Foundations of the Scaling Law
Empirical scaling laws relate validation loss to model size, training exposure and compute, providing a quantitative basis for allocating resources. This technical note explains these relationships through three linked analyses. Geometric approximation connects spatial resolution, intrinsic dimension and smoothness to function error; the loss function determines how that error transfers to task risk. Spectral analysis combines feature visibility, covariance eigenvalues and target alignment to describe how additional data reduce unresolved predictive mass. Constrained optimization then gives the parameter-data balance and the training-inference trade-off. We also examine emergence and grok...
| Authors | Tianyu Zhao, Zhixiang Ren |
|---|---|
| Primary category | cs.LG |
| Published | 20 October 2026 |
| Last updated at source | 20 October 2026 |
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