On the Optimizer Dependence of Neural Scaling Laws

The scaling exponent $\alpha$ in neural scaling laws $L(N) \propto N^{-\alpha}$ is commonly treated as a fixed constant set by architecture and data. We present evidence that $\alpha$ depends systematically on the optimizer. In controlled random-feature regression experiments---the canonical theoretical framework for neural scaling---we measure $\alpha$ across five optimizer variants and six spectral conditions. Preconditioned optimizers consistently yield steeper scaling (larger $\alpha$), with the $\alpha$-shift growing monotonically with the spectral decay rate $s$ of the data covariance. At $s \approx 1.0$ (characteristic of natural language), the full natural gradient achieves $\alpha \approx 0.31$ versus $\alpha \approx 0.12$ for gradient descent---a $2.6\times$ improvement that, within the random-feature model, compounds with each model-size doubling. Whether and how this exponent shift transfers to large-scale LLM training---where recent evidence suggests the advantage may attenuate with scale---remains an important open question. Our results imply that scaling-law forecasts should account for optimizer choice, and we provide a spectral diagnostic predicting when advanced optimizers will pay off.