Tianyu Zhao

Classical neural scaling laws write excess risk as a power law in capacity and data and assume the compute-optimal allocation exponent stays fixed across scales. This analysis uses Mellin–Barnes contour integration to split finite-scale departures along the capacity axis into a three-term expansion, reduces multi-domain mixing to scalar water-filling via KKT duality, and shows the allocation exponent drifts in proportion to capacity and data exponent variances. Across 45 operator spectra, Pythia, Chinchilla's 245 runs and RegMix's 17 domains, the estimators give holdout error and cut high-compute allocation regret.

Classical neural scaling laws write excess risk as a power law in capacity and data and assume the compute-optimal allocation exponent stays fixed across scales. This analysis uses Mellin–Barnes contour integration to split finite-scale departures along the capacity axis into a three-term expansion, reduces multi-domain mixing to scalar water-filling via KKT duality, and shows the allocation exponent drifts in proportion to capacity and data exponent variances. Across 45 operator spectra, Pythia, Chinchilla's 245 runs and RegMix's 17 domains, the estimators give holdout error and cut high-compute allocation regret.