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Adversarial examples have recently drawn considerable attention in the fieldof machine learning due to the fact that small perturbations in the data canresult in major performance degradation. This phenomenon is usually modeled bya malicious adversary that can apply perturbations to the data in a constrainedfashion, such as being bounded in a certain norm. In this paper, we study thisproblem when the adversary is constrained by the $\ell_0$ norm; i.e., it canperturb a certain number of coordinates in the input, but has no limit on howmuch it can perturb those coordinates. Due to the combinatorial nature of thissetting, we need to go beyond the standard techniques in robust machinelearning to address this problem. We consider a binary classification scenariowhere $d$ noisy data samples of the true label are provided to us afteradversarial perturbations. We introduce a classification method which employs anonlinear component called truncation, and show in an asymptotic scenario, aslong as the adversary is restricted to perturb no more than $\sqrt{d}$ datasamples, we can almost achieve the optimal classification error in the absenceof the adversary, i.e. we can completely neutralize adversary's effect.Surprisingly, we observe a phase transition in the sense that using a converseargument, we show that if the adversary can perturb more than $\sqrt{d}$coordinates, no classifier can do better than a random guess.

Binary Classification Under $\ell_0$ Attacks for General Noise Distribution