# Analysis of classifiers' robustness to adversarial perturbations

arXiv: https://arxiv.org/abs/1502.02590 · HTML: https://mlsanity.com/paper/1502.02590/

## Key facts

- Cited by 76 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Feb 2015.

## Built on by (most cited first, 30 of 76)

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- [PRIME: A few primitives can boost robustness to common corruptions](https://mlsanity.com/paper/2112.13547/) (2021)
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- [Regularization properties of adversarially-trained linear regression](https://mlsanity.com/paper/2310.10807/) (2023)
- [A Unified Game-Theoretic Interpretation of Adversarial Robustness](https://mlsanity.com/paper/2111.03536/) (2021)
- [SPADE: A Spectral Method for Black-Box Adversarial Robustness Evaluation](https://mlsanity.com/paper/2102.03716/) (2021)
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- [Directive Explanations for Actionable Explainability in Machine Learning Applications](https://mlsanity.com/paper/2102.02671/) (2021)
- [GAMA: Generative Adversarial Multi-Object Scene Attacks](https://mlsanity.com/paper/2209.09502/) (2022)
- [Understanding Generalization in Adversarial Training via the Bias-Variance Decomposition](https://mlsanity.com/paper/2103.09947/) (2021)
- [A.I. Robustness: a Human-Centered Perspective on Technological Challenges and Opportunities](https://mlsanity.com/paper/2210.08906/) (2022)
- [Cross-Task Attack: A Self-Supervision Generative Framework Based on Attention Shift](https://mlsanity.com/paper/2407.13700/) (2024)
- [Building Robust Ensembles via Margin Boosting](https://mlsanity.com/paper/2206.03362/) (2022)
- [Learning Provably Robust Estimators for Inverse Problems via Jittering](https://mlsanity.com/paper/2307.12822/) (2023)
- [Adversarial Ink: Componentwise Backward Error Attacks on Deep Learning](https://mlsanity.com/paper/2306.02918/) (2023)
- [On the Minimal Adversarial Perturbation for Deep Neural Networks with Provable Estimation Error](https://mlsanity.com/paper/2201.01235/) (2022)
- [On the Computability of Robust PAC Learning](https://mlsanity.com/paper/2406.10161/) (2024)
- [Bridging the Gap Between Adversarial Robustness and Optimization Bias](https://mlsanity.com/paper/2102.08868/) (2021)
- [Adversarial attacks on neural networks through canonical Riemannian foliations](https://mlsanity.com/paper/2203.00922/) (2022)
- [Software Testing for Machine Learning](https://mlsanity.com/paper/2205.00210/) (2022)
- [Leveraging Local Patch Differences in Multi-Object Scenes for Generative Adversarial Attacks](https://mlsanity.com/paper/2209.09883/) (2022)
- [How many dimensions are required to find an adversarial example?](https://mlsanity.com/paper/2303.14173/) (2023)
- [Thundernna: a white box adversarial attack](https://mlsanity.com/paper/2111.12305/) (2021)
- [Sample Complexity Bounds for Robustly Learning Decision Lists against Evasion Attacks](https://mlsanity.com/paper/2205.06127/) (2022)
- [Investigating the Corruption Robustness of Image Classifiers with Random Lp-norm Corruptions](https://mlsanity.com/paper/2305.05400/) (2023)
- [In-distribution adversarial attacks on object recognition models using gradient-free search](https://mlsanity.com/paper/2106.16198/) (2021)
- [Tightening the Approximation Error of Adversarial Risk with Auto Loss Function Search](https://mlsanity.com/paper/2111.05063/) (2021)
- [A Survey of Neural Network Robustness Assessment in Image Recognition](https://mlsanity.com/paper/2404.08285/) (2024)
- [Intriguing Properties of Robust Classification](https://mlsanity.com/paper/2412.04245/) (2024)
- [Attack Transferability Characterization for Adversarially Robust Multi-label Classification](https://mlsanity.com/paper/2106.15360/) (2021)

## Abstract (arXiv)

The goal of this paper is to analyze an intriguing phenomenon recently discovered in deep networks, namely their instability to adversarial perturbations (Szegedy et. al., 2014). We provide a theoretical framework for analyzing the robustness of classifiers to adversarial perturbations, and show fundamental upper bounds on the robustness of classifiers. Specifically, we establish a general upper bound on the robustness of classifiers to adversarial perturbations, and then illustrate the obtained upper bound on the families of linear and quadratic classifiers. In both cases, our upper bound depends on a distinguishability measure that captures the notion of difficulty of the classification task. Our results for both classes imply that in tasks involving small distinguishability, no classifier in the considered set will be robust to adversarial perturbations, even if a good accuracy is achieved. Our theoretical framework moreover suggests that the phenomenon of adversarial instability is due to the low flexibility of classifiers, compared to the difficulty of the classification task (captured by the distinguishability). Moreover, we show the existence of a clear distinction between the robustness of a classifier to random noise and its robustness to adversarial perturbations. Specifically, the former is shown to be larger than the latter by a factor that is proportional to √d (with d being the signal dimension) for linear classifiers. This result gives a theoretical explanation for the discrepancy between the two robustness properties in high dimensional problems, which was empirically observed in the context of neural networks. To the best of our knowledge, our results provide the first theoretical work that addresses the phenomenon of adversarial instability recently observed for deep networks. Our analysis is complemented by experimental results on controlled and real-world data.