Effective Frontiers: A Unification of Neural Scaling Laws

Neural scaling laws govern the prediction power-law improvement of test loss with respect to model capacity ($N$), datasize ($D$), and compute ($C$). However, existing theoretical explanations often rely on specific architectures or complex kernel methods, lacking intuitive universality. In this paper, we propose a unified framework that abstracts general learning tasks as the progressive coverage of patterns from a long-tail (Zipfian) distribution. We introduce the Effective Frontier ($k_\star$), a threshold in the pattern rank space that separates learned knowledge from the unlearned tail. We prove that reducible loss is asymptotically determined by the probability mass of the tail a resource-dependent frontier truncation. Based on our framework, we derive the precise scaling laws for $N$, $D$, and $C$, attributing them to capacity, coverage, and optimization bottlenecks, respectively. Furthermore, we unify these mechanisms via a Max-Bottleneck principle, demonstrating that the Kaplan and Chinchilla scaling laws are not contradictory, but equilibrium solutions to the same constrained optimization problem under different active bottlenecks.