Quotient Gradient System Adversarial Training: Eliminating Robust Overfitting With a Nonlinear Dynamical System Approach

Adversarial training is widely considered the most effective method for improving model robustness, yet the robust overfitting hinders its performance. In this paper, we investigate remedies for robust overfitting and argue that effective strategies typically fall into two categories: confidence calibration and flat minimum methods, both of which are commonly formulated as regularised adversarial training approaches. We show that these methods can be unified under a constrained optimisation framework, revealing that flat minimum approaches can be interpreted as adaptive confidence calibration. To solve this constrained optimisation problem, we propose a novel adversarial training framework based on a powerful nonlinear dynamical system technique called the quotient gradient system (QGS). Specifically, we introduce QGS adversarial training (QGSAT) framework and develop two numerical methods termed the QGS method assisted by confidence calibration and the QGS method assisted by a flat minimum . We theoretically prove that QGSAT enforces effective regularisation by ensuring convergence to the feasible region, and achieves optimal regularisation under convexity when the region is empty. Extensive numerical studies demonstrate that QGSAT mitigates robust overfitting and outperforms existing regularised adversarial training methods. The code is publicly available at: https://github.com/yj373/QGSAT .