# Effects of Exponential Gaussian Distribution on (Double Sampling) Randomized Smoothing

arXiv: https://arxiv.org/abs/2406.02309 · HTML: https://mlsanity.com/paper/2406.02309/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 36 of its 48 references that are in the map.
- First posted on arXiv 4 Jun 2024.

## Builds on (most cited first, 30 of 36)

- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [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)
- [Certified Adversarial Robustness via Randomized Smoothing](https://mlsanity.com/paper/1902.02918/) (2019)
- [Obfuscated Gradients Give a False Sense of Security: Circumventing Defenses to Adversarial Examples](https://mlsanity.com/paper/1802.00420/) (2018)
- [Adversarial Machine Learning at Scale](https://mlsanity.com/paper/1611.01236/) (2016)
- [Provable defenses against adversarial examples via the convex outer adversarial polytope](https://mlsanity.com/paper/1711.00851/) (2017)
- [Adversarial Examples Are Not Easily Detected: Bypassing Ten Detection Methods](https://mlsanity.com/paper/1705.07263/) (2017)
- [Certified Robustness to Adversarial Examples with Differential Privacy](https://mlsanity.com/paper/1802.03471/) (2018)
- [Provably Robust Deep Learning via Adversarially Trained Smoothed Classifiers](https://mlsanity.com/paper/1906.04584/) (2019)
- [Adversarial Risk and the Dangers of Evaluating Against Weak Attacks](https://mlsanity.com/paper/1802.05666/) (2018)
- [Certified Adversarial Robustness with Additive Noise](https://mlsanity.com/paper/1809.03113/) (2018)
- [Semidefinite relaxations for certifying robustness to adversarial examples](https://mlsanity.com/paper/1811.01057/) (2018)
- [Randomized Smoothing of All Shapes and Sizes](https://mlsanity.com/paper/2002.08118/) (2020)
- [Scaling provable adversarial defenses](https://mlsanity.com/paper/1805.12514/) (2018)
- [(De)Randomized Smoothing for Certifiable Defense against Patch Attacks](https://mlsanity.com/paper/2002.10733/) (2020)
- [Consistency Regularization for Certified Robustness of Smoothed Classifiers](https://mlsanity.com/paper/2006.04062/) (2020)
- [Tight Certificates of Adversarial Robustness for Randomly Smoothed Classifiers](https://mlsanity.com/paper/1906.04948/) (2019)
- [Curse of Dimensionality on Randomized Smoothing for Certifiable Robustness](https://mlsanity.com/paper/2002.03239/) (2020)
- [TSS: Transformation-Specific Smoothing for Robustness Certification](https://mlsanity.com/paper/2002.12398/) (2020)
- [SmoothMix: Training Confidence-calibrated Smoothed Classifiers for Certified Robustness](https://mlsanity.com/paper/2111.09277/) (2021)
- [Detection as Regression: Certified Object Detection by Median Smoothing](https://mlsanity.com/paper/2007.03730/) (2020)
- [Black-Box Certification with Randomized Smoothing: A Functional Optimization Based Framework](https://mlsanity.com/paper/2002.09169/) (2020)
- [Certified Neural Network Watermarks with Randomized Smoothing](https://mlsanity.com/paper/2207.07972/) (2022)
- [Improved, Deterministic Smoothing for L_1 Certified Robustness](https://mlsanity.com/paper/2103.10834/) (2021)
- [Random Smoothing Might be Unable to Certify ℓ_∞ Robustness for High-Dimensional Images](https://mlsanity.com/paper/2002.03517/) (2020)
- [Scalable Certified Segmentation via Randomized Smoothing](https://mlsanity.com/paper/2107.00228/) (2021)
- [Certified Robustness of Community Detection against Adversarial Structural Perturbation via Randomized Smoothing](https://mlsanity.com/paper/2002.03421/) (2020)
- [Intriguing Properties of Input-dependent Randomized Smoothing](https://mlsanity.com/paper/2110.05365/) (2021)
- [GSmooth: Certified Robustness against Semantic Transformations via Generalized Randomized Smoothing](https://mlsanity.com/paper/2206.04310/) (2022)

## Built on by (most cited first, 1 of 1)

- [CAMP in the Odyssey: Provably Robust Reinforcement Learning with Certified Radius Maximization](https://mlsanity.com/paper/2501.17667/) (2025)

## Abstract (arXiv)

Randomized Smoothing (RS) is currently a scalable certified defense method providing robustness certification against adversarial examples. Although significant progress has been achieved in providing defenses against ℓp adversaries, the interaction between the smoothing distribution and the robustness certification still remains vague. In this work, we comprehensively study the effect of two families of distributions, named Exponential Standard Gaussian (ESG) and Exponential General Gaussian (EGG) distributions, on Randomized Smoothing and Double Sampling Randomized Smoothing (DSRS). We derive an analytic formula for ESG's certified radius, which converges to the origin formula of RS as the dimension d increases. Additionally, we prove that EGG can provide tighter constant factors than DSRS in providing Omega(√d) lower bounds of ℓ2 certified radius, and thus further addresses the curse of dimensionality in RS. Our experiments on real-world datasets confirm our theoretical analysis of the ESG distributions, that they provide almost the same certification under different exponents η for both RS and DSRS. In addition, EGG brings a significant improvement to the DSRS certification, but the mechanism can be different when the classifier properties are different. Compared to the primitive DSRS, the increase in certified accuracy provided by EGG is prominent, up to 6.4% on ImageNet.