# Minimax rates of convergence for nonparametric regression under adversarial attacks

arXiv: https://arxiv.org/abs/2410.09402 · HTML: https://mlsanity.com/paper/2410.09402/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 37 of its 73 references that are in the map.
- First posted on arXiv 12 Oct 2024.

## Builds on (most cited first, 30 of 37)

- [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)
- [Towards Evaluating the Robustness of Neural Networks](https://mlsanity.com/paper/1608.04644/) (2016)
- [Theoretically Principled Trade-off between Robustness and Accuracy](https://mlsanity.com/paper/1901.08573/) (2019)
- [Certified Adversarial Robustness via Randomized Smoothing](https://mlsanity.com/paper/1902.02918/) (2019)
- [DeepFool: a simple and accurate method to fool deep neural networks](https://mlsanity.com/paper/1511.04599/) (2015)
- [Obfuscated Gradients Give a False Sense of Security: Circumventing Defenses to Adversarial Examples](https://mlsanity.com/paper/1802.00420/) (2018)
- [Robustness May Be at Odds with Accuracy](https://mlsanity.com/paper/1805.12152/) (2018)
- [The Limitations of Deep Learning in Adversarial Settings](https://mlsanity.com/paper/1511.07528/) (2015)
- [Evasion Attacks against Machine Learning at Test Time](https://mlsanity.com/paper/1708.06131/) (2017)
- [A Survey of Deep Learning Techniques for Autonomous Driving](https://mlsanity.com/paper/1910.07738/) (2019)
- [Adversarially Robust Generalization Requires More Data](https://mlsanity.com/paper/1804.11285/) (2018)
- [Certified Defenses against Adversarial Examples](https://mlsanity.com/paper/1801.09344/) (2018)
- [Rademacher Complexity for Adversarially Robust Generalization](https://mlsanity.com/paper/1810.11914/) (2018)
- [Adversarial Training Can Hurt Generalization](https://mlsanity.com/paper/1906.06032/) (2019)
- [VC Classes are Adversarially Robustly Learnable, but Only Improperly](https://mlsanity.com/paper/1902.04217/) (2019)
- [Precise Tradeoffs in Adversarial Training for Linear Regression](https://mlsanity.com/paper/2002.10477/) (2020)
- [Distributionally Robust Losses for Latent Covariate Mixtures](https://mlsanity.com/paper/2007.13982/) (2020)
- [Lower Bounds on Adversarial Robustness from Optimal Transport](https://mlsanity.com/paper/1909.12272/) (2019)
- [Precise Statistical Analysis of Classification Accuracies for Adversarial Training](https://mlsanity.com/paper/2010.11213/) (2020)
- [Scaleable input gradient regularization for adversarial robustness](https://mlsanity.com/paper/1905.11468/) (2019)
- [Adversarial Learning Guarantees for Linear Hypotheses and Neural Networks](https://mlsanity.com/paper/2004.13617/) (2020)
- [Provable tradeoffs in adversarially robust classification](https://mlsanity.com/paper/2006.05161/) (2020)
- [The Curious Case of Adversarially Robust Models: More Data Can Help, Double Descend, or Hurt Generalization](https://mlsanity.com/paper/2002.11080/) (2020)
- [Analyzing the Robustness of Nearest Neighbors to Adversarial Examples](https://mlsanity.com/paper/1706.03922/) (2017)
- [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)
- [Adversarially Robust Estimate and Risk Analysis in Linear Regression](https://mlsanity.com/paper/2012.10278/) (2020)
- [Overparameterized Linear Regression under Adversarial Attacks](https://mlsanity.com/paper/2204.06274/) (2022)

## Built on by (most cited first, 2 of 2)

- [On damage of interpolation to adversarial robustness in regression](https://mlsanity.com/paper/2601.16070/) (2026)
- [Adversarial Robustness of NTK Neural Networks](https://mlsanity.com/paper/2604.25965/) (2026)

## Abstract (arXiv)

Recent research shows the susceptibility of machine learning models to adversarial attacks, wherein minor but maliciously chosen perturbations of the input can significantly degrade model performance. In this paper, we theoretically analyse the limits of robustness against such adversarial attacks in a nonparametric regression setting, by examining the minimax rates of convergence in an adversarial sup-norm. Our work reveals that the minimax rate under adversarial attacks in the input is the same as sum of two terms: one represents the minimax rate in the standard setting without adversarial attacks, and the other reflects the maximum deviation of the true regression function value within the target function class when subjected to the input perturbations. The optimal rates under the adversarial setup can be achieved by an adversarial plug-in procedure constructed from a minimax optimal estimator in the corresponding standard setting. Two specific examples are given to illustrate the established minimax results.