The Limitations of Adversarial Training and the Blind-Spot Attack

Introduction

Since the discovery of adversarial examples in deep neural networks (DNNs) (Szegedy et al., 2013), adversarial training under the robustness optimization framework (Madry et al., 2018; Sinha et al., 2018) has become one of the most effective methods to defend against adversarial examples. A recent study by Athalye et al. (2018) showed that adversarial training does not rely on obfuscated gradients and delivers promising results for defending adversarial examples on small datasets. Adversarial training approximately solves the following min-max optimization problem:

The effectiveness of adversarial training is measured by the robustness on the test set. However, the adversarial training process itself is done on the training set. Suppose we can optimize (1) perfectly, then certified robustness may be obtained on those training data points. However, if the empirical distribution of training dataset differs from the true data distribution, a test point drawn from the true data distribution might lie in a low probability region in the empirical distribution of training dataset and is not “covered” by the adversarial training procedure. For datasets that are relatively simple and have low intrinsic dimensions (MNIST, Fashion MNIST, etc), we can obtain enough training examples to make sure adversarial training covers most part of the data distribution. For high dimensional datasets (CIFAR, ImageNet), adversarial training have been shown difficult (Kurakin et al., 2016; Tramèr et al., 2018) and only limited success was obtained.

A recent attack proposed by Song et al. (2018) shows that adversarial training can be defeated when the input image is produced by a generative model (for example, a generative adversarial network) rather than selected directly from the test examples. The generated images are well recognized by humans and thus valid images in the ground-truth data distribution. In our interpretation, this attack effective finds the “blind-spots” in the input space that the training data do not well cover.

For higher dimensional datasets, we hypothesize that many test images already fall into these blind-spots of training data and thus adversarial training only obtains a moderate level of robustness. It is interesting to see that for those test images that adversarial training fails to defend, if their distances (in some metrics) to the training dataset are indeed larger. In our paper, we try to explain the success of robust optimization based adversarial training and show the limitations of this approach when the test points are slightly off the empirical distribution of training data. Our main contributions are:

We show that on the original set of test images, the effectiveness of adversarial training is highly correlated with the distance (in some distance metrics) from the test image to the manifold of training images. For MNIST and Fashion MNIST datasets, most test images are close to the training data and very good robustness is observed on these points. For CIFAR, there is a clear trend that the adversarially trained network gradually loses its robustness property when the test images are further away from training data.

We identify a new class of attacks, “blind-spot attacks”, where the input image resides in a “blind-spot” of the empirical distribution of training data (far enough from any training examples in some embedding space) but is still in the ground-truth data distribution (well recognized by humans and correctly classified by the model). Adversarial training cannot provide good robustness on these blind-spots and their adversarial examples have small distortions.

We show that blind-spots can be easily found on a few strong defense models including Madry et al. (2018), Wong & Kolter (2018) and Sinha et al. (2018). We propose a few simple transformations (slightly changing contrast and background), that do not noticeably affect the accuracy of adversarially trained MNIST and Fashion MNIST models, but these models become vulnerable to adversarial attacks on these sets of transformed input images. These transformations effectively move the test images slightly out of the manifold of training images, which does not affect generalization but poses a challenge for robust learning.

Our results imply that current adversarial training procedures cannot scale to datasets with a large (intrinsic) dimension, where any practical amount of training data cannot cover all the blind-spots. This explains the limited success for applying adversarial training on ImageNet dataset, where many test images can be sufficiently far away from the empirical distribution of training dataset.

Related Works

Adversarial examples in DNNs have brought great threats to the deep learning-based AI applications such as autonomous driving and face recognition. Therefore, defending against adversarial examples is an urgent task before we can safely deploy deep learning models to a wider range of applications. Following the emergence of adversarial examples, various defense methods have been proposed, such as defensive distillation by Papernot et al. (2016) and feature squeezing by Xu et al. (2017). Some of these defense methods have been proven vulnerable or ineffective under strong attack methods such as C&W in Carlini & Wagner (2017). Another category of recent defense methods is based on gradient masking or obfuscated gradient (Buckman et al. (2018); Ma et al. (2018); Guo et al. (2017); Song et al. (2017); Samangouei et al. (2018)), but these methods are also successfully evaded by the stronger BPDA attack (Athalye et al. (2018)). Randomization in DNNs (Dhillon et al., 2018; Xie et al., 2017; Liu et al., 2018) is also used to reduce the success rate of adversarial attacks, however, it usually incurs additional computational costs and still cannot fully defend against an adaptive attacker (Athalye et al., 2018; Athalye & Sutskever, 2017).

