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 (NNs), 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 in 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.