Neural Scaling Laws Rooted in the Data Distribution

TL;DR

This paper proposes a mathematical model based on percolation theory to explain the universal power-law neural scaling laws observed across various neural network architectures and tasks, unifying prior theories.

Contribution

It introduces a novel percolation-based framework that describes natural datasets and explains the emergence of neural scaling laws, linking them to data distribution properties.

Findings

Two criticality regimes yield optimal power-law scaling.

The regimes relate to data subtasks and data manifolds.

The theory is validated with toy dataset experiments.

Abstract

Deep neural networks exhibit empirical neural scaling laws, with error decreasing as a power law with increasing model or data size, across a wide variety of architectures, tasks, and datasets. This universality suggests that scaling laws may result from general properties of natural learning tasks. We develop a mathematical model intended to describe natural datasets using percolation theory. Two distinct criticality regimes emerge, each yielding optimal power-law neural scaling laws. These regimes, corresponding to power-law-distributed discrete subtasks and a dominant data manifold, can be associated with previously proposed theories of neural scaling, thereby grounding and unifying prior works. We test the theory by training regression models on toy datasets derived from percolation theory simulations. We suggest directions for quantitatively predicting language model scaling.