Adversarial attacks of neural network classifiers (NNC) and the use of random
noises in these methods have stimulated a large number of works in recent
years. However, despite all the previous investigations, existing approaches
that rely on random noises to fool NNC have fallen far short of
the-state-of-the-art adversarial methods performances. In this paper, we fill
this gap by introducing stochastic sparse adversarial attacks (SSAA), standing
as simple, fast and purely noise-based targeted and untargeted attacks of NNC.
SSAA offer new examples of sparse (or $L_0$) attacks for which only few methods
have been proposed previously. These attacks are devised by exploiting a
small-time expansion idea widely used for Markov processes. Experiments on
small and large datasets (CIFAR-10 and ImageNet) illustrate several advantages
of SSAA in comparison with the-state-of-the-art methods. For instance, in the
untargeted case, our method called voting folded Gaussian attack (VFGA) scales
efficiently to ImageNet and achieves a significantly lower $L_0$ score than
SparseFool (up to $frac{1}{14}$ lower) while being faster. In the targeted
setting, VFGA achives appealing results on ImageNet and is significantly much
faster than Carlini-Wagner $L_0$ attack.

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