Why the Maximum Second Derivative of Activations Matters for Adversarial Robustness

This work investigates the critical role of activation function curvature -- quantified by the maximum second derivative $\max|\sigma''|$ -- in adversarial robustness. Using the Recursive Curvature-Tunable Activation Family (RCT-AF), which enables precise control over curvature through parameters $\alpha$ and $\beta$, we systematically analyze this relationship. Our study reveals a fundamental trade-off: insufficient curvature limits model expressivity, while excessive curvature amplifies the normalized Hessian diagonal norm of the loss, leading to sharper minima that hinder robust generalization. This results in a non-monotonic relationship where optimal adversarial robustness consistently occurs when $\max|\sigma''|$ falls within 4 to 10, a finding that holds across diverse network architectures, datasets, and adversarial training methods. We provide theoretical insights into how activation curvature affects the diagonal elements of the hessian matrix of the loss, and experimentally demonstrate that the normalized Hessian diagonal norm exhibits a U-shaped dependence on $\max|\sigma''|$, with its minimum within the optimal robustness range, thereby validating the proposed mechanism.