mlsanityData to 7 Oct 2026

The Geometry of Adversarial Training in Binary Classification

arXiv 2111.13613 ↗First posted 26 Nov 2021cs.LG math.AP math.MGcited by 20

Not on the map. Why?

Abstract (arXiv)

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 L1+ (nonlocal) operatornameTV, a form frequently studied in image analysis and graph-based learning. A rich geometric structure is revealed by this reformulation which in turn allows us to establish a series of properties of optimal solutions of the original problem, including the existence of minimal and maximal solutions (interpreted in a suitable sense), and the existence of regular solutions (also interpreted in a suitable sense). In addition, we highlight how the connection between adversarial training and perimeter minimization problems provides a novel, directly interpretable, statistical motivation for a family of regularized risk minimization problems involving perimeter/total variation. The majority of our theoretical results are independent of the distance used to define adversarial attacks.

Key facts. Cited by 20 papers in the map (data through 7 Oct 2026). Builds on 23 of its 65 references that are in the map. First posted on arXiv 26 Nov 2021.

Builds on

23 of 65 references.

References · 23

  1. 2014Explaining and Harnessing Adversarial Examplescited by 7,287
  2. 2017Towards Deep Learning Models Resistant to Adversarial Attackscited by 6,326
  3. 2017Improving the Adversarial Robustness and Interpretability of Deep Neural Networks by Regularizing their Input Gradientscited by 236
  4. 2017Optimal approximation of piecewise smooth functions using deep ReLU neural networkscited by 176
  5. 2018Robustness via curvature regularization, and vice versacited by 174
  6. 2020Learned convex regularizers for inverse problemscited by 54
  7. 2019Lower Bounds on Adversarial Robustness from Optimal Transportcited by 52
  8. 2019Scaleable input gradient regularization for adversarial robustnesscited by 45
  9. 2021Distributionally Robust Learningcited by 44
  10. 2015A Unified Gradient Regularization Family for Adversarial Examplescited by 43
  11. 2019Adversarial Risk via Optimal Transport and Optimal Couplingscited by 36
  12. 2021CLIP: Cheap Lipschitz Training of Neural Networkscited by 29
11 more
  1. 2021Statistical Analysis of Wasserstein Distributionally Robust Estimatorscited by 27
  2. 2014Continuum limit of total variation on point cloudscited by 24
  3. 2021Mixed Nash Equilibria in the Adversarial Examples Gamecited by 22
  4. 2020Adversarial Classification: Necessary conditions and geometric flowscited by 19
  5. 2018Improved robustness to adversarial examples using Lipschitz regularization of the losscited by 13
  6. 2021On the regularized risk of distributionally robust learning over deep neural networkscited by 8
  7. 2014Consistency of Cheeger and Ratio Graph Cutscited by 6
  8. 2018Adversarially Robust Training through Structured Gradient Regularizationcited by 5
  9. 2016A new analytical approach to consistency and overfitting in regularized empirical risk minimizationcited by 4
  10. 2018Solution Paths of Variational Regularization Methods for Inverse Problemscited by 4
  11. 2017Combining Contrast Invariant L1 Data Fidelities with Nonlinear Spectral Image Decompositioncited by 1

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 6; 2023: 5; 2024: 5; 2025: 2; 2026: 2.

20 papers.

Sort:

  1. 2022Probabilistically Robust Learning: Balancing Average- and Worst-case Performancecited by 36
  2. 2022Existence and Minimax Theorems for Adversarial Surrogate Risks in Binary Classificationcited by 15
  3. 2022Towards Consistency in Adversarial Classificationcited by 12
  4. 2022Gamma-convergence of a nonlocal perimeter arising in adversarial machine learningcited by 10
  5. 2023On the existence of solutions to adversarial training in multiclass classificationcited by 9
  6. 2023The Adversarial Consistency of Surrogate Risks for Binary Classificationcited by 6
  7. 2023On adversarial robustness and the use of Wasserstein ascent-descent dynamics to enforce itcited by 4
  8. 2024A mean curvature flow arising in adversarial trainingcited by 4
  9. 2022The Consistency of Adversarial Training for Binary Classificationcited by 3
  10. 2023It begins with a boundary: A geometric view on probabilistically robust learningcited by 3
  11. 2024A Notion of Uniqueness for the Adversarial Bayes Classifiercited by 3
  12. 2023Non-Asymptotic Bounds for Adversarial Excess Risk under Misspecified Modelscited by 2
8 more
  1. 2024Do stable neural networks exist for classification problems? -- A new view on stability in AIcited by 2
  2. 2024Adversarial Consistency and the Uniqueness of the Adversarial Bayes Classifiercited by 2
  3. 2022Eikonal depth: an optimal control approach to statistical depthscited by 1
  4. 2024Uniform Convergence of Adversarially Robust Classifierscited by 1
  5. 2025Adversarial Surrogate Risk Bounds for Binary Classificationcited by 1
  6. 2025Lower Bounds on Adversarial Robustness for Multiclass Classification with General Loss Functionscited by 1
  7. 2026Duality for the Adversarial Total Variationcited by 0
  8. 2026Statistical Inference for Adversarial Training: Central Limit Theorems via Optimal Transportcited by 0