Scaling Laws and Spectra of Shallow Neural Networks in the Feature Learning Regime
read the original abstract
Neural scaling laws underlie many of the recent advances in deep learning, yet their theoretical understanding remains largely confined to linear models. In this work, we present a systematic analysis of scaling laws for quadratic and diagonal neural networks in the feature learning regime. Leveraging connections with matrix compressed sensing and LASSO, we derive a detailed phase diagram for the scaling exponents of the excess risk as a function of sample complexity and weight decay. This analysis uncovers crossovers between distinct scaling regimes and plateau behaviors, mirroring phenomena widely reported in the empirical neural scaling literature. Furthermore, we establish a precise link between these regimes and the spectral properties of the trained network weights, which we characterize in detail. As a consequence, we provide a theoretical validation of recent empirical observations connecting the emergence of power-law tails in the weight spectrum with network generalization performance, yielding an interpretation from first principles.
This paper has not been read by Pith yet.
Forward citations
Cited by 7 Pith papers
-
Sharp feature-learning transitions and Bayes-optimal neural scaling laws in extensive-width networks
In extensive-width networks, features are recovered sequentially through sharp phase transitions, yielding an effective width k_c that unifies Bayes-optimal generalization error scaling as Θ(k_c d / n).
-
Spectral Dynamics in Deep Networks: Feature Learning, Outlier Escape, and Learning Rate Transfer
A two-level DMFT predicts width-consistent outlier escape and hyperparameter transfer under μP in deep networks, with bulk restructuring dominating for tasks with many outputs.
-
A Fourier perspective on the learning dynamics of neural networks: from sample complexities to mechanistic insights
Neural networks prioritize amplitude over phase in Fourier space during training on translation-invariant data; power-law spectra accelerate phase learning despite not aiding classification.
-
Spectral Dynamics in Deep Networks: Feature Learning, Outlier Escape, and Learning Rate Transfer
A two-level DMFT tracks bulk and outlier spectral dynamics in wide networks, predicting width-consistent outlier growth and hyperparameter transfer under muP scaling for deep linear nets while noting bulk restructurin...
-
Asymmetric Scaling Laws from Sparse Features
A sparse-activation model predicts double-descent loss with distinct under- and over-parameterized scaling exponents set by sparsity, plus a compute-optimal frontier favoring dataset growth.
-
HTMuon: Improving Muon via Heavy-Tailed Spectral Correction
HTMuon modifies Muon to produce heavier-tailed updates and weight spectra via HT-SR theory, yielding up to 0.98 lower perplexity on LLaMA pretraining and serving as a plug-in for other Muon variants.
-
There Will Be a Scientific Theory of Deep Learning
A mechanics of the learning process is emerging in deep learning theory, characterized by dynamics, coarse statistics, and falsifiable predictions across idealized settings, limits, laws, hyperparameters, and universa...
discussion (0)
Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.