# A Scalable Approach for Safe and Robust Learning via Lipschitz-Constrained Networks

arXiv: https://arxiv.org/abs/2506.23977 · HTML: https://mlsanity.com/paper/2506.23977/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 24 of its 28 references that are in the map.
- First posted on arXiv 30 Jun 2025.

## Builds on (most cited first, 24 of 24)

- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Sequence to Sequence Learning with Neural Networks](https://mlsanity.com/paper/1409.3215/) (2014)
- [Spectrally-normalized margin bounds for neural networks](https://mlsanity.com/paper/1706.08498/) (2017)
- [Distillation as a Defense to Adversarial Perturbations against Deep Neural Networks](https://mlsanity.com/paper/1511.04508/) (2015)
- [Lipschitz regularity of deep neural networks: analysis and efficient estimation](https://mlsanity.com/paper/1805.10965/) (2018)
- [Safe Learning in Robotics: From Learning-Based Control to Safe Reinforcement Learning](https://mlsanity.com/paper/2108.06266/) (2021)
- [Regularisation of Neural Networks by Enforcing Lipschitz Continuity](https://mlsanity.com/paper/1804.04368/) (2018)
- [Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks](https://mlsanity.com/paper/1906.04893/) (2019)
- [Globally-Robust Neural Networks](https://mlsanity.com/paper/2102.08452/) (2021)
- [Safety Verification and Robustness Analysis of Neural Networks via Quadratic Constraints and Semidefinite Programming](https://mlsanity.com/paper/1903.01287/) (2019)
- [Training robust neural networks using Lipschitz bounds](https://mlsanity.com/paper/2005.02929/) (2020)
- [Direct Parameterization of Lipschitz-Bounded Deep Networks](https://mlsanity.com/paper/2301.11526/) (2023)
- [Efficiently Computing Local Lipschitz Constants of Neural Networks via Bound Propagation](https://mlsanity.com/paper/2210.07394/) (2022)
- [Imitation Learning with Stability and Safety Guarantees](https://mlsanity.com/paper/2012.09293/) (2020)
- [Randomized Sketches of Convex Programs with Sharp Guarantees](https://mlsanity.com/paper/1404.7203/) (2014)
- [Certified Robustness via Dynamic Margin Maximization and Improved Lipschitz Regularization](https://mlsanity.com/paper/2310.00116/) (2023)
- [A Berkeley View of Systems Challenges for AI](https://mlsanity.com/paper/1712.05855/) (2017)
- [Verification of Non-Linear Specifications for Neural Networks](https://mlsanity.com/paper/1902.09592/) (2019)
- [Neural network training under semidefinite constraints](https://mlsanity.com/paper/2201.00632/) (2022)
- [COSMO: A conic operator splitting method for convex conic problems](https://mlsanity.com/paper/1901.10887/) (2019)
- [Chordal and factor-width decompositions for scalable semidefinite and polynomial optimization](https://mlsanity.com/paper/2107.02379/) (2021)
- [Chordal Sparsity for SDP-based Neural Network Verification](https://mlsanity.com/paper/2206.03482/) (2022)
- [On Robust Reinforcement Learning with Lipschitz-Bounded Policy Networks](https://mlsanity.com/paper/2405.11432/) (2024)
- [Learning Neural Networks under Input-Output Specifications](https://mlsanity.com/paper/2202.11246/) (2022)

## Abstract (arXiv)

Certified robustness is a critical property for deploying neural networks (NN) in safety-critical applications. A principle approach to achieving such guarantees is to constrain the global Lipschitz constant of the network. However, accurate methods for Lipschitz-constrained training often suffer from non-convex formulations and poor scalability due to reliance on global semidefinite programs (SDPs). In this letter, we propose a convex training framework that enforces global Lipschitz constraints via semidefinite relaxation. By reparameterizing the NN using loop transformation, we derive a convex admissibility condition that enables tractable and certifiable training. While the resulting formulation guarantees robustness, its scalability is limited by the size of global SDP. To overcome this, we develop a randomized subspace linear matrix inequalities (RS-LMI) approach that decomposes the global constraints into sketched layerwise constraints projected onto low-dimensional subspaces, yielding a smooth and memory-efficient training objective. Empirical results on MNIST, CIFAR-10, and ImageNet demonstrate that the proposed framework achieves competitive accuracy with significantly improved Lipschitz bounds and runtime performance.