 of its 54 references that are in the map.
- First posted on arXiv 20 Dec 2025.

## Builds on (most cited first, 29 of 29)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
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- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [Mamba: Linear-Time Sequence Modeling with Selective State Spaces](https://mlsanity.com/paper/2312.00752/) (2023)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Training Compute-Optimal Large Language Models](https://mlsanity.com/paper/2203.15556/) (2022)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Efficiently Modeling Long Sequences with Structured State Spaces](https://mlsanity.com/paper/2111.00396/) (2021)
- [On Large-Batch Training for Deep Learning: Generalization Gap and Sharp Minima](https://mlsanity.com/paper/1609.04836/) (2016)
- [RWKV: Reinventing RNNs for the Transformer Era](https://mlsanity.com/paper/2305.13048/) (2023)
- [Simplified State Space Layers for Sequence Modeling](https://mlsanity.com/paper/2208.04933/) (2022)
- [Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient Descent](https://mlsanity.com/paper/1902.06720/) (2019)
- [Self-Normalizing Neural Networks](https://mlsanity.com/paper/1706.02515/) (2017)
- [Deep Learning Scaling is Predictable, Empirically](https://mlsanity.com/paper/1712.00409/) (2017)
- [Beyond neural scaling laws: beating power law scaling via data pruning](https://mlsanity.com/paper/2206.14486/) (2022)
- [Scaling Laws for Autoregressive Generative Modeling](https://mlsanity.com/paper/2010.14701/) (2020)
- [An Investigation into Neural Net Optimization via Hessian Eigenvalue Density](https://mlsanity.com/paper/1901.10159/) (2019)
- [A Dynamical Model of Neural Scaling Laws](https://mlsanity.com/paper/2402.01092/) (2024)
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- [How Feature Learning Can Improve Neural Scaling Laws](https://mlsanity.com/paper/2409.17858/) (2024)
- [From Lazy to Rich: Exact Learning Dynamics in Deep Linear Networks](https://mlsanity.com/paper/2409.14623/) (2024)
- [Dynamical Isometry is Achieved in Residual Networks in a Universal Way for any Activation Function](https://mlsanity.com/paper/1809.08848/) (2018)
- [Functional Scaling Laws in Kernel Regression: Loss Dynamics and Learning Rate Schedules](https://mlsanity.com/paper/2509.19189/) (2025)
- [Provable Scaling Laws of Feature Emergence from Learning Dynamics of Grokking](https://mlsanity.com/paper/2509.21519/) (2025)
- [AuroraLong: Bringing RNNs Back to Efficient Open-Ended Video Understanding](https://mlsanity.com/paper/2507.02591/) (2025)
- [A Generalized Spectral Framework to Expain Neural Scaling and Compression Dynamics](https://mlsanity.com/paper/2511.07892/) (2025)
- [A Framework for Overparameterized Learning](https://mlsanity.com/paper/2205.13507/) (2022)

## Abstract (arXiv)

Empirical power--law scaling has been widely observed across modern deep learning systems, yet its theoretical origins and scope of validity remain incompletely understood. The Generalized Resolution--Shell Dynamics (GRSD) framework models learning as spectral energy transport across logarithmic resolution shells, providing a coarse--grained dynamical description of training. Within GRSD, power--law scaling corresponds to a particularly simple renormalized shell dynamics; however, such behavior is not automatic and requires additional structural properties of the learning process. In this work, we identify a set of sufficient conditions under which the GRSD shell dynamics admits a renormalizable coarse--grained description. These conditions constrain the learning configuration at multiple levels, including boundedness of gradient propagation in the computation graph, weak functional incoherence at initialization, controlled Jacobian evolution along training, and log--shift invariance of renormalized shell couplings. We further show that power--law scaling does not follow from renormalizability alone, but instead arises as a rigidity consequence: once log--shift invariance is combined with the intrinsic time--rescaling covariance of gradient flow, the renormalized GRSD velocity field is forced into a power--law form.