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Theoretical Analysis of Adversarial Learning: A Minimax Approach

arXiv 1811.05232 ↗First posted 13 Nov 2018stat.ML cs.LGcited by 31

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Abstract (arXiv)

Here we propose a general theoretical method for analyzing the risk bound in the presence of adversaries. Specifically, we try to fit the adversarial learning problem into the minimax framework. We first show that the original adversarial learning problem can be reduced to a minimax statistical learning problem by introducing a transport map between distributions. Then, we prove a new risk bound for this minimax problem in terms of covering numbers under a weak version of Lipschitz condition. Our method can be applied to multi-class classification problems and commonly used loss functions such as the hinge and ramp losses. As some illustrative examples, we derive the adversarial risk bounds for SVMs, deep neural networks, and PCA, and our bounds have two data-dependent terms, which can be optimized for achieving adversarial robustness.

Key facts. Cited by 31 papers in the map (data through 14 Oct 2026). First posted on arXiv 13 Nov 2018.

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2021: 5; 2022: 8; 2023: 10; 2024: 5; 2025: 2; 2026: 1.

31 papers.

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  1. 2021Improving Robustness using Generated Datacited by 234
  2. 2024Robust Prompt Optimization for Defending Language Models Against Jailbreaking Attackscited by 156
  3. 2021Exploring Memorization in Adversarial Trainingcited by 55
  4. 2021Adversarial Robustness with Semi-Infinite Constrained Learningcited by 39
  5. 2022The Many Faces of Adversarial Riskcited by 28
  6. 2022Exploring the Relationship between Architecture and Adversarially Robust Generalizationcited by 23
  7. 2021Mixed Nash Equilibria in the Adversarial Examples Gamecited by 22
  8. 2022Federated Adversarial Learning: A Framework with Convergence Analysiscited by 17
  9. 2023Outlier-Robust Wasserstein DROcited by 16
  10. 2022Toward Learning Robust and Invariant Representations with Alignment Regularization and Data Augmentationcited by 11
  11. 2023Distilling Out-of-Distribution Robustness from Vision-Language Foundation Modelscited by 9
  12. 2023Nash Equilibria, Regularization and Computation in Optimal Transport-Based Distributionally Robust Optimizationcited by 8
19 more
  1. 2023Understanding Noise-Augmented Training for Randomized Smoothingcited by 7
  2. 2023Towards Understanding the Mechanism of Contrastive Learning via Similarity Structure: A Theoretical Analysiscited by 6
  3. 2023On adversarial robustness and the use of Wasserstein ascent-descent dynamics to enforce itcited by 4
  4. 2024Exploring Adversarial Robustness of Deep State Space Modelscited by 4
  5. 2022AdAUC: End-to-end Adversarial AUC Optimization Against Long-tail Problemscited by 3
  6. 2023Generating Less Certain Adversarial Examples Improves Robust Generalizationcited by 3
  7. 2025Wasserstein Distributionally Robust Nonparametric Regressioncited by 3
  8. 2021Attack Transferability Characterization for Adversarially Robust Multi-label Classificationcited by 2
  9. 2022AdvCat: Domain-Agnostic Robustness Assessment for Cybersecurity-Critical Applications with Categorical Inputscited by 2
  10. 2023Non-Asymptotic Bounds for Adversarial Excess Risk under Misspecified Modelscited by 2
  11. 2024Minimax rates of convergence for nonparametric regression under adversarial attackscited by 2
  12. 2022Finding Dynamics Preserving Adversarial Winning Ticketscited by 1
  13. 2022Provable Robust Saliency-based Explanationscited by 1
  14. 2023On Achieving Optimal Adversarial Test Errorcited by 1
  15. 2024Tropical Decision Boundaries for Neural Networks Are Robust Against Adversarial Attackscited by 1
  16. 2023Robust Ranking Explanationscited by 0
  17. 2024Robust optimization for adversarial learning with finite sample complexity guaranteescited by 0
  18. 2025Adversarial generalization of unfolding (model-based) networkscited by 0
  19. 2026Risk-Averse Wasserstein Distributionally Robust Online Learningcited by 0