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

Researchers establish a theoretical connection between the Law of Robustness and robust generalization in machine learning, proving that Lipschitz constants maintain consistent scaling properties across both global and localized function classes. This work resolves an open problem by demonstrating how overparameterization requirements for robust interpolation relate to statistical learning guarantees for test performance.

This theoretical computer science paper addresses a fundamental question in machine learning robustness: whether models trained to fit data robustly (with bounded Lipschitz constants) will maintain that robustness on unseen test data. The research bridges two previously separate concepts—the Law of Robustness, which establishes lower bounds on model complexity needed for robust interpolation, and robust generalization error, which measures whether training performance translates to test performance.

The significance lies in formalizing the relationship between these concepts using Rademacher complexity, a standard tool in statistical learning theory. The authors prove that at the global scale, the Lipschitz requirement remains $\Omega(n^{1/d})$, where $n$ is sample size and $d$ is dimensionality. At localized scales—examining function classes with small empirical error—the scaling changes based on perturbation radius and concentration bounds. This distinction provides practitioners with nuanced guidance: generalization requirements vary depending on whether models operate across full function classes or restricted subsets.

For the machine learning community, this work strengthens theoretical foundations for adversarial robustness research, which has grown increasingly important as AI systems face real-world security challenges. The results validate that overparameterization isn't merely an empirical phenomenon but a mathematical necessity for robust learning. However, the practical impact remains limited to advancing theoretical understanding rather than enabling new algorithms or system improvements. Researchers in adversarial machine learning will find this formally rigorous framework useful for future work, though practitioners building robust systems will see limited immediate application beyond confirming existing intuitions about model scaling.

• →Researchers formally link the Law of Robustness with robust generalization using Rademacher complexity analysis
• →Lipschitz constant scaling requirements remain consistent at global scale but vary at local scales with perturbation radius
• →The work confirms overparameterization is mathematically necessary for robust interpolation across arbitrary data distributions
• →Analysis differentiates between global and local function class behavior, providing nuanced complexity bounds
• →Results advance theoretical foundations for adversarial robustness but offer limited immediate practical applications