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Numerous methods for crafting adversarial examples were proposed recentlywith high success rate. Since most existing machine learning based classifiersnormalize images into some continuous, real vector, domain firstly, attacksoften craft adversarial examples in such domain. However, "adversarial"examples may become benign after denormalizing them back into the discreteinteger domain, known as the discretization problem. This problem was mentionedin some work, but has received relatively little attention.  In this work, we first conduct a comprehensive study of existing methods andtools for crafting. We theoretically analyze 34 representative methods andempirically study 20 representative open source tools for crafting adversarialimages. Our study reveals that the discretization problem is far more seriousthan originally thought. This suggests that the discretization problem shouldbe taken into account seriously when crafting adversarial examples andmeasuring attack success rate. As a first step towards addressing this problemin black-box scenario, we propose a black-box method which reduces theadversarial example searching problem to a derivative-free optimizationproblem. Our method is able to craft adversarial images by derivative-freesearch in the discrete integer domain. Experimental results show that ourmethod is comparable to recent white-box methods (e.g., FGSM, BIM and C\&W) andachieves significantly higher success rate in terms of adversarial examples inthe discrete integer domain than recent black-box methods (e.g., ZOO, NES-PGDand Bandits). Moreover, our method is able to handle models that isnon-differentiable and successfully break the winner of NIPS 2017 competitionon defense with 95\% success rate. Our results suggest that discreteoptimization algorithms open up a promising area of research into effectiveblack-box attacks.

Taking Care of The Discretization Problem: A Comprehensive Study of the  Discretization Problem and A Black-Box Adversarial Attack in Discrete Integer  Domain