We propose a general approach to evaluating the performance of robust estimators based on adversarial losses under misspecified models. We first show that adversarial risk is equivalent to the risk induced by a distributional adversarial attack under certain smoothness conditions. This ensures that the adversarial training procedure is well-defined. To evaluate the generalization performance of the adversarial estimator, we study the adversarial excess risk. Our proposed analysis method includes investigations on both generalization error and approximation error. We then establish non-asymptotic upper bounds for the adversarial excess risk associated with Lipschitz loss functions. In addition, we apply our general results to adversarial training for classification and regression problems. For the quadratic loss in nonparametric regression, we show that the adversarial excess risk bound can be improved over those for a general loss.

翻译：我们提出了一种通用方法，用于评估误指定模型下基于对抗损失的稳健估计量性能。我们首先证明，在特定光滑性条件下，对抗性风险等价于由分布对抗攻击诱导的风险。这确保了对抗训练过程的良好定义性。为评估对抗估计量的泛化性能，我们研究了对抗性超风险。所提出的分析方法包括对泛化误差与逼近误差的考量。随后，我们建立了与Lipschitz损失函数相关的对抗性超风险的非渐近上界。此外，我们将一般性结论应用于分类与回归问题的对抗训练。针对非参数回归中的二次损失，我们证明对抗性超风险界可优于一般损失下的对应结果。