From Zipf's Law to Neural Scaling through Heaps' Law and Hilberg's Hypothesis

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.

Neural scaling laws establish a predictable relationship between model performance and data or compute, offering crucial guidance for resource allocation in new domains and tasks. Yet such laws are most needed precisely where they are hardest to obtain: fitting one for a new model task pair demands expensive sweeps that typically exhaust the very compute budget the law is meant to economize. This paper poses the research question of how to develop generalizable scaling laws: laws fit once on a well-resourced source domain and reliably transported to new domains where running a full sweep is infeasible, which requires a fundamental understanding of when and why scaling properties change. We address this by identifying the right invariants: scaling laws are preserved under bijective (information-preserving) transformations of the data and modified in predictable, information-theoretically grounded ways under non-bijective transformations that lower its information resolution $ρ$: a single axis along which a law fit in one domain can be transported to another. We validate this across language, vision, and speech, and demonstrate two cross-domain applications: predicting scaling for language models trained on electronic health records from laws fit on general text, and predicting time-series classification scaling under varying levels of noise injection, recovering the data-scaling exponents to within $3\%$ error.

Neural scaling laws describe how pre-training loss decays as power laws with training time, model size, and compute. This position paper argues that the exponents of these power laws are fixed by generic mechanisms: a one-third time scaling due to the strong nonlinearity of Softmax, an inverse width scaling due to representational superposition, and an inverse depth scaling due to ensemble averaging of Transformer layers. These mechanisms are robust to a wide range of data structures and architectural details, placing current large language models in a universality class with fixed exponents. The coefficients, however, are expected to be sensitive to data and architecture details, and directly determine practical quantities such as the optimal model shape and the compute-optimal frontier. We therefore argue that understanding the coefficients is the key to near-term performance improvements, and that a closer examination of the current universality class may reveal pathways to better universality classes.