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arXiv:2610.36113

Scaling and Renormalization in Neural Networks: An Effective Renormalization-Group Description of Grokking

Min Chen, Junqi Wang, Zeguan Wu, Wenhao Huang, Junyu Liu

cs.LGcs.AIhep-thquant-phSubmitted Oct 26, 2026• Announced Oct 28, 2026
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Abstract

Grokking is a delayed transition from memorization to generalization in deep neural networks (NNs). We examine whether scaling and renormalization group (RG) theories can describe this dynamics, establishing a quantitative physical framework for neural generalization transitions. For modular addition on a two-layer NN, an ensemble of trained models defines an order parameter (the squared weight norm) and fluctuation proxies for the heat capacity and the susceptibility, with training epoch and weight decay serving as thermal and field control coordinates. To bridge neural architectures to real-space RG, we construct an exact coupling matrix via a second-order expansion of the loss with respect to per-neuron norms and use its Fiedler vector to define an effective geometry. This spectral ordering yields a monotonic spatial decay of coupling strength that is absent under random permutations. We then execute real-space RG by block-spin aggregating adjacent neurons and parameterizing the coupling matrix into off-diagonal band strengths. Linearizing this local RG map yields relevant scaling exponents and five derived critical exponents, with direct empirical estimates matching RG predictions within 1-10%. Overall, our framework provides a consistent effective description of grokking, suggesting the scaling and renormalization group formalism as a tool to investigate NN dynamics.

Comments
22 pages, 10 figures

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Grokking is a delayed transition from memorization to generalization in deep neural networks (NNs). We examine whether scaling and renormalization group (RG) theories can describe this dynamics, establishing a quantitative physical framework for neural generalization transitions. For modular addition on a two-layer NN, an ensemble of trained models defines an order parameter (the squared weight norm) and fluctuation proxies for the heat capacity and the susceptibility, with training epoch and weight decay serving as thermal and field control coordinates. To bridge neural architectures to real-space RG, we construct an exact coupling matrix via a second-order expansion of the loss with respect to per-neuron norms and use its Fiedler vector to define an effective geometry. This spectral ordering yields a monotonic spatial decay of coupling strength that is absent under random permutations. We then execute real-space RG by block-spin aggregating adjacent neurons and parameterizing the coupling matrix into off-diagonal band strengths. Linearizing this local RG map yields relevant scaling exponents and five derived critical exponents, with direct empirical estimates matching RG predictions within 1-10%. Overall, our framework provides a consistent effective description of grokking, suggesting the scaling and renormalization group formalism as a tool to investigate NN dynamics.
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We investigate the analogy between the renormalization group (RG) and deep neural networks, wherein subsequent layers of neurons are analogous to successive steps along the RG. In particular, we quantify the flow of information by explicitly computing the relative entropy or Kullback-Leibler divergence in both the one- and two-dimensional Ising models under decimation RG, as well as in a feedforward neural network as a function of depth. We observe qualitatively identical behavior characterized by the monotonic increase to a parameter-dependent asymptotic value. On the quantum field theory side, the monotonic increase confirms the connection between the relative entropy and the c-theorem. For the neural networks, the asymptotic behavior may have implications for various information maximization methods in machine learning, as well as for disentangling compactness and generalizability. Furthermore, while both the two-dimensional Ising model and the random neural networks we consider exhibit non-trivial critical points, the relative entropy appears insensitive to the phase structure of either system. In this sense, more refined probes are required in order to fully elucidate the flow of information in these models.
Deep neural network architectures often consist of repetitive structural elements. We introduce an approach that reveals these patterns and can be broadly applied to the study of deep learning. Similarly to how a power strip helps untangle and organize complex cable connections, this approach treats neurons as additional degrees of freedom in interactions, simplifying the structure and enhancing the intuitive understanding of interactions within deep neural networks. Furthermore, it reveals the translational symmetry of deep neural networks, which simplifies the application of the renormalization group transformation, a method that effectively analyzes the scaling behavior of the system. By utilizing translational symmetry and renormalization group transformations, we can analyze critical phenomena. This approach may open new avenues for studying deep neural networks using statistical physics.
Even though deep learning has proved to be very powerful as the core method of machine learning, theoretical understanding behind its success is still unclear. It is been pointed out in recent years that the behaviour of deep neural networks is reminiscent of a fundamental framework in statistical physics: the renormalization group (RG). Motivated by this analogy, we develop an analytical method of directly obtaining the trained weights W resulting from a training set of 1D-Ising samples with coupling J and temperature T. The found relation W(J) drives a flow that takes 1D-Ising configurations at non-zero temperature to the critical point at T = 0. This behaviour is opposite to what a typical RG-flow dictates.
The key idea behind the renormalization group (RG) transformation is that properties of physical systems with very different microscopic makeups can be characterized by a few universal parameters.However, finding a systematic way to construct RG transformation for particular systems remains difficult due to the many possible choices of the weight factors in the RG procedure.Here we show, by identifying the conditional distribution in the restricted Boltzmann machine and the weight factor distribution in the RG procedure, that a valid real-space RG transformation can be learned without prior knowledge of the physical system.This neural Monte Carlo RG algorithm allows for direct computation of the RG flow and critical exponents.Our results establish a solid connection between the RG transformation in physics and the deep architecture in machine learning, paving the way for further interdisciplinary research.