How Width and Data Shape Generalization Scaling Laws in Quadratic Neural Networks
Researchers have mapped how neural network generalization scales jointly with model width and training data in a finite-sample regime, moving beyond the infinite-width and online-learning assumptions that dominate existing theory. By analyzing regularized loss minimization in quadratic two-layer networks with structured data, the work reveals distinct scaling phases and explicit error bounds tied to sample count, parameter count, and regularization strength. This bridges a gap between practical feature-learning models and rigorous scaling law characterization, offering practitioners and theorists a clearer picture of the width-data tradeoff that governs real training dynamics.
Modelwire context
ExplainerThe paper's core contribution is finite-sample error bounds that explicitly track the width-data tradeoff in a tractable setting. Most prior scaling law theory either assumes infinite width (where data is the only variable) or online learning (where samples arrive one at a time). This work pins down how both dimensions interact when both are finite and bounded.
This connects directly to the PAC-Bayesian control paper from the same day, which also extends finite-sample learning guarantees to a domain (closed-loop control) where traditional theory fell short. Both papers share a common thread: moving formal ML rigor into regimes practitioners actually operate in, rather than asymptotic limits. The nuclear physics interpretability work from yesterday also echoes this pattern, showing that structured inductive bias (whether from physics or from controlled architecture) can outperform black-box scaling when the problem has exploitable structure. Here, quadratic networks are that controlled setting.
If follow-up work extends these bounds to deeper networks or non-quadratic activations within the next 12 months and maintains similar explicitness about the width-data tradeoff, that signals the framework is generalizable. If the bounds remain tight only for quadratic networks, the contribution stays narrow and mostly of theoretical interest rather than predictive for real training dynamics.
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