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

TL;DR

This paper establishes a theoretical connection between the law of robustness and robust generalization, showing that the Lipschitz constant's role in robust interpolation is fundamental and experimentally validated on MNIST.

Contribution

The paper explicitly links robust generalization error to the Lipschitz constant and Rademacher complexity, providing bounds that unify previous theoretical results and empirical observations.

Findings

Robust generalization bounds depend on the Lipschitz constant and dataset size.

Empirical results on MNIST support the theoretical predictions about Lipschitz scaling.

The necessary Lipschitz constant for robust interpolation aligns with Wu et al. (2023) predictions.

Abstract

Bubeck and Sellke (2021) pose as an open problem the connection between the law of robustness and robust generalization. The law of robustness states that overparameterization is necessary for models to interpolate robustly; in particular, robust interpolation requires the learned function to be Lipschitz. Robust generalization asks whether small robust training loss implies small robust test loss. We resolve this problem by explicitly connecting the two for arbitrary data distributions. Specifically, we introduce a nontrivial notion of robust generalization error and convert it into a lower bound on the expected Rademacher complexity of the induced robust loss class. Our bounds recover the $\Omega(n^{1/d})$ regime of Wu et al. (2023) and show that, up to constants, robust generalization does not change the order of the Lipschitz constant required for smooth interpolation. We conduct experiments to probe the predicted scaling with dataset size and model capacity, testing whether empirical behavior aligns more closely with the predictions of Bubeck and Sellke (2021) or Wu et al. (2023). For MNIST, we find that the lower-bound Lipschitz constant scales on the order predicted by Wu et al. (2023). Informally, to obtain low robust generalization error, the Lipschitz constant must lie in a range that we bound, and the allowable perturbation radius is linked to the Lipschitz scale.