Measuring the Transferability of ℓ∞ Attacks by the ℓ2 Norm

Abstract

Deep neural networks could be fooled by adversarial examples with trivial differences to original samples. To keep the difference imperceptible in human eyes, researchers bound the adversarial perturbations by the ℓ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</inf> norm, which is now commonly served as the standard to align the strength of different attacks for a fair comparison. However, we propose that using the ℓ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</inf> norm alone is not sufficient in measuring the attack strength, because even with a fixed ℓ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</inf> distance, the ℓ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> distance also greatly affects the attack transferability between models. Through the discovery, we reach more in-depth understandings towards the attack mechanism, i.e., several existing methods attack black-box models better partly because they craft perturbations with 70% to 130% larger ℓ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> distances. Since larger perturbations naturally lead to better transferability, we thereby advocate that the strength of attacks should be simultaneously measured by both the ℓ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</inf> and ℓ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> norm. Our proposal is firmly supported by extensive experiments on ImageNet dataset from 7 attacks, 4 white-box models, and 9 black-box models.