# On adversarial robustness and the use of Wasserstein ascent-descent dynamics to enforce it

arXiv: https://arxiv.org/abs/2301.03662 · HTML: https://mlsanity.com/paper/2301.03662/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 41 of its 60 references that are in the map.
- First posted on arXiv 9 Jan 2023.

## Builds on (most cited first, 30 of 41)

- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Adversarial examples in the physical world](https://mlsanity.com/paper/1607.02533/) (2016)
- [Theoretically Principled Trade-off between Robustness and Accuracy](https://mlsanity.com/paper/1901.08573/) (2019)
- [Robustness May Be at Odds with Accuracy](https://mlsanity.com/paper/1805.12152/) (2018)
- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [Certifying Some Distributional Robustness with Principled Adversarial Training](https://mlsanity.com/paper/1710.10571/) (2017)
- [Certified Defenses against Adversarial Examples](https://mlsanity.com/paper/1801.09344/) (2018)
- [On Gradient Descent Ascent for Nonconvex-Concave Minimax Problems](https://mlsanity.com/paper/1906.00331/) (2019)
- [Improving the Adversarial Robustness and Interpretability of Deep Neural Networks by Regularizing their Input Gradients](https://mlsanity.com/paper/1711.09404/) (2017)
- [You Only Propagate Once: Accelerating Adversarial Training via Maximal Principle](https://mlsanity.com/paper/1905.00877/) (2019)
- [Robustness via curvature regularization, and vice versa](https://mlsanity.com/paper/1811.09716/) (2018)
- [Wasserstein Distributionally Robust Optimization: Theory and Applications in Machine Learning](https://mlsanity.com/paper/1908.08729/) (2019)
- [Solving a Class of Non-Convex Min-Max Games Using Iterative First Order Methods](https://mlsanity.com/paper/1902.08297/) (2019)
- [Near-Optimal Algorithms for Minimax Optimization](https://mlsanity.com/paper/2002.02417/) (2020)
- [Lower Bounds on Adversarial Robustness from Optimal Transport](https://mlsanity.com/paper/1909.12272/) (2019)
- [Scaleable input gradient regularization for adversarial robustness](https://mlsanity.com/paper/1905.11468/) (2019)
- [Unbalanced Optimal Transport, from Theory to Numerics](https://mlsanity.com/paper/2211.08775/) (2022)
- [Distributionally Robust Learning](https://mlsanity.com/paper/2108.08993/) (2021)
- [A Unified Gradient Regularization Family for Adversarial Examples](https://mlsanity.com/paper/1511.06385/) (2015)
- [On the Convergence of Gradient Descent Training for Two-layer ReLU-networks in the Mean Field Regime](https://mlsanity.com/paper/2005.13530/) (2020)
- [Adversarial Risk via Optimal Transport and Optimal Couplings](https://mlsanity.com/paper/1912.02794/) (2019)
- [Theoretical Analysis of Adversarial Learning: A Minimax Approach](https://mlsanity.com/paper/1811.05232/) (2018)
- [A mean-field analysis of two-player zero-sum games](https://mlsanity.com/paper/2002.06277/) (2020)
- [The Many Faces of Adversarial Risk](https://mlsanity.com/paper/2201.08956/) (2022)
- [Statistical Analysis of Wasserstein Distributionally Robust Estimators](https://mlsanity.com/paper/2108.02120/) (2021)
- [On the Existence of the Adversarial Bayes Classifier (Extended Version)](https://mlsanity.com/paper/2112.01694/) (2021)
- [Continuum limit of total variation on point clouds](https://mlsanity.com/paper/1403.6355/) (2014)
- [Can Shallow Neural Networks Beat the Curse of Dimensionality? A mean field training perspective](https://mlsanity.com/paper/2005.10815/) (2020)

## Built on by (most cited first, 4 of 4)

- [Strategic Distribution Shift of Interacting Agents via Coupled Gradient Flows](https://mlsanity.com/paper/2307.01166/) (2023)
- [Analysis of Kernel Mirror Prox for Measure Optimization](https://mlsanity.com/paper/2403.00147/) (2024)
- [Mirror Descent-Ascent for mean-field min-max problems](https://mlsanity.com/paper/2402.08106/) (2024)
- [Gradient Flow Sampler-based Distributionally Robust Optimization](https://mlsanity.com/paper/2510.25956/) (2025)

## Abstract (arXiv)

We propose iterative algorithms to solve adversarial problems in a variety of supervised learning settings of interest. Our algorithms, which can be interpreted as suitable ascent-descent dynamics in Wasserstein spaces, take the form of a system of interacting particles. These interacting particle dynamics are shown to converge toward appropriate mean-field limit equations in certain large number of particles regimes. In turn, we prove that, under certain regularity assumptions, these mean-field equations converge, in the large time limit, toward approximate Nash equilibria of the original adversarial learning problems. We present results for nonconvex-nonconcave settings, as well as for nonconvex-concave ones. Numerical experiments illustrate our results.