AI Insight
Mathematicians have derived predictable bounds for the Lipschitz constants of deep ReLU neural networks with randomly initialized weights, finding that stability behavior differs markedly depending on the mathematical norm used. For networks using He initialization, the research shows that Lipschitz constants follow different patterns in two regimes: for p-norms where p is between 1 and 2, stability scales similarly to Euclidean norms, while for p between 2 and infinity, it follows a different Gaussian distribution pattern. The bounds derived are tight, differing only by logarithmic factors in network depth for zero-bias networks, and by factors logarithmic in width and linear in depth for networks with biases.
Why it matters
This work provides mathematical foundations for predicting neural network stability during training and deployment, which is crucial for ensuring reliable performance in safety-critical applications like medical diagnosis or autonomous systems. Understanding these stability limits could help practitioners choose appropriate network architectures and initialization schemes to avoid training instabilities or unpredictable behavior.
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⚠️ Preprint – Noch nicht peer-reviewed
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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 $Vert gVert_{p’}$, 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.
Source: Near-optimal estimates for the $ell^p$-Lipschitz constants of deep random ReLU neural networks