Improving the robustness of deep neural networks (DNNs) to adversarial examples is an important yet challenging problem for secure deep learning. Across existing defense techniques, adversarial training with Projected Gradient Decent (PGD) is amongst the most effective. Adversarial training solves a min-max optimization problem, with the \textit{inner maximization} generating adversarial examples by maximizing the classification loss, and the \textit{outer minimization} finding model parameters by minimizing the loss on adversarial examples generated from the inner maximization. A criterion that measures how well the inner maximization is solved is therefore crucial for adversarial training. In this paper, we propose such a criterion, namely First-Order Stationary Condition for constrained optimization (FOSC), to quantitatively evaluate the convergence quality of adversarial examples found in the inner maximization. With FOSC, we find that to ensure better robustness, it is essential to use adversarial examples with better convergence quality at the \textit{later stages} of training. Yet at the early stages, high convergence quality adversarial examples are not necessary and may even lead to poor robustness. Based on these observations, we propose a \textit{dynamic} training strategy to gradually increase the convergence quality of the generated adversarial examples, which significantly improves the robustness of adversarial training. Our theoretical and empirical results show the effectiveness of the proposed method.

本文提出了一种用于评估内部最大值的解决方法——FOSC，以量化评估在内部最大化期间找到的对抗性例子的收敛质量，结果表明，在训练后期使用收敛质量更好的对抗性例子是确保更好的稳健性所必需的，而在训练初期，高收敛质量的对抗性例子则并不是必需的，甚至可能导致差的鲁棒性，因此，我们提出了一种渐进式的动态训练策略，逐步提高生成的对抗性例子的收敛质量，从而显著提高了对抗训练的鲁棒性。

对抗训练的收敛性和鲁棒性