# Fast and Stable Interval Bounds Propagation for Training Verifiably Robust Models

arXiv: https://arxiv.org/abs/1906.00628 · HTML: https://mlsanity.com/paper/1906.00628/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Jun 2019.

## Built on by (most cited first, 5 of 5)

- [Continual Learning with Guarantees via Weight Interval Constraints](https://mlsanity.com/paper/2206.07996/) (2022)
- [HyperCube: Implicit Field Representations of Voxelized 3D Models](https://mlsanity.com/paper/2110.05770/) (2021)
- [Interval Bound Interpolation for Few-shot Learning with Few Tasks](https://mlsanity.com/paper/2204.03511/) (2022)
- [Is Certifying ℓp Robustness Still Worthwhile?](https://mlsanity.com/paper/2310.09361/) (2023)
- [SHIELD: Secure Hypernetworks for Incremental Expansion Learning Defense](https://mlsanity.com/paper/2506.08255/) (2025)

## Abstract (arXiv)

We present an efficient technique, which allows to train classification networks which are verifiably robust against norm-bounded adversarial attacks. This framework is built upon the work of Gowal et al., who applies the interval arithmetic to bound the activations at each layer and keeps the prediction invariant to the input perturbation. While that method is faster than competitive approaches, it requires careful tuning of hyper-parameters and a large number of epochs to converge. To speed up and stabilize training, we supply the cost function with an additional term, which encourages the model to keep the interval bounds at hidden layers small. Experimental results demonstrate that we can achieve comparable (or even better) results using a smaller number of training iterations, in a more stable fashion. Moreover, the proposed model is not so sensitive to the exact specification of the training process, which makes it easier to use by practitioners.