  For small training set sizes $P$, the generalization error of wide neural
networks is well-approximated by the error of an infinite width neural network
(NN), either in the kernel or mean-field/feature-learning regime. However,
after a critical sample size $P^*$, we empirically find the finite-width
network generalization becomes worse than that of the infinite width network.
In this work, we empirically study the transition from infinite-width behavior
to this variance limited regime as a function of sample size $P$ and network
width $N$. We find that finite-size effects can become relevant for very small
dataset sizes on the order of $P^* \sim \sqrt{N}$ for polynomial regression
with ReLU networks. We discuss the source of these effects using an argument
based on the variance of the NN's final neural tangent kernel (NTK). This
transition can be pushed to larger $P$ by enhancing feature learning or by
ensemble averaging the networks. We find that the learning curve for regression
with the final NTK is an accurate approximation of the NN learning curve. Using
this, we provide a toy model which also exhibits $P^* \sim \sqrt{N}$ scaling
and has $P$-dependent benefits from feature learning.
