Deriving Neural Scaling Laws from the Statistics of Natural Language

Despite the fact that experimental neural scaling laws have substantially guided empirical progress in large-scale machine learning, no existing theory can quantitatively predict the exponents of these important laws for any modern LLM trained on any natural language dataset. We provide the first such theory in the case of data-limited scaling laws. We isolate two key statistical properties of language that alone can predict neural scaling exponents: (i) the decay of pairwise token correlations with time separation between token pairs, and (ii) the decay of the next-token conditional entropy with the length of the conditioning context. We further derive a simple formula in terms of these statistics that predicts data-limited neural scaling exponents from first principles without any free parameters or synthetic data models. Our theory exhibits a remarkable match with experimentally measured neural scaling laws obtained from training GPT-2 and LLaMA style models from scratch on two qualitatively different benchmarks, TinyStories and WikiText.

Large language models improve in predictable ways when they are trained on more data. Thus far, researchers have mostly discovered these patterns empirically, by running expensive experiments. Where do these scaling laws come from? Can be predicted directly from the structure of language itself? In this work, we show that two simple measurements of text are enough to predict how quickly a language model improves as it sees more data. The first measures how related tokens are to one another across distance in a sentence or document. The second measures how much easier it becomes to guess the next token as the model is given more preceding tokens. Using these measurements, we derive a formula that predicts scaling laws without needing fitted parameters or simplified artificial datasets.