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Title: Batch Normalization is a Cause of Adversarial Vulnerability

Paper content:
Batch normalization (BN) is often used in an attempt to stabilize and accelerate training in deep neural networks.
In many cases it indeed decreases the number of parameter updates required to achieve low training error.
However, it also reduces robustness to small adversarial input perturbations and common corruptions by double-digit percentages, as we show on five standard datasets.
Furthermore, we find that substituting weight decay for BN is sufficient to nullify a relationship between adversarial vulnerability and the input dimension.
A recent mean-field analysis found that BN induces gradient explosion when used on multiple layers, but this cannot fully explain the vulnerability we observe, given that it occurs already for a single BN layer.
We argue that the actual cause is the tilting of the decision boundary with respect to the nearest-centroid classifier along input dimensions of low variance.
As a result, the constant introduced for numerical stability in the BN step acts as an important hyperparameter that can be tuned to recover some robustness at the cost of standard test accuracy.
We explain this mechanism explicitly on a linear ``toy model and show in experiments that it still holds for nonlinear ``real-world models.
BN is a standard component of modern deep neural networks, and tends to make the training process less sensitive to the choice of hyperparameters in many cases (Ioffe & Szegedy, 2015) .
While ease of training is desirable for model developers, an important concern among stakeholders is that of model robustness during deployment to plausible, previously unseen inputs.
The adversarial examples phenomenon has exposed unstable predictions across state-of-the-art models (Szegedy et al., 2014) .
This has led to a variety of methods that aim to improve robustness, but doing so effectively remains a challenge (Athalye et al., 2018; Schott et al., 2019; Hendrycks & Dietterich, 2019; Jacobsen et al., 2019a) .
We believe that a prerequisite to developing methods that increase robustness is an understanding of factors that reduce it.
Approaches for improving robustness often begin with existing neural network architectures-that use BN-and patch them against specific attacks, e.g., through inclusion of adversarial examples during training (Szegedy et al., 2014; Goodfellow et al., 2015; Kurakin et al., 2017; Madry et al., 2018 ).
An implicit assumption is that BN itself does not reduce robustness, however, recent initialization-time analyses have shown that it causes exploding gradients, and increased sensitivity to input perturbations as the network depth increases (Yang et al., 2019; Labatie, 2019) .
In this work, we consider the impact of BN in practical scenarios in terms of robustness to common corruptions (Hendrycks & Dietterich, 2019) and adversarial examples (Szegedy et al., 2014) , finding that BN induced sensitivity remains a concern even in cases where its use appears benign on the basis of clean test accuracy, and when only one BN layer is used.
The frequently made observation that adversarial vulnerability can scale with the input dimension (Goodfellow et al., 2015; Gilmer et al., 2018; Simon-Gabriel et al., 2019) highlights the importance of identifying regularizers as more than merely a way to improve test accuracy.
In particular, BN was a confounding factor in Simon-Gabriel et al. (2019) , making the results of their initialization-time analysis hold after training.
By adding 2 regularization and removing BN, we show that there is no inherent relationship between adversarial vulnerability and the input dimension.
We found that there is no free lunch with batch norm when model robustness is a concern: the accelerated training properties and occasionally higher clean test accuracy come at the cost of increased vulnerability, both to additive noise and for adversarial perturbations.
We have shown that there is no inherent relationship between the input dimension and vulnerability.
Our results highlight the importance of identifying the disparate mechanisms of regularization techniques.

assistant: text: Batch normalization reduces robustness at test-time to common corruptions and adversarial examples.