# Convex training of Lipschitz-regularized shallow neural networks

arXiv: https://arxiv.org/abs/2606.19652 · HTML: https://mlsanity.com/paper/2606.19652/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 24 references that are in the map.
- First posted on arXiv 17 Jun 2026.

## Builds on (most cited first, 15 of 15)

- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [End-to-end Autonomous Driving: Challenges and Frontiers](https://mlsanity.com/paper/2306.16927/) (2023)
- [Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks](https://mlsanity.com/paper/1906.04893/) (2019)
- [SoK: Certified Robustness for Deep Neural Networks](https://mlsanity.com/paper/2009.04131/) (2020)
- [Training robust neural networks using Lipschitz bounds](https://mlsanity.com/paper/2005.02929/) (2020)
- [Neural Networks are Convex Regularizers: Exact Polynomial-time Convex Optimization Formulations for Two-layer Networks](https://mlsanity.com/paper/2002.10553/) (2020)
- [Fast Convex Optimization for Two-Layer ReLU Networks: Equivalent Model Classes and Cone Decompositions](https://mlsanity.com/paper/2202.01331/) (2022)
- [Verification of Neural Network Behaviour: Formal Guarantees for Power System Applications](https://mlsanity.com/paper/1910.01624/) (2019)
- [Global Optimality Beyond Two Layers: Training Deep ReLU Networks via Convex Programs](https://mlsanity.com/paper/2110.05518/) (2021)
- [Efficient Global Optimization of Two-Layer ReLU Networks: Quadratic-Time Algorithms and Adversarial Training](https://mlsanity.com/paper/2201.01965/) (2022)
- [Convolutional Neural Networks as 2-D systems](https://mlsanity.com/paper/2303.03042/) (2023)
- [Chordal Sparsity for SDP-based Neural Network Verification](https://mlsanity.com/paper/2206.03482/) (2022)
- [On the sensitivity of pose estimation neural networks: rotation parameterizations, Lipschitz constants, and provable bounds](https://mlsanity.com/paper/2203.09937/) (2022)

## Abstract (arXiv)

In this work, we introduce a training procedure for shallow neural networks that promotes robustness against adversarial attacks. We solve a non-convex Lipschitz-regularized training program by introducing a convex restriction that can be efficiently solved to global optimality. Our approach can be employed as a post-processing step by taking a pre-trained network as an initial solution to then solving the convex program whose optimal network is guaranteed to be no worse than the initial one. We illustrate the improvements of our training procedure with experiments using real world datasets for regression tasks under an adversarial setting. We show numerically that solving our proposed convex program yields networks with lower objective values on the Lipschitz-regularized program compared to existing methods. Additionally, we show that on certain datasets, networks obtained using our convex training program are both more accurate and robust with respect to adversarial attacks.