Depth Dependence of $\mu$P Learning Rates in ReLU MLPs

In this short note we consider random fully connected ReLU networks of width $n$ and depth $L$ equipped with a mean-field weight initialization. Our purpose is to study the dependence on $n$ and $L$ of the maximal update ($\mu$P) learning rate, the largest learning rate for which the mean squared change in pre-activations after one step of gradient descent remains uniformly bounded at large $n,L$. As in prior work on $\mu$P of Yang et. al., we find that this maximal update learning rate is independent of $n$ for all but the first and last layer weights. However, we find that it has a non-trivial dependence of $L$, scaling like $L^{-3\mathrm{/}2}\mathrm{.}$