A Rescaling-Invariant Lipschitz Bound Based on Path-Metrics for Modern ReLU Network Parameterizations

Antoine Gonon, Nicolas Brisebarre, Elisa Riccietti, Rémi Gribonval
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:20047-20074, 2025.

Abstract

Robustness with respect to weight perturbations underpins guarantees for generalization, pruning and quantization. Existing guarantees rely on Lipschitz bounds in parameter space, cover only plain feed-forward MLPs, and break under the ubiquitous neuron-wise rescaling symmetry of ReLU networks. We prove a new Lipschitz inequality expressed through the $\ell^{1}$-path-metric of the weights. The bound is (i) rescaling-invariant by construction and (ii) applies to any ReLU-DAG architecture with any combination of convolutions, skip connections, pooling, and frozen (inference-time) batch-normalization —thus encompassing ResNets, U-Nets, VGG-style CNNs, and more. By respecting the network’s natural symmetries, the new bound strictly sharpens prior parameter-space bounds and can be computed in two forward passes. To illustrate its utility, we derive from it a symmetry-aware pruning criterion and show—through a proof-of-concept experiment on a ResNet-18 trained on ImageNet—that its pruning performance matches that of classical magnitude pruning, while becoming totally immune to arbitrary neuron-wise rescalings.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-gonon25a, title = {A Rescaling-Invariant {L}ipschitz Bound Based on Path-Metrics for Modern {R}e{LU} Network Parameterizations}, author = {Gonon, Antoine and Brisebarre, Nicolas and Riccietti, Elisa and Gribonval, R\'{e}mi}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {20047--20074}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/gonon25a/gonon25a.pdf}, url = {https://proceedings.mlr.press/v267/gonon25a.html}, abstract = {Robustness with respect to weight perturbations underpins guarantees for generalization, pruning and quantization. Existing guarantees rely on Lipschitz bounds in parameter space, cover only plain feed-forward MLPs, and break under the ubiquitous neuron-wise rescaling symmetry of ReLU networks. We prove a new Lipschitz inequality expressed through the $\ell^{1}$-path-metric of the weights. The bound is (i) rescaling-invariant by construction and (ii) applies to any ReLU-DAG architecture with any combination of convolutions, skip connections, pooling, and frozen (inference-time) batch-normalization —thus encompassing ResNets, U-Nets, VGG-style CNNs, and more. By respecting the network’s natural symmetries, the new bound strictly sharpens prior parameter-space bounds and can be computed in two forward passes. To illustrate its utility, we derive from it a symmetry-aware pruning criterion and show—through a proof-of-concept experiment on a ResNet-18 trained on ImageNet—that its pruning performance matches that of classical magnitude pruning, while becoming totally immune to arbitrary neuron-wise rescalings.} }
Endnote
%0 Conference Paper %T A Rescaling-Invariant Lipschitz Bound Based on Path-Metrics for Modern ReLU Network Parameterizations %A Antoine Gonon %A Nicolas Brisebarre %A Elisa Riccietti %A Rémi Gribonval %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-gonon25a %I PMLR %P 20047--20074 %U https://proceedings.mlr.press/v267/gonon25a.html %V 267 %X Robustness with respect to weight perturbations underpins guarantees for generalization, pruning and quantization. Existing guarantees rely on Lipschitz bounds in parameter space, cover only plain feed-forward MLPs, and break under the ubiquitous neuron-wise rescaling symmetry of ReLU networks. We prove a new Lipschitz inequality expressed through the $\ell^{1}$-path-metric of the weights. The bound is (i) rescaling-invariant by construction and (ii) applies to any ReLU-DAG architecture with any combination of convolutions, skip connections, pooling, and frozen (inference-time) batch-normalization —thus encompassing ResNets, U-Nets, VGG-style CNNs, and more. By respecting the network’s natural symmetries, the new bound strictly sharpens prior parameter-space bounds and can be computed in two forward passes. To illustrate its utility, we derive from it a symmetry-aware pruning criterion and show—through a proof-of-concept experiment on a ResNet-18 trained on ImageNet—that its pruning performance matches that of classical magnitude pruning, while becoming totally immune to arbitrary neuron-wise rescalings.
APA
Gonon, A., Brisebarre, N., Riccietti, E. & Gribonval, R.. (2025). A Rescaling-Invariant Lipschitz Bound Based on Path-Metrics for Modern ReLU Network Parameterizations. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:20047-20074 Available from https://proceedings.mlr.press/v267/gonon25a.html.

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