Practical Scaling Laws: Converting Compute into Performance in a Data-Constrained World

The scaling laws guiding modern model training were calibrated for a single regime: data-rich, single-epoch pretraining. The dominant such scaling law form, Chinchilla's L = E + A/Nα + B/Dβ, has three structural limitations outside that regime: it diverges as unique data shrinks instead of saturating at the uninformed baseline; it cannot represent overfitting when capacity exceeds the data; and it conflates total examples seen with unique examples available. We propose a closed-form extension, L(N, D, T) = E + (L0 - E),h/(1+h) with h = a/Nα + b/Tβ + c,Nγ/Dδ, that decomposes loss into undercapacity, undertraining, and overfitting terms. It saturates between the irreducible loss E and an uninformed baseline L0 fixed by the loss type, and reduces to Chinchilla in the data-rich, single-epoch limit. We validate it on four multi-epoch experiments spanning four architecture families (MLPs, ResNets, Fourier neural operators, and transformers) across vision, scientific ML, and language domains, and refit it to five published LLM scaling-law grids. Extrapolating to higher compute and larger unique data than seen at fit time, our form achieves state-of-the-art RMSE on every published LLM grid we evaluate and on most cells of our constructed experiments. Once calibrated, the form admits a cost-aware allocation that recovers Chinchilla's optimum when data is free and shifts toward smaller corpora and more epochs as data grows expensive.