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Title: A Perturbation Analysis of Input Transformations for Adversarial Attacks

Paper content:
The existence of adversarial examples, or intentional mis-predictions constructed from small changes to correctly predicted examples, is one of the most significant challenges in neural network research today.
Ironically, many new defenses are based on a simple observation - the adversarial inputs themselves are not robust and small perturbations to the attacking input often recover the desired prediction.
While the intuition is somewhat clear, a detailed understanding of this phenomenon is missing from the research literature.
This paper presents a comprehensive experimental analysis of when and why perturbation defenses work and potential mechanisms that could explain their effectiveness (or ineffectiveness) in different settings.
The attacks on Convolutional Neural Networks, such as Carlini & Wanger or PGD (Madry et al., 2017) , generate strategically placed modifications that can be easily dominated by different types of perturbations resulting in correct predictions (Dziugaite et al., 2016; Roth et al., 2019) .
This suggests that the standard adversarial examples are not robust.
Many defense techniques explicitly leverage this property and can be retrospectively interpreted as perturbations of the input images.
However, a detailed understanding of this phenomenon is lacking from the research literature including:
(1) what types of perturbations work and what is their underlying mechanism, (2) whether all attacks exhibit this property, and (3) possible counter-measures attackers can employ to defeat perturbation defenses.
We can interpret a large number of recent defenses as a type of input perturbations, for example, feature squeezing (Xu et al., 2017) , frequency or JPEG compression (Dziugaite et al., 2016) , randomized smoothing (Cohen et al., 2019) , and perturbation of network structure or the inputs randomly (JafarniaJahromi et al., 2018; Zhang & Liang, 2019; Guo et al., 2017) .
The defense techniques exhibit very similar gains in robustness.
To show it, we start with a simple model where every example is passed through a lossy channel (stochastic or deterministic) prior to model inference.
This channel induces a small perturbation to the input.
We optimize the perturbation to be small enough as not to affect the prediction accuracy on clean examples but large enough to dominate any adversarial attack.
We find that this trade-off is surprisingly consistent across very different families of input perturbations, where the relationship between channel distortion (the L 2 distance between channel input and output) and robustness is very similar.
Why are some state-of-the-art attacks are sensitive to perturbation-based defenses?
We find that many attacks execute an optimization procedure that finds an adversarial image that is very close to the original image in terms of of L 1 , L 2 , or L ∞ norm.
The resultant optimum, i.e., the adversarial image, tends to exhibit a higher level of instability than natural examples, which we demonstrate from the perspective of a first-order and second-order analysis.
By instability we mean that small perturbations of the example can affect the prediction confidences.
The unification of perturbation-based defense also gives us some insight into how an attacker might avoid them.
Our experiments suggest that all the perturbation based defenses are vulnerable to the same types of attack strategies.
We argue that the optimization procedure in the attacker should find the smallest distance from the original image that closes the recovery window.
In fact, we can devise a generic attacker that attacks a particularly strong lossy channel, based on the additive Laplace noise, and adaptive attacks designed on this channel are often successful against other defenses.
This result implies that for many input perturbation defenses the attacker need not be fully adaptive, i.e., they do not need to know exactly what kind of transformation is used to defend the network.
The non-adaptive attacks are not robust since small changes to the adversarial input often recover the original label.
This is an obvious corollary to the very existence of adversarial examples that by definition are relatively close to correctly predicted examples in the input space.
Random perturbations of the input can dominate the strategically placed perturbations synthesized by an attack.
In fact, the results are consistent across both deterministic and stochastic channels that degrade the fidelity of the input example.
From the perspective of the attacker, the recovery window can be closed to make the perturbation based recovery techniques ineffective.
A PERTURBATION ANALYSIS: ADDENDUM
From the Cauchy-Schwarz inequality:
From the definition of maximum eigenvalue :

assistant: text: The paper studies the robustness of adversarial attacks to transformations of their input.