Geometry Is Not Robustness: A Trajectory-Level Study of PGD Evaluation

Abstract

Projected Gradient Descent (PGD) is widely used to evaluate adversarial robustness, typically via final adversarial accuracy, which does not capture model behaviour throughout the attack. Recent work proposes trajectory-level diagnostics, such as loss evolution, gradient alignment, and steps-to-failure, for deeper insight into adversarial optimisation dynamics. However, whether these diagnostics reliably indicate robustness strength remains unclear. We conduct a trajectory-level investigation of PGD attacks on convolutional neural networks trained on Fashion-MNIST. We compare clean-trained and adversarially-trained models across multiple robustness regimes, using rigorous 20-step PGD evaluations with random initialisation and multiple restarts for robustness measurement, and single-initialisation trajectory recording for diagnostics. We record full PGD trajectories across 3000 clean-correct samples per model and analyse loss evolution, gradient alignment, and failure timing across attack iterations. Our results reveal a clear robustness hierarchy across models; however, trajectory metrics do not contribute equally to its identification. Mean loss trajectories and gradient alignment patterns appear quantitatively similar across adversarially-trained models with substantially different robust accuracies. In contrast, steps-to-failure distributions provide a clearer separation of robustness regimes, directly reflecting functional resistance to adversarial perturbation. These findings indicate that trajectory-level diagnostics describe optimisation geometry but do not independently measure adversarial robustness. Their interpretability depends on robustness regime, attack strength, and multi-metric evaluation. Trajectory-level analysis should be a complementary diagnostic tool, interpreted in context, rather than a replacement for standard robustness measurements.