  State-of-the-art neural networks require extreme computational power to
train. It is therefore natural to wonder whether they are optimally trained.
Here we apply a recent advancement in stochastic thermodynamics which allows
bounding the speed at which one can go from the initial weight distribution to
the final distribution of the fully trained network, based on the ratio of
their Wasserstein-2 distance and the entropy production rate of the dynamical
process connecting them. Considering both gradient-flow and Langevin training
dynamics, we provide analytical expressions for these speed limits for linear
and linearizable neural networks e.g. Neural Tangent Kernel (NTK). Remarkably,
given some plausible scaling assumptions on the NTK spectra and spectral
decomposition of the labels -- learning is optimal in a scaling sense. Our
results are consistent with small-scale experiments with Convolutional Neural
Networks (CNNs) and Fully Connected Neural networks (FCNs) on CIFAR-10, showing
a short highly non-optimal regime followed by a longer optimal regime.
