Decision making and learning in the presence of uncertainty has attracted significant attention in view of the increasing need to achieve robust and reliable operations. In the case where uncertainty stems from the presence of adversarial attacks this need is becoming more prominent. In this paper we focus on linear and nonlinear classification problems and propose a novel adversarial training method for robust classifiers, inspired by Support Vector Machine (SVM) margins. We view robustness under a data driven lens, and derive finite sample complexity bounds for both linear and non-linear classifiers in binary and multi-class scenarios. Notably, our bounds match natural classifiers' complexity. Our algorithm minimizes a worst-case surrogate loss using Linear Programming (LP) and Second Order Cone Programming (SOCP) for linear and non-linear models. Numerical experiments on the benchmark MNIST and CIFAR10 datasets show our approach's comparable performance to state-of-the-art methods, without needing adversarial examples during training. Our work offers a comprehensive framework for enhancing binary linear and non-linear classifier robustness, embedding robustness in learning under the presence of adversaries.

通过受支持向量机（SVM）边界启发，本文提出了一种新的对抗训练方法来提高鲁棒分类器对线性和非线性分类问题中不确定性的处理能力。我们通过数据驱动的视角来解释鲁棒性，并推导了二元和多类情景下线性和非线性分类器的有限样本复杂度界限。我们的算法通过线性规划（LP）和二阶锥规划（SOCP）来最小化最坏情况下的代理损失函数，适用于线性和非线性模型。在MNIST和CIFAR10基准数据集上的数值实验显示，我们的方法与最先进的方法相比具有可比性能，在训练过程中不需要对抗性示例。我们的工作提供了一个全面的框架，用于增强二元线性和非线性分类器的鲁棒性，在面对对手时保持学习的鲁棒性。

关于具有有限样本复杂度保证的对抗学习的鲁棒优化