# On the existence of solutions to adversarial training in multiclass classification

arXiv: https://arxiv.org/abs/2305.00075 · HTML: https://mlsanity.com/paper/2305.00075/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 41 references that are in the map.
- First posted on arXiv 28 Apr 2023.

## Builds on (most cited first, 22 of 22)

- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Wild Patterns: Ten Years After the Rise of Adversarial Machine Learning](https://mlsanity.com/paper/1712.03141/) (2017)
- [Precise Tradeoffs in Adversarial Training for Linear Regression](https://mlsanity.com/paper/2002.10477/) (2020)
- [Lower Bounds on Adversarial Robustness from Optimal Transport](https://mlsanity.com/paper/1909.12272/) (2019)
- [Adversarial Risk via Optimal Transport and Optimal Couplings](https://mlsanity.com/paper/1912.02794/) (2019)
- [Adversarial Example Games](https://mlsanity.com/paper/2007.00720/) (2020)
- [The Many Faces of Adversarial Risk](https://mlsanity.com/paper/2201.08956/) (2022)
- [On the Existence of the Adversarial Bayes Classifier (Extended Version)](https://mlsanity.com/paper/2112.01694/) (2021)
- [Mixed Nash Equilibria in the Adversarial Examples Game](https://mlsanity.com/paper/2102.06905/) (2021)
- [The Multimarginal Optimal Transport Formulation of Adversarial Multiclass Classification](https://mlsanity.com/paper/2204.12676/) (2022)
- [The Geometry of Adversarial Training in Binary Classification](https://mlsanity.com/paper/2111.13613/) (2021)
- [Adversarial Classification: Necessary conditions and geometric flows](https://mlsanity.com/paper/2011.10797/) (2020)
- [Existence and Minimax Theorems for Adversarial Surrogate Risks in Binary Classification](https://mlsanity.com/paper/2206.09098/) (2022)
- [Gamma-convergence of a nonlocal perimeter arising in adversarial machine learning](https://mlsanity.com/paper/2211.15223/) (2022)
- [Nash Equilibria and Pitfalls of Adversarial Training in Adversarial Robustness Games](https://mlsanity.com/paper/2210.12606/) (2022)
- [Mumford-Shah functionals on graphs and their asymptotics](https://mlsanity.com/paper/1906.09521/) (2019)
- [From graph cuts to isoperimetric inequalities: Convergence rates of Cheeger cuts on data clouds](https://mlsanity.com/paper/2004.09304/) (2020)
- [A Method Based on Total Variation for Network Modularity Optimization using the MBO Scheme](https://mlsanity.com/paper/1304.4679/) (2013)
- [A new analytical approach to consistency and overfitting in regularized empirical risk minimization](https://mlsanity.com/paper/1607.00274/) (2016)
- [Stochastic Block Models are a Discrete Surface Tension](https://mlsanity.com/paper/1806.02485/) (2018)
- [The Consistency of Adversarial Training for Binary Classification](https://mlsanity.com/paper/2206.09099/) (2022)
- [Total variation based community detection using a nonlinear optimization approach](https://mlsanity.com/paper/1907.08048/) (2019)

## Built on by (most cited first, 9 of 9)

- [The Adversarial Consistency of Surrogate Risks for Binary Classification](https://mlsanity.com/paper/2305.09956/) (2023)
- [It begins with a boundary: A geometric view on probabilistically robust learning](https://mlsanity.com/paper/2305.18779/) (2023)
- [Adversarial Consistency and the Uniqueness of the Adversarial Bayes Classifier](https://mlsanity.com/paper/2404.17358/) (2024)
- [Genetic Column Generation for Computing Lower Bounds for Adversarial Classification](https://mlsanity.com/paper/2406.08331/) (2024)
- [Uniform Convergence of Adversarially Robust Classifiers](https://mlsanity.com/paper/2406.14682/) (2024)
- [Lower Bounds on Adversarial Robustness for Multiclass Classification with General Loss Functions](https://mlsanity.com/paper/2510.01969/) (2025)
- [Unveiling the Role of Randomization in Multiclass Adversarial Classification: Insights from Graph Theory](https://mlsanity.com/paper/2503.14299/) (2025)
- [Duality for the Adversarial Total Variation](https://mlsanity.com/paper/2604.18540/) (2026)
- [Statistical Inference for Adversarial Training: Central Limit Theorems via Optimal Transport](https://mlsanity.com/paper/2609.22240/) (2026)

## Abstract (arXiv)

We study three models of the problem of adversarial training in multiclass classification designed to construct robust classifiers against adversarial perturbations of data in the agnostic-classifier setting. We prove the existence of Borel measurable robust classifiers in each model and provide a unified perspective of the adversarial training problem, expanding the connections with optimal transport initiated by the authors in previous work and developing new connections between adversarial training in the multiclass setting and total variation regularization. As a corollary of our results, we prove the existence of Borel measurable solutions to the agnostic adversarial training problem in the binary classification setting, a result that improves results in the literature of adversarial training, where robust classifiers were only known to exist within the enlarged universal σ-algebra of the feature space.