Adversarial training can achieve robustness against adversarial perturbations and has been widely used in machine learning models. This paper delivers a non-asymptotic consistency analysis of the adversarial training procedure under $\ell_\infty$-perturbation in high-dimensional linear regression. It will be shown that the associated convergence rate of prediction error can achieve the minimax rate up to a logarithmic factor in the high-dimensional linear regression on the class of sparse parameters. Additionally, the group adversarial training procedure is analyzed. Compared with classic adversarial training, it will be proved that the group adversarial training procedure enjoys a better prediction error upper bound under certain group-sparsity patterns.

翻译：对抗训练能够实现对对抗性扰动的鲁棒性，并已广泛应用于机器学习模型。本文对高维线性回归中$\ell_\infty$扰动下的对抗训练过程进行了非渐近一致性分析。结果表明，在稀疏参数类的高维线性回归中，预测误差的收敛速率在达到极小极大速率时仅相差一个对数因子。此外，本文还分析了分组对抗训练过程。与经典对抗训练相比，研究证明在特定群组稀疏模式下，分组对抗训练过程具有更优的预测误差上界。