# The Geometry of Adversarial Training in Binary Classification

@article{Bungert2021TheGO, title={The Geometry of Adversarial Training in Binary Classification}, author={Leon Bungert and Nicol{\'a}s Garc{\'i}a Trillos and Ryan W. Murray}, journal={ArXiv}, year={2021}, volume={abs/2111.13613} }

We establish an equivalence between a family of adversarial training problems for non-parametric binary classification and a family of regularized risk minimization problems where the regularizer is a nonlocal perimeter functional. The resulting regularized risk minimization problems admit exact convex relaxations of the type L+ (nonlocal) TV, a form frequently studied in image analysis and graph-based learning. A rich geometric structure is revealed by this reformulation which in turn allows…

## 3 Citations

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## References

SHOWING 1-10 OF 65 REFERENCES

Robustness via Curvature Regularization, and Vice Versa

- Computer Science2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)
- 2019

It is shown in particular that adversarial training leads to a significant decrease in the curvature of the loss surface with respect to inputs, leading to a drastically more "linear" behaviour of the network.

Adversarial Classification: Necessary conditions and geometric flows

- Computer Science, MathematicsArXiv
- 2020

A version of adversarial classification where an adversary is empowered to corrupt data inputs up to some distance $\varepsilon$ is studied, using tools from variational analysis to derive a geometric evolution equation which can be used to track the change in classification boundaries as $\vARPSilon$ varies.

The Many Faces of Adversarial Risk

- Computer ScienceNeurIPS
- 2021

The technical tools derive from optimal transport, robust statistics, functional analysis, and game theory, and generalizing Strassen’s theorem to the unbalanced optimal transport setting with applications to adversarial classification with unequal priors and proving the existence of a pure Nash equilibrium in the two-player game between the adversary and the algorithm.

Lower Bounds on Adversarial Robustness from Optimal Transport

- Computer ScienceNeurIPS
- 2019

While progress has been made in understanding the robustness of machine learning classifiers to test-time adversaries (evasion attacks), fundamental questions remain unresolved. In this paper, we use…

Improved robustness to adversarial examples using Lipschitz regularization of the loss

- Computer Science, MathematicsArXiv
- 2018

This work augments AT with worst case adversarial training (WCAT) which improves adversarial robustness by 11% over the current state-of-the-art result in the $\ell_2$ norm on CIFAR-10, and obtains verifiable average case and worst case robustness guarantees.

A Unified Gradient Regularization Family for Adversarial Examples

- Computer Science2015 IEEE International Conference on Data Mining
- 2015

A family of gradient regularization methods that effectively penalize the gradient of loss function w.r.t. inputs are developed and achieved the best accuracy on MNIST data (without data augmentation) and competitive performance on CIFAR-10 data.

Towards Deep Learning Models Resistant to Adversarial Attacks

- Computer ScienceICLR
- 2018

This work studies the adversarial robustness of neural networks through the lens of robust optimization, and suggests the notion of security against a first-order adversary as a natural and broad security guarantee.

Improving the Adversarial Robustness and Interpretability of Deep Neural Networks by Regularizing their Input Gradients

- Computer ScienceAAAI
- 2018

It is demonstrated that regularizing input gradients makes them more naturally interpretable as rationales for model predictions, and also exhibits robustness to transferred adversarial examples generated to fool all of the other models.

Improving Gradient Regularization using Complex-Valued Neural Networks

- Computer ScienceICML
- 2021

Experimental results show that the performance of gradient regularized CVNN surpasses that of real-valued neural networks with comparable storage and computational complexity and that the properties of the CVNN parameter derivatives resist decrease of performance on the standard objective that is caused by competition with the gradient regularization objective.

CLIP: Cheap Lipschitz Training of Neural Networks

- Computer ScienceSSVM
- 2021

A variational regularization method named CLIP is investigated for controlling the Lipschitz constant of a neural network, which can easily be integrated into the training procedure and compared with a weight regularization approach.