  It is necessary to improve the performance of some special classes or to
particularly protect them from attacks in adversarial learning. This paper
proposes a framework combining cost-sensitive classification and adversarial
learning together to train a model that can distinguish between protected and
unprotected classes, such that the protected classes are less vulnerable to
adversarial examples. We find in this framework an interesting phenomenon
during the training of deep neural networks, called Min-Max property, that is,
the absolute values of most parameters in the convolutional layer approach zero
while the absolute values of a few parameters are significantly larger becoming
bigger. Based on this Min-Max property which is formulated and analyzed in a
view of random distribution, we further build a new defense model against
adversarial examples for adversarial robustness improvement. An advantage of
the built model is that it does no longer need adversarial training, and thus,
has a higher computational efficiency than most existing models of needing
adversarial training. It is experimentally confirmed that, regarding the
average accuracy of all classes, our model is almost as same as the existing
models when an attack does not occur and is better than the existing models
when an attack occurs. Specifically, regarding the accuracy of protected
classes, the proposed model is much better than the existing models when an
attack occurs.
