# Does Order Matter : Connecting The Law of Robustness to Robust Generalization

arXiv: https://arxiv.org/abs/2602.20971 · HTML: https://mlsanity.com/paper/2602.20971/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 40 references that are in the map.
- First posted on arXiv 24 Feb 2026.

## Builds on (most cited first, 30 of 32)

- [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)
- [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)
- [Deep Double Descent: Where Bigger Models and More Data Hurt](https://mlsanity.com/paper/1912.02292/) (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)
- [Unlabeled Data Improves Adversarial Robustness](https://mlsanity.com/paper/1905.13736/) (2019)
- [Lipschitz regularity of deep neural networks: analysis and efficient estimation](https://mlsanity.com/paper/1805.10965/) (2018)
- [Measuring the Intrinsic Dimension of Objective Landscapes](https://mlsanity.com/paper/1804.08838/) (2018)
- [Adversarially Robust Generalization Requires More Data](https://mlsanity.com/paper/1804.11285/) (2018)
- [Parseval Networks: Improving Robustness to Adversarial Examples](https://mlsanity.com/paper/1704.08847/) (2017)
- [Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks](https://mlsanity.com/paper/1906.04893/) (2019)
- [Lipschitz-Margin Training: Scalable Certification of Perturbation Invariance for Deep Neural Networks](https://mlsanity.com/paper/1802.04034/) (2018)
- [Evaluating the Robustness of Neural Networks: An Extreme Value Theory Approach](https://mlsanity.com/paper/1801.10578/) (2018)
- [A Universal Law of Robustness via Isoperimetry](https://mlsanity.com/paper/2105.12806/) (2021)
- [Rademacher Complexity for Adversarially Robust Generalization](https://mlsanity.com/paper/1810.11914/) (2018)
- [Adversarially Robust Generalization Just Requires More Unlabeled Data](https://mlsanity.com/paper/1906.00555/) (2019)
- [A law of robustness for two-layers neural networks](https://mlsanity.com/paper/2009.14444/) (2020)
- [Understanding Robust Overfitting of Adversarial Training and Beyond](https://mlsanity.com/paper/2206.08675/) (2022)
- [Rethinking Lipschitz Neural Networks and Certified Robustness: A Boolean Function Perspective](https://mlsanity.com/paper/2210.01787/) (2022)
- [Deep neural networks have an inbuilt Occam's razor](https://mlsanity.com/paper/2304.06670/) (2023)
- [Why Robust Generalization in Deep Learning is Difficult: Perspective of Expressive Power](https://mlsanity.com/paper/2205.13863/) (2022)
- [Local Rademacher Complexity Bounds based on Covering Numbers](https://mlsanity.com/paper/1510.01463/) (2015)
- [A Law of Robustness beyond Isoperimetry](https://mlsanity.com/paper/2202.11592/) (2022)
- [The Lipschitz-Variance-Margin Tradeoff for Enhanced Randomized Smoothing](https://mlsanity.com/paper/2309.16883/) (2023)
- [On the Clean Generalization and Robust Overfitting in Adversarial Training from Two Theoretical Views: Representation Complexity and Training Dynamics](https://mlsanity.com/paper/2306.01271/) (2023)
- [Stability and Generalization in Free Adversarial Training](https://mlsanity.com/paper/2404.08980/) (2024)
- [A direct proof of a unified law of robustness for Bregman divergence losses](https://mlsanity.com/paper/2405.16639/) (2024)

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

- [Adversarially Robust PAC Learning with Optimal VC Rates](https://mlsanity.com/paper/2609.24260/) (2026)

## Abstract (arXiv)

Bubeck and Selke (2021) propose the connection between the Law of Robustness and robust generalization error as an open problem. The Law of Robustness states that overparameterization is necessary for models to interpolate robustly, i.e., the interpolating function is required to be Lipschitz. Wu et al. (2023) extend this law to arbitrary data distributions, proving that the Lipschitz constant satisfies L = Omega(n1/d). Robust generalization, on the other hand, asks whether small robust training loss implies small robust test loss. This can be studied using statistical learning techniques such as Rademacher complexities, where a bound on the Rademacher complexity of the robust loss class implies a bound on the Lipschitzness of the function class. We use this connection to explicitly link the two for arbitrary data distributions. (i) We prove that the order of the Lipschitz bound remains the same when considering the global Rademacher complexity of robust loss classes. (ii) At the local scale, i.e., for subsets of functions with small empirical error, the order of the Lipschitz bound changes with the perturbation radius ρ and the localized concentration term √r/n.