Lipschitz verification of neural networks through training

• URL: http://arxiv.org/abs/2603.28113v1
• Date: Mon, 30 Mar 2026 07:20:50 GMT
• Title: Lipschitz verification of neural networks through training
• Abstract summary: We show that penalizing the trivial bound during training forces it to become tight, thereby effectively regularizing the true Lipschitz constant.<n>Our approach avoids the runtime complexity of advanced verification while achieving strong results.
• Score: 3.8708566551482537
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The global Lipschitz constant of a neural network governs both adversarial robustness and generalization. Conventional approaches to ``certified training" typically follow a train-then-verify paradigm: they train a network and then attempt to bound its Lipschitz constant. Because the efficient ``trivial bound" (the product of the layerwise Lipschitz constants) is exponentially loose for arbitrary networks, these approaches must rely on computationally expensive techniques such as semidefinite programming, mixed-integer programming, or branch-and-bound. We propose a different paradigm: rather than designing complex verifiers for arbitrary networks, we design networks to be verifiable by the fast trivial bound. We show that directly penalizing the trivial bound during training forces it to become tight, thereby effectively regularizing the true Lipschitz constant. To achieve this, we identify three structural obstructions to a tight trivial bound (dead neurons, bias terms, and ill-conditioned weights) and introduce architectural mitigations, including a novel notion of norm-saturating polyactivations and bias-free sinusoidal layers. Our approach avoids the runtime complexity of advanced verification while achieving strong results: we train robust networks on MNIST with Lipschitz bounds that are small (orders of magnitude lower than comparable works) and tight (within 10% of the ground truth). The experimental results validate the theoretical guarantees, support the proposed mechanisms, and extend empirically to diverse activations and non-Euclidean norms.

Related papers

• ExLipBaB: Exact Lipschitz Constant Computation for Piecewise Linear Neural Networks [0.0]
  Lipschitz constant can be leveraged to derive robustness guarantees.<n>We propose a generalization of the LipBaB algorithm to compute exact Lipschitz constants for arbitrary piecewise linear neural networks.
  arXiv  Detail & Related papers  (2026-02-17T11:09:41Z)
• A Near Complete Nonasymptotic Generalization Theory For Multilayer Neural Networks: Beyond the Bias-Variance Tradeoff [57.25901375384457]
  We propose a nonasymptotic generalization theory for multilayer neural networks with arbitrary Lipschitz activations and general Lipschitz loss functions.<n>In particular, it doens't require the boundness of loss function, as commonly assumed in the literature.<n>We show the near minimax optimality of our theory for multilayer ReLU networks for regression problems.
  arXiv  Detail & Related papers  (2025-03-03T23:34:12Z)
• Efficient Bound of Lipschitz Constant for Convolutional Layers by Gram Iteration [122.51142131506639]
  We introduce a precise, fast, and differentiable upper bound for the spectral norm of convolutional layers using circulant matrix theory. We show through a comprehensive set of experiments that our approach outperforms other state-of-the-art methods in terms of precision, computational cost, and scalability. It proves highly effective for the Lipschitz regularization of convolutional neural networks, with competitive results against concurrent approaches.
  arXiv  Detail & Related papers  (2023-05-25T15:32:21Z)
• Efficiently Computing Local Lipschitz Constants of Neural Networks via Bound Propagation [79.13041340708395]
  Lipschitz constants are connected to many properties of neural networks, such as robustness, fairness, and generalization. Existing methods for computing Lipschitz constants either produce relatively loose upper bounds or are limited to small networks. We develop an efficient framework for computing the $ell_infty$ local Lipschitz constant of a neural network by tightly upper bounding the norm of Clarke Jacobian.
  arXiv  Detail & Related papers  (2022-10-13T22:23:22Z)
• Rethinking Lipschitz Neural Networks for Certified L-infinity Robustness [33.72713778392896]
  We study certified $ell_infty$ from a novel perspective of representing Boolean functions. We develop a unified Lipschitz network that generalizes prior works, and design a practical version that can be efficiently trained.
  arXiv  Detail & Related papers  (2022-10-04T17:55:27Z)
• Training Certifiably Robust Neural Networks with Efficient Local Lipschitz Bounds [99.23098204458336]
  Certified robustness is a desirable property for deep neural networks in safety-critical applications. We show that our method consistently outperforms state-of-the-art methods on MNIST and TinyNet datasets.
  arXiv  Detail & Related papers  (2021-11-02T06:44:10Z)
• Robust Implicit Networks via Non-Euclidean Contractions [63.91638306025768]
  Implicit neural networks show improved accuracy and significant reduction in memory consumption. They can suffer from ill-posedness and convergence instability. This paper provides a new framework to design well-posed and robust implicit neural networks.
  arXiv  Detail & Related papers  (2021-06-06T18:05:02Z)
• Lipschitz Bounded Equilibrium Networks [3.2872586139884623]
  This paper introduces new parameterizations of equilibrium neural networks, i.e. networks defined by implicit equations. The new parameterization admits a Lipschitz bound during training via unconstrained optimization. In image classification experiments we show that the Lipschitz bounds are very accurate and improve robustness to adversarial attacks.
  arXiv  Detail & Related papers  (2020-10-05T01:00:40Z)
• On Lipschitz Regularization of Convolutional Layers using Toeplitz Matrix Theory [77.18089185140767]
  Lipschitz regularity is established as a key property of modern deep learning. computing the exact value of the Lipschitz constant of a neural network is known to be NP-hard. We introduce a new upper bound for convolutional layers that is both tight and easy to compute.
  arXiv  Detail & Related papers  (2020-06-15T13:23:34Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.