Hinted Scaling Laws

Abstract

Scaling laws predict the outcome of large, expensive training runs by extrapolating trends from smaller, cheaper runs. On downstream accuracy evaluations, these predictions fail when the weaker models produced by the cheaper runs score near zero accuracy, because these scores give no trend to extrapolate to larger scales. We introduce hinted scaling laws, which use privileged information to turn a benchmark into a continuum of easier versions so that even the small models score above zero. We find that accuracy rises smoothly and predictably with the hint level, enabling new types of scaling laws that operate below the noise threshold. On the Delphi pretraining ladder, a hinted scaling law fit on models trained with at most $10^{20}$ FLOPs forecasts the GSM8K accuracy of a $10^{23}$ FLOP model (25.9%) to within 3 percentage points; a standard scaling law under-forecasts it by 13 percentage points when fit on the same models and by 12 even with models up to $10^{22}$ FLOPs in the fit. Across both this pretraining ladder and a collection of post-trained model families, hinted scaling laws reduce the median forecasting error on GSM8K, AIME and HLE from 21 percentage points to 4.3. Hinting also reduces the number of test examples needed to tell whether one model is better than another by widening the accuracy gaps between models while approximately preserving their order: a single hinted rollout per problem ranks nearly indistinguishable models as accurately as five unhinted rollouts. Our results suggest that hinting is a general primitive for improving the accuracy and cost of downstream scaling laws and other prediction tasks in model development.