Prescriptive Scaling Laws for Data Constrained Training

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 $L = E + A/N^α+ B/D^β$, 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, $L(N, D, T) = E + (L_0 - E)\,h/(1+h)$ with $h = a/N^α+ b/T^β+ c\,N^γ/D^δ$, that decomposes loss into undercapacity, undertraining, and overfitting terms. It saturates between the irreducible loss $E$ and an uninformed baseline $L_0$ 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.

Classical compute-optimal scaling laws assume an unbounded supply of fresh pretraining data, yet pretraining is increasingly entering a regime in which compute grows faster than the availability of high-quality data. We propose Compute-Data (CD) scaling laws, a unified framework that bridges compute-optimal scaling, where data scales freely with compute, and data-optimal scaling, where the corpus is fixed while compute can grow without bound. CD scaling extends classical scaling laws by introducing a token-effectiveness function, $η$, which quantifies the value of a derived token-produced, for example, through multi-epoch repetition or paraphrasing-relative to a fresh token, ranging from a perfect substitute to having no value. We fit $η$ for two data-expansion strategies, multi-epoch repetition and paraphrasing, across model sizes from 14M to 600M parameters using the Dolma-3 corpus. We find that token effectiveness is far from constant: it depends jointly on model size, the tokens-per-parameter ratio, and the amount of derived data, and it saturates as the corpus is expanded. The functional form of $η$ implies diminishing returns when substituting compute for data as either model size or data availability increases. It also partitions training into three operational regimes---compute-bound, data-bound, and model-bound---and shows that classical compute-optimal allocation is suboptimal across most practically relevant settings.

Upweighting high-quality data in LLM pretraining often improves performance, but in datalimited regimes, especially under overtraining, stronger upweighting increases repetition and can degrade performance. However, standard scaling laws do not reliably extrapolate across mixture recipes or under repetitions, making the selection for optimal data recipes at scaling underdetermined. To solve this, we introduce InfoLaw (Information Scaling Laws), a data-aware scaling framework that predicts loss from consumed tokens, model size, data mixture weights, and repetition. The key idea is to model pretraining as information accumulation, where quality controls information density and repetition induces scaledependent diminishing returns. We first collect the model performance after training on datasets that vary in scale, quality distribution, and repetition level. Then we build up the modeling for information so that information accurately predicts those model performance. InfoLaw predicts performance on unseen data recipes and larger scale runs (up to 7B, 425B tokens) with 0.15% mean and 0.96% max absolute error in loss, and it extrapolates reliably across overtraining levels, enabling efficient data-recipe selection under varying compute budgets.