# Second Order Optimization for Adversarial Robustness and Interpretability

arXiv: https://arxiv.org/abs/2009.04923 · HTML: https://mlsanity.com/paper/2009.04923/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Sep 2020.

## Built on by (most cited first, 7 of 7)

- [Toward Transparent AI: A Survey on Interpreting the Inner Structures of Deep Neural Networks](https://mlsanity.com/paper/2207.13243/) (2022)
- [Enhancing Adversarial Training with Second-Order Statistics of Weights](https://mlsanity.com/paper/2203.06020/) (2022)
- [Superposition as Lossy Compression: Measure with Sparse Autoencoders and Connect to Adversarial Vulnerability](https://mlsanity.com/paper/2512.13568/) (2025)
- [Characterizing Model Robustness via Natural Input Gradients](https://mlsanity.com/paper/2409.20139/) (2024)
- [Hessian-Free Second-Order Adversarial Examples for Adversarial Learning](https://mlsanity.com/paper/2207.01396/) (2022)
- [Ensemble Adversarial Defense via Integration of Multiple Dispersed Low Curvature Models](https://mlsanity.com/paper/2403.16405/) (2024)
- [SORA: Free Second-Order Attacks in Fast Adversarial Training](https://mlsanity.com/paper/2606.00738/) (2026)

## Abstract (arXiv)

Deep neural networks are easily fooled by small perturbations known as adversarial attacks. Adversarial Training (AT) is a technique aimed at learning features robust to such attacks and is widely regarded as a very effective defense. However, the computational cost of such training can be prohibitive as the network size and input dimensions grow. Inspired by the relationship between robustness and curvature, we propose a novel regularizer which incorporates first and second order information via a quadratic approximation to the adversarial loss. The worst case quadratic loss is approximated via an iterative scheme. It is shown that using only a single iteration in our regularizer achieves stronger robustness than prior gradient and curvature regularization schemes, avoids gradient obfuscation, and, with additional iterations, achieves strong robustness with significantly lower training time than AT. Further, it retains the interesting facet of AT that networks learn features which are well-aligned with human perception. We demonstrate experimentally that our method produces higher quality human-interpretable features than other geometric regularization techniques. These robust features are then used to provide human-friendly explanations to model predictions.