  We provide quantitative bounds measuring the $L^2$ difference in function
space between the trajectory of a finite-width network trained on finitely many
samples from the idealized kernel dynamics of infinite width and infinite data.
An implication of the bounds is that the network is biased to learn the top
eigenfunctions of the Neural Tangent Kernel not just on the training set but
over the entire input space. This bias depends on the model architecture and
input distribution alone and thus does not depend on the target function which
does not need to be in the RKHS of the kernel. The result is valid for deep
architectures with fully connected, convolutional, and residual layers.
Furthermore the width does not need to grow polynomially with the number of
samples in order to obtain high probability bounds up to a stopping time. The
proof exploits the low-effective-rank property of the Fisher Information Matrix
at initialization, which implies a low effective dimension of the model (far
smaller than the number of parameters). We conclude that local capacity control
from the low effective rank of the Fisher Information Matrix is still
underexplored theoretically.
