Beyond the Universal Law of Robustness: Sharper Laws for Random Features and Neural Tangent Kernels
This work provides sharper theoretical insights into model robustness, which is crucial for developing more secure AI systems, though it is incremental by refining existing laws.
The paper tackles the problem of adversarial robustness in machine learning by analyzing two models, random features and neural tangent kernels (NTK), proving that random features are not robust under any over-parameterization while NTK with even activations meets robustness bounds, addressing a prior conjecture.
Machine learning models are vulnerable to adversarial perturbations, and a thought-provoking paper by Bubeck and Sellke has analyzed this phenomenon through the lens of over-parameterization: interpolating smoothly the data requires significantly more parameters than simply memorizing it. However, this "universal" law provides only a necessary condition for robustness, and it is unable to discriminate between models. In this paper, we address these gaps by focusing on empirical risk minimization in two prototypical settings, namely, random features and the neural tangent kernel (NTK). We prove that, for random features, the model is not robust for any degree of over-parameterization, even when the necessary condition coming from the universal law of robustness is satisfied. In contrast, for even activations, the NTK model meets the universal lower bound, and it is robust as soon as the necessary condition on over-parameterization is fulfilled. This also addresses a conjecture in prior work by Bubeck, Li and Nagaraj. Our analysis decouples the effect of the kernel of the model from an "interaction matrix", which describes the interaction with the test data and captures the effect of the activation. Our theoretical results are corroborated by numerical evidence on both synthetic and standard datasets (MNIST, CIFAR-10).