Scaling Laws and Spectra of Shallow Neural Networks in the Feature Learning Regime
Researchers present a theoretical framework analyzing scaling laws for shallow neural networks in the feature learning regime, deriving phase diagrams that connect sample complexity and weight decay to risk exponents. The work bridges empirical observations in deep learning with rigorous mathematical analysis, establishing links between weight spectrum properties and generalization performance through matrix compressed sensing and LASSO theory.
This research addresses a fundamental gap in deep learning theory by extending scaling law analysis beyond linear models to quadratic and diagonal neural networks. While empirical scaling laws have driven recent AI advances, theoretical understanding has lagged significantly. The authors leverage mathematical tools from compressed sensing to characterize how neural networks transition between different scaling regimes as a function of data availability and regularization strength.
The phase diagram framework represents a substantial contribution because it explains previously observed phenomena—crossovers between scaling regimes and plateau behaviors—through first principles mathematics. These phenomena have been extensively documented in empirical studies but lacked rigorous theoretical justification. By connecting scaling exponents to spectral properties of trained weights, the research validates earlier empirical findings that power-law tails in weight spectra correlate with better generalization.
For the AI research community, this work provides theoretical grounding that could inform architecture design and hyperparameter selection. Understanding the precise relationship between weight decay, sample size, and performance enables more principled optimization strategies. The connections drawn with matrix compressed sensing open pathways for applying established signal processing theory to neural network analysis.
The research indicates that theoretical foundations for neural scaling are becoming increasingly rigorous, potentially accelerating progress in understanding when and why deep learning systems succeed. Future work likely builds on these phase diagrams to characterize deeper networks and more complex architectures, moving toward practical applicability for model development.
- →Researchers derive detailed phase diagrams characterizing how neural network scaling exponents vary with sample complexity and weight decay regularization
- →The analysis establishes mathematical connections between weight spectrum properties and generalization performance through compressed sensing theory
- →The work theoretically validates empirical observations linking power-law tails in weight spectra to improved network generalization
- →Phase transitions between scaling regimes explain previously empirical phenomena using rigorous mathematical foundations
- →Results apply to quadratic and diagonal neural networks, extending theoretical understanding beyond existing linear model analyses