An effective defense method is adversarial training, which trains the model with adversarial examples freshly generated during the entire training process. First introduced by Goodfellow et al. , adversarial training demonstrates the state-of-the-art defending performance. Madry et al. (2018) formulated the adversarial training procedure into a min-max robust optimization problem and has achieved state-of-the-art defending performance on MNIST and CIFAR datasets. Several attacks have been proposed to attack the model release by Madry et al. (2018). On the MNIST testset, so far the best attack by Zheng et al. (2018) can only reduce the test accuracy from 98% to 88%. Analysis by Athalye et al. (2018) shows that this adversarial training framework does not rely on obfuscated gradient and truly increases model robustness; gradient based attacks with random starts can only achieve less than 10% success rate with given distortion constraints and are unable to penetrate this defense. On the other hand, attacking adversarial training using generative models have also been investigated; both Xiao et al. (2018) and Song et al. (2018) propose to use GANs to produce adversarial examples in black-box and white-box settings, respectively.

Finally, a few certified defense methods (Raghunathan et al., 2018; Sinha et al., 2018; Wong & Kolter, 2018) were proposed, which are able to provably increase model robustness. Besides adversarial training, in our paper we also consider several certified defenses which can achieve relatively good performance (i.e., test accuracy on natural images does not drop significantly and training is computationally feasible), and can be applied to medium-sized networks with multiple layers. Notably, Sinha et al. (2018) analyzes adversarial training using distributional robust optimization techniques. Wong & Kolter (2018) and Wong et al. (2018) proposed a robustness certificate based on the dual of a convex relaxation for ReLU networks, and used it for training to provably increase robustness. During training, certified defense methods can provably guarantee that the model is robust on training examples; however, on unseen test examples a non-vacuous robustness generalization guarantee is hard to obtain.

2 Analyzing Adversarial Examples

Along with the attack-defense arms race, some insightful findings have been discovered to understand the natural of adversarial examples, both theoretically and experimentally. Schmidt et al. (2018a) show that even for a simple data distribution of two class-conditional Gaussians, robust generalization requires significantly larger number of samples than standard generalization. Cullina et al. (2018) extend the well-known PAC learning theory to the case with adversaries, and derive the adversarial VC-dimension which can be either larger or smaller than the standard VC-dimension. Bubeck et al. (2018b) conjecture that a robust classifier can be computationally intractable to find, and give a proof for the computation hardness under statistical query (SQ) model. Recently, Bubeck et al. (2018a) prove a computational hardness result under a standard cryptographic assumption. Additionally, finding the safe area approximately is computationally hard according to Katz et al. (2017) and Weng et al. (2018). Mahloujifar et al. (2018) explain the prevalence of adversarial examples by making a connection to the “concentration of measure” phenomenon in metric measure spaces. Su et al. (2018) conduct large scale experiments on ImageNet and find a negative correlation between robustness and accuracy. Tsipras et al. (2019) discover that data examples consist of robust and non-robust features and adversarial training tends to find robust features that have strongly-correlations with the labels.

Both adversarial training and certified defenses significantly improve robustness on training data, but it is still unknown if the trained model has good robust generalization property. Typically, we evaluate the robustness of a model by computing an upper bound of error on the test set; specifically, given a norm bounded distortion $\epsilon$, we verify if each image in test set has a robustness certificate (Zhang et al., 2018; Dvijotham et al., 2018; Singh et al., 2018). There might exist test images that are still within the capability of standard generalization (i.e., correctly classified by DNNs with high confidence, and well recognized by humans), but behaves badly in robust generalization (i.e., adversarial examples can be easily found with small distortions). Our paper complements those existing findings by showing the strong correlation between the effectiveness of adversarial defenses (both adversarial training and some certified defenses) and the distance between training data and test points. Additionally, we show that a tiny shift in input distribution (which may or may not be detectable in embedding space) can easily destroy the robustness property of an robust model.

Methodology

To verify the correlation between the effectiveness of adversarial training and how close a test point is to the manifold of training dataset, we need to propose a reasonable distance metric between a test example and a set of training examples. However, defining a meaningful distance metric for high dimensional image data is a challenging problem. Naively using an Euclidean distance metric in the input space of images works poorly as it does not reflect the true distance between the images on their ground-truth manifold. One strategy is to use (kernel-)PCA, t-SNE (Maaten & Hinton, 2008), or UMAP (McInnes & Healy, 2018) to reduce the dimension of training data to a low dimensional space, and then define distance in that space. These methods are sufficient for small and simple datasets like MNIST, but for more general and complicated dataset like CIFAR, extracting a meaningful low-dimensional manifold directly on the input space can be really challenging.

On the other hand, using a DNN to extract features of input images and measuring the distance in the deep feature embedding space has demonstrated better performance in many applications (Hu et al., 2014; 2015), since DNN models can capture the manifold of image data much better than simple methods such as PCA or t-SNE. Although we can form an empirical distribution using kernel density estimation (KDE) on the deep feature embedding space and then obtain probability densities for test points, our experience showed that KDE work poorly in this case because the features extracted by DNNs are still high dimensional (hundreds or thousands dimensions).

Taking the above considerations into account, we propose a simple and intuitive distance metric using deep feature embeddings and $k$-nearest neighbour. Given a feature extraction neural network $h(x)$, a set of $n$ training data points $\mathcal{X}_{\text{train}}=\{x^{1}_{\text{train}},x^{2}_{\text{train}},\cdots,x^{n}_{\text{train}}\}$, and a set of $m$ test data points $\mathcal{X}_{\text{test}}=\{x_{\text{test}}^{1},x_{\text{test}}^{2},\cdots,x_{\text{test}}^{m}\}$ from the true data distribution, for each $j\in[m]$, we define the following distance between $x_{\text{test}}^{j}$ and $\mathcal{X}_{\text{train}}$:

In other words, we average the embedding space distance of $k$ nearest neighbors of $x_{j}$ in the training dataset. This simple metric is non-parametric and we found that the results are not sensitive to the selection of $k$; also, for naturally trained and adversarially trained feature extractors, the distance metrics obtained by different feature extractors reveal very similar correlations with the effectiveness of adversarial training.

2 Measuring the distance between training and test datasets

We are also interested to investigate the “distance” between the training dataset and the test dataset to gain some insights on how adversarial training performs on the entire test set. Unlike the setting in Section 3.1, this requires to compute a divergence between two empirical data distributions.

Given $n$ training data points $\mathcal{X}_{\text{train}}=\{x^{1}_{\text{train}},x^{2}_{\text{train}},\cdots,x^{n}_{\text{train}}\}$ and $m$ test data points $\mathcal{X}_{\text{test}}=\{x_{\text{test}}^{1},x_{\text{test}}^{2},\cdots,x_{\text{test}}^{m}\}$, we first apply a neural feature extractor $h$ to them, which is the same as in Section 3.1. Then, we apply a non-linear projection (in our case, we use t-SNE) to project both $h(x^{i}_{\text{train}})$ and $h(x^{j}_{\text{test}})$ to a low dimensional space, and obtain $\bar{x}^{i}_{\text{train}}=\text{proj}(h(x^{i}_{\text{train}}))$ and $\bar{x}^{j}_{\text{test}}=\text{proj}(h(x^{j}_{\text{test}}))$. The dataset after feature extraction and projection is denoted as $\bar{\mathcal{X}}_{\text{train}}$ and $\bar{\mathcal{X}}_{\text{test}}$. Because $\bar{x}^{i}_{\text{train}}$ and $\bar{x}^{j}_{\text{test}}$ are low dimensional, we can use kernel density estimation (KDE) to form empirical distributions $\bar{p}_{\text{train}}$ and $\bar{p}_{\text{test}}$ for them. We use $p_{\text{train}}$ and $p_{\text{test}}$ to denote the true distributions. Then, we approximate the K-L divergence between $p_{\text{train}}$ and $p_{\text{test}}$ via a numerical integration of Eq.(3):

where $\bar{p}_{\text{train}}(x)=\frac{1}{n}\sum_{i=1}^{n}K(x-\bar{x}^{i}_{\text{train}};H)$ and $\bar{p}_{\text{test}}(x)=\frac{1}{m}\sum_{j=1}^{m}K(x-\bar{x}^{j}_{\text{test}};H)$ are the KDE density functions. $K$ is the kernel function (specifically, we use the Gaussian kernel) and $H$ is the bandwidth parameter automatically selected by Scott’s rule (Scott, 2015). $V$ is chosen as a box bounding all training and test data points. For a multi-class dataset, we compute the aforementioned KDE and K-L divergence for each class separately. We should emphasize that this method only gives us a rough characterization which might help us understand the limitations of adversarial training. The true divergence between general training and test distributions in high dimensional space is not accessible in our setting.

3 The Blind-Spot Attack: a new class of adversarial attacks

Inspired by our findings of the negative correlation between the effectiveness of adversarial training and the distance between a test image and training dataset, we identify a new class of adversarial attacks called “blind-spot attacks”, where we find input images that are “far enough” from any existing training examples such that:

They are still drawn from the ground-truth data distribution (i.e. well recognized by humans) and classified correctly by the model (within the generalization capability of the model);

Adversarial training cannot provide good robustness properties on these images, and we can easily find their adversarial examples with small distortions using a simple gradient based attack.

Importantly, blind-spot images are not adversarial images themselves. However, after performing adversarial attacks, we can find their adversarial examples with small distortions, despite adversarial training. In other words, we exploit the weakness in a model’s robust generalization capability.

We find that these blind-spots are prevalent and can be easily found without resorting to complex generative models like in Song et al. (2018). For the MNIST dataset which Madry et al. (2018), Wong & Kolter (2018) and Sinha et al. (2018) demonstrate the strongest defense results so far, we propose a simple transformation to find the blind-spots in these models. We simