  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 $n^{-\beta}$ learning curves
for arbitrary power $\beta>0$, and determine whether power laws are universal
or depend on the data distribution.
