Hard-label Manifolds: Unexpected Advantages of Query Efficiency for Finding On-manifold Adversarial Examples

Designing deep networks robust to adversarial examples remains an open problem. Likewise, recent zeroth order hard-label attacks on image classification models have shown comparable performance to their first-order, gradient-level alternatives. It was recently shown in the gradient-level setting that regular adversarial examples leave the data manifold, while their on-manifold counterparts are in fact generalization errors. In this paper, we argue that query efficiency in the zeroth-order setting is connected to an adversary's traversal through the data manifold. To explain this behavior, we propose an information-theoretic argument based on a noisy manifold distance oracle, which leaks manifold information through the adversary's gradient estimate. Through numerical experiments of manifold-gradient mutual information, we show this behavior acts as a function of the effective problem dimensionality and number of training points. On real-world datasets and multiple zeroth-order attacks using dimension-reduction, we observe the same universal behavior to produce samples closer to the data manifold. This results in up to two-fold decrease in the manifold distance measure, regardless of the model robustness. Our results suggest that taking the manifold-gradient mutual information into account can thus inform better robust model design in the future, and avoid leakage of the sensitive data manifold.