Learning Curve Theory

Recently a number of empirical "universal" scaling law papers have been published, most notably by OpenAI. `Scaling laws' refers to power-law decreases of training or test error w.r.t. more data, larger neural networks, and/or more compute. In this work we focus on scaling w.r.t. data size n. Theoretical understanding of this phenomenon is largely lacking, except in finite-dimensional models for which error typically decreases with n^−1/2 or n^−1, where n is the sample size. We develop and theoretically analyse the simplest possible (toy) model that can exhibit n^−β learning curves for arbitrary power β>0, and determine whether power laws are universal or depend on the data distribution.