Near-optimal estimates for the $\ell^p$-Lipschitz constants of deep random ReLU neural networks

This paper studies the $\ell^p$-Lipschitz constants of ReLU neural networks $\Phi: \mathbb{R}^d \to \mathbb{R}$ with random parameters for $p \in [1,\infty]$. The distribution of the weights follows a variant of the He initialization and the biases are drawn from symmetric distributions. We derive high probability upper and lower bounds for wide networks that differ at most by a factor that is logarithmic in the network's width and linear in its depth. In the special case of shallow networks, we obtain matching bounds. Remarkably, the behavior of the $\ell^p$-Lipschitz constant varies significantly between the regimes $ p \in [1,2) $ and $ p \in [2,\infty] $. For $p \in [2,\infty]$, the $\ell^p$-Lipschitz constant behaves similarly to $\Vert g\Vert_{p'}$, where $g \in \mathbb{R}^d$ is a $d$-dimensional standard Gaussian vector and $1/p + 1/p' = 1$. In contrast, for $p \in [1,2)$, the $\ell^p$-Lipschitz constant aligns more closely to $\Vert g \Vert_{2}$.