# Evaluating robustness of support vector machines with the Lagrangian dual approach

arXiv: https://arxiv.org/abs/2306.02639 · HTML: https://mlsanity.com/paper/2306.02639/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 30 of its 64 references that are in the map.
- First posted on arXiv 5 Jun 2023.

## Builds on (most cited first, 30 of 30)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Adversarial examples in the physical world](https://mlsanity.com/paper/1607.02533/) (2016)
- [Obfuscated Gradients Give a False Sense of Security: Circumventing Defenses to Adversarial Examples](https://mlsanity.com/paper/1802.00420/) (2018)
- [Evasion Attacks against Machine Learning at Test Time](https://mlsanity.com/paper/1708.06131/) (2017)
- [Square Attack: a query-efficient black-box adversarial attack via random search](https://mlsanity.com/paper/1912.00049/) (2019)
- [Provable defenses against adversarial examples via the convex outer adversarial polytope](https://mlsanity.com/paper/1711.00851/) (2017)
- [HopSkipJumpAttack: A Query-Efficient Decision-Based Attack](https://mlsanity.com/paper/1904.02144/) (2019)
- [Certified Defenses against Adversarial Examples](https://mlsanity.com/paper/1801.09344/) (2018)
- [Towards Fast Computation of Certified Robustness for ReLU Networks](https://mlsanity.com/paper/1804.09699/) (2018)
- [Adversarial Risk and the Dangers of Evaluating Against Weak Attacks](https://mlsanity.com/paper/1802.05666/) (2018)
- [A Dual Approach to Scalable Verification of Deep Networks](https://mlsanity.com/paper/1803.06567/) (2018)
- [Reachability Analysis of Deep Neural Networks with Provable Guarantees](https://mlsanity.com/paper/1805.02242/) (2018)
- [Lipschitz constant estimation of Neural Networks via sparse polynomial optimization](https://mlsanity.com/paper/2004.08688/) (2020)
- [Low Frequency Adversarial Perturbation](https://mlsanity.com/paper/1809.08758/) (2018)
- [Learning perturbation sets for robust machine learning](https://mlsanity.com/paper/2007.08450/) (2020)
- [An SMT-Based Approach for Verifying Binarized Neural Networks](https://mlsanity.com/paper/2011.02948/) (2020)
- [Perfectly Parallel Fairness Certification of Neural Networks](https://mlsanity.com/paper/1912.02499/) (2019)
- [Partition-based formulations for mixed-integer optimization of trained ReLU neural networks](https://mlsanity.com/paper/2102.04373/) (2021)
- [Lagrangian Decomposition for Neural Network Verification](https://mlsanity.com/paper/2002.10410/) (2020)
- [Efficient Exact Verification of Binarized Neural Networks](https://mlsanity.com/paper/2005.03597/) (2020)
- [Analyzing Deep Neural Networks with Symbolic Propagation: Towards Higher Precision and Faster Verification](https://mlsanity.com/paper/1902.09866/) (2019)
- [Scalable Verification of Quantized Neural Networks (Technical Report)](https://mlsanity.com/paper/2012.08185/) (2020)
- [Adversarial Feature Selection against Evasion Attacks](https://mlsanity.com/paper/2005.12154/) (2020)
- [Improved Gradient based Adversarial Attacks for Quantized Networks](https://mlsanity.com/paper/2003.13511/) (2020)
- [Robustness Verification of Support Vector Machines](https://mlsanity.com/paper/1904.11803/) (2019)
- [Security Evaluation of Support Vector Machines in Adversarial Environments](https://mlsanity.com/paper/1401.7727/) (2014)
- [Spanning Attack: Reinforce Black-box Attacks with Unlabeled Data](https://mlsanity.com/paper/2005.04871/) (2020)
- [QNNVerifier: A Tool for Verifying Neural Networks using SMT-Based Model Checking](https://mlsanity.com/paper/2111.13110/) (2021)

## Abstract (arXiv)

Adversarial examples bring a considerable security threat to support vector machines (SVMs), especially those used in safety-critical applications. Thus, robustness verification is an essential issue for SVMs, which can provide provable robustness against various kinds of adversary attacks. The evaluation results obtained through the robustness verification can provide a safe guarantee for the use of SVMs. The existing verification method does not often perform well in verifying SVMs with nonlinear kernels. To this end, we propose a method to improve the verification performance for SVMs with nonlinear kernels. We first formalize the adversarial robustness evaluation of SVMs as an optimization problem. Then a lower bound of the original problem is obtained by solving the Lagrangian dual problem of the original problem. Finally, the adversarial robustness of SVMs is evaluated concerning the lower bound. We evaluate the adversarial robustness of SVMs with linear and nonlinear kernels on the MNIST and Fashion-MNIST datasets. The experimental results show that the percentage of provable robustness obtained by our method on the test set is better than that of the state-of-the-art.