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This research establishes near-optimal bounds for the Lipschitz constants of random ReLU networks, revealing a distinct mathematical divergence between $p \in [1,2)$ and $p \in [2,\infty]$ regimes. These findings provide critical theoretical insights into how network width and depth influence stability properties specifically in zero-bias versus non-zero-bias architectures. Such rigorous analysis is essential for advancing the understanding of generalization and robustness in deep learning theory.
Near-optimal estimates for the $\ell^p$-Lipschitz constants of deep random ReLU neural networks
arXiv:2506.19695v2 Announce Type: replace-cross Abstract: 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. In the case of zero-bias networks, 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 depth. 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 $\Vert g \Vert_{2}$. We extend our analysis to networks with possibly non-zero biases drawn from arbitrary symmetric distributions. In this case, we obtain high probability upper and lower bounds that differ at most by a factor that is logarithmic in the network's width and linear in its depth.
Abstract: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. In the case of zero-bias networks, 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 depth. 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 $\Vert g \Vert_{2}$. We extend our analysis to networks with possibly non-zero biases drawn from arbitrary symmetric distributions. In this case, we obtain high probability upper and lower bounds that differ at most by a factor that is logarithmic in the network's width and linear in its depth.
Machine Learning (stat.ML); Machine Learning (cs.LG); Probability (math.PR)
arXiv:2506.19695 [stat.ML]
(or arXiv:2506.19695v2 [stat.ML] for this version)
https://doi.org/10.48550/arXiv.2506.19695 Focus to learn more arXiv-issued DOI via DataCite
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