# Quadratic Upper Bound for Boosting Robustness

arXiv: https://arxiv.org/abs/2601.13645 · HTML: https://mlsanity.com/paper/2601.13645/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 38 of its 45 references that are in the map.
- First posted on arXiv 20 Jan 2026.

## Builds on (most cited first, 30 of 38)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [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)
- [Wide Residual Networks](https://mlsanity.com/paper/1605.07146/) (2016)
- [Identity Mappings in Deep Residual Networks](https://mlsanity.com/paper/1603.05027/) (2016)
- [Theoretically Principled Trade-off between Robustness and Accuracy](https://mlsanity.com/paper/1901.08573/) (2019)
- [Reliable evaluation of adversarial robustness with an ensemble of diverse parameter-free attacks](https://mlsanity.com/paper/2003.01690/) (2020)
- [An overview of gradient descent optimization algorithms](https://mlsanity.com/paper/1609.04747/) (2016)
- [Adversarial Machine Learning at Scale](https://mlsanity.com/paper/1611.01236/) (2016)
- [Ensemble Adversarial Training: Attacks and Defenses](https://mlsanity.com/paper/1705.07204/) (2017)
- [Fast is better than free: Revisiting adversarial training](https://mlsanity.com/paper/2001.03994/) (2020)
- [Square Attack: a query-efficient black-box adversarial attack via random search](https://mlsanity.com/paper/1912.00049/) (2019)
- [Adversarial Training for Free!](https://mlsanity.com/paper/1904.12843/) (2019)
- [Overfitting in adversarially robust deep learning](https://mlsanity.com/paper/2002.11569/) (2020)
- [Minimally distorted Adversarial Examples with a Fast Adaptive Boundary Attack](https://mlsanity.com/paper/1907.02044/) (2019)
- [Understanding and Improving Fast Adversarial Training](https://mlsanity.com/paper/2007.02617/) (2020)
- [Adversarial Distributional Training for Robust Deep Learning](https://mlsanity.com/paper/2002.05999/) (2020)
- [Understanding Catastrophic Overfitting in Single-step Adversarial Training](https://mlsanity.com/paper/2010.01799/) (2020)
- [Guided Adversarial Attack for Evaluating and Enhancing Adversarial Defenses](https://mlsanity.com/paper/2011.14969/) (2020)
- [Make Some Noise: Reliable and Efficient Single-Step Adversarial Training](https://mlsanity.com/paper/2202.01181/) (2022)
- [Single-step Adversarial training with Dropout Scheduling](https://mlsanity.com/paper/2004.08628/) (2020)
- [Subspace Adversarial Training](https://mlsanity.com/paper/2111.12229/) (2021)
- [Jacobian Adversarially Regularized Networks for Robustness](https://mlsanity.com/paper/1912.10185/) (2019)
- [Boosting Fast Adversarial Training with Learnable Adversarial Initialization](https://mlsanity.com/paper/2110.05007/) (2021)
- [Prior-Guided Adversarial Initialization for Fast Adversarial Training](https://mlsanity.com/paper/2207.08859/) (2022)
- [Improving Fast Adversarial Training with Prior-Guided Knowledge](https://mlsanity.com/paper/2304.00202/) (2023)
- [Reliably fast adversarial training via latent adversarial perturbation](https://mlsanity.com/paper/2104.01575/) (2021)
- [Fast Adversarial Training with Adaptive Step Size](https://mlsanity.com/paper/2206.02417/) (2022)
- [Understanding and Combating Robust Overfitting via Input Loss Landscape Analysis and Regularization](https://mlsanity.com/paper/2212.04985/) (2022)
- [On the Over-Memorization During Natural, Robust and Catastrophic Overfitting](https://mlsanity.com/paper/2310.08847/) (2023)

## Abstract (arXiv)

Fast adversarial training (FAT) aims to enhance the robustness of models against adversarial attacks with reduced training time, however, FAT often suffers from compromised robustness due to insufficient exploration of adversarial space. In this paper, we develop a loss function to mitigate the problem of degraded robustness under FAT. Specifically, we derive a quadratic upper bound (QUB) on the adversarial training (AT) loss function and propose to utilize the bound with existing FAT methods. Our experimental results show that applying QUB loss to the existing methods yields significant improvement of robustness. Furthermore, using various metrics, we demonstrate that this improvement is likely to result from the smoothened loss landscape of the resulting model.