On Lipschitz Explosion in Deep Neural Networks with Normalization: Consequences for Optimization and Robustness

The Lipschitz constant of a neural network is a basic stability quantity appearing in optimization guarantees, generalization bounds, and robustness certificates. Deep networks naturally admit two such notions, input Lipschitzness, measuring sensitivity to data perturbations, and parameter Lipschitzness, measuring sensitivity to weight perturbations. We prove that, for deep networks with normalization layers, with batch normalization as the canonical example, both Lipschitz constants can grow exponentially with depth and hence with parameter dimension, even under strong per-layer norm control such as $$\begin{gathered} |W_{\ell } \\ {|}_2\le 1 \end{gathered}$$, and already for linear activations. Thus, a mechanism widely used to stabilize training can create severe worst-case instabilities that are invisible from layerwise norm bounds alone. Parameter-Lipschitz explosion turns Lipschitz-dependent nonsmooth optimization guarantees into exponential-in-dimension bounds for deep normalized networks, while input-Lipschitz explosion makes worst-case Lipschitz-based generalization and robustness certificates vacuous. The latter also yields a concrete rank-separation mechanism for adversarial vulnerability. The theory predicts that perturbations introducing new singular directions should be amplified much more strongly than equal-energy perturbations that remain within the input's original singular subspace. Experiments on MNIST, Fashion-MNIST, and CIFAR-10 support this prediction, showing that rank-creating perturbations cause substantially sharper drops in accuracy, confidence, and margins than same-subspace perturbations.