Improving Neural Network Robustness via Persistency of Excitation

Improving adversarial robustness of neural networks remains a major challenge. Fundamentally, training a neural network via gradient descent is a parameter estimation problem. In adaptive control, maintaining persistency of excitation (PoE) is integral to ensuring convergence of parameter estimates in dynamical systems to their true values. We show that parameter estimation with gradient descent can be modeled as a sampling of an adaptive linear time-varying continuous system. Leveraging this model, and with inspiration from Model-Reference Adaptive Control (MRAC), we prove a sufficient condition to constrain gradient descent updates to reference persistently excited trajectories converging to the true parameters. The sufficient condition is achieved when the learning rate is less than the inverse of the Lipschitz constant of the gradient of loss function. We provide an efficient technique for estimating the corresponding Lipschitz constant in practice using extreme value theory. Our results in both standard and adversarial training illustrate that networks trained with the PoE-motivated learning rate schedule have similar clean accuracy but are significantly more robust to adversarial attacks than models trained using current state-of-the-art heuristics.