Loss-to-Loss Prediction: Scaling Laws for All Datasets
Authors: name: David Brandfonbrener, name: Nikhil Anand, name: Nikhil Vyas, name: Eran Malach, name: Sham Kakade

While scaling laws provide a reliable methodology for predicting train loss
across compute scales for a single data distribution, less is known about how
these predictions should change as we change the distribution. In this paper,
we derive a strategy for predicting one loss from another and apply it to
predict across different pre-training datasets and from pre-training data to
downstream task data. Our predictions extrapolate well even at 20x the largest
FLOP budget used to fit the curves. More precisely, we find that there are
simple shifted power law relationships between (1) the train losses of two
models trained on two separate datasets when the models are paired by training
compute (train-to-train), (2) the train loss and the test loss on any
downstream distribution for a single model (train-to-test), and (3) the test
losses of two models trained on two separate train datasets (test-to-test). The
results hold up for pre-training datasets that differ substantially (some are
entirely code and others have no code at all) and across a variety of
downstream tasks. Finally, we find that in some settings these shifted power
law relationships can yield more accurate predictions than extrapolating
single-dataset scaling laws.