Adversarial training augments the training set with perturbations to improve
the robust error (over worst-case perturbations), but it often leads to an
increase in the standard error (on unperturbed test inputs). Previous
explanations for this tradeoff rely on the assumption that no predictor in the
hypothesis class has low standard and robust error. In this work, we precisely
characterize the effect of augmentation on the standard error in linear
regression when the optimal linear predictor has zero standard and robust
error. In particular, we show that the standard error could increase even when
the augmented perturbations have noiseless observations from the optimal linear
predictor. We then prove that the recently proposed robust self-training (RST)
estimator improves robust error without sacrificing standard error for
noiseless linear regression. Empirically, for neural networks, we find that RST
with different adversarial training methods improves both standard and robust
error for random and adversarial rotations and adversarial $\ell_\infty$
perturbations in CIFAR-10.

本研究通过对线性回归中对最优线性预测器进行噪声扰动的方法，准确描述了数据增广对于标准误差的影响，并证明了最近提出的鲁棒自学习估计器（RST）可以在不损失标准误差的情况下提高鲁棒误差。经验证，使用不同的对抗训练方法进行 RST 可以改善 CIFAR-10 数据集上的标准误差和鲁棒误差。

理解和减轻稳健性与准确性之间的权衡

Understanding and Mitigating the Tradeoff Between Robustness and  Accuracy

Recent work has developed methods for learning deep network classifiers that
are provably robust to norm-bounded adversarial perturbation; however, these
methods are currently only possible for relatively small feedforward networks.
In this paper, in an effort to scale these approaches to substantially larger
models, we extend previous work in three main directions. First, we present a
technique for extending these training procedures to much more general
networks, with skip connections (such as ResNets) and general nonlinearities;
the approach is fully modular, and can be implemented automatically (analogous
to automatic differentiation). Second, in the specific case of $\ell_\infty$
adversarial perturbations and networks with ReLU nonlinearities, we adopt a
nonlinear random projection for training, which scales linearly in the number
of hidden units (previous approaches scaled quadratically). Third, we show how
to further improve robust error through cascade models. On both MNIST and CIFAR
data sets, we train classifiers that improve substantially on the state of the
art in provable robust adversarial error bounds: from 5.8% to 3.1% on MNIST
(with $\ell_\infty$ perturbations of $\epsilon=0.1$), and from 80% to 36.4% on
CIFAR (with $\ell_\infty$ perturbations of $\epsilon=2/255$). Code for all
experiments in the paper is available at
this https URL

本文提出了一种技术，能够将深度学习分类器的防御性能从较小的前馈神经网络拓展到更广泛的网络结构，同时采用非线性随机投影的方式进行训练，并通过级联模型进一步提高分类器的鲁棒性能。在 MNIST 和 CIFAR 数据集上进行实验，证明了该方法在可证明的抗干扰错误率上有着明显的提升。