# Towards Understanding the Dynamics of the First-Order Adversaries

arXiv: https://arxiv.org/abs/2010.10650 · HTML: https://mlsanity.com/paper/2010.10650/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Oct 2020.

## Built on by (most cited first, 7 of 7)

- [When and How Mixup Improves Calibration](https://mlsanity.com/paper/2102.06289/) (2021)
- [Adversarial Training Helps Transfer Learning via Better Representations](https://mlsanity.com/paper/2106.10189/) (2021)
- [FIFA: Making Fairness More Generalizable in Classifiers Trained on Imbalanced Data](https://mlsanity.com/paper/2206.02792/) (2022)
- [Reinforcement Learning with Stepwise Fairness Constraints](https://mlsanity.com/paper/2211.03994/) (2022)
- [Adversarial Training for Gradient Descent: Analysis Through its Continuous-time Approximation](https://mlsanity.com/paper/2105.08037/) (2021)
- [Dynamic Efficient Adversarial Training Guided by Gradient Magnitude](https://mlsanity.com/paper/2103.03076/) (2021)
- [When Muon Optimizer Meets Adversarial Training: A Theoretical and Empirical Study](https://mlsanity.com/paper/2605.26929/) (2026)

## Abstract (arXiv)

An acknowledged weakness of neural networks is their vulnerability to adversarial perturbations to the inputs. To improve the robustness of these models, one of the most popular defense mechanisms is to alternatively maximize the loss over the constrained perturbations (or called adversaries) on the inputs using projected gradient ascent and minimize over weights. In this paper, we analyze the dynamics of the maximization step towards understanding the experimentally observed effectiveness of this defense mechanism. Specifically, we investigate the non-concave landscape of the adversaries for a two-layer neural network with a quadratic loss. Our main result proves that projected gradient ascent finds a local maximum of this non-concave problem in a polynomial number of iterations with high probability. To our knowledge, this is the first work that provides a convergence analysis of the first-order adversaries. Moreover, our analysis demonstrates that, in the initial phase of adversarial training, the scale of the inputs matters in the sense that a smaller input scale leads to faster convergence of adversarial training and a "more regular" landscape. Finally, we show that these theoretical findings are in excellent agreement with a series of experiments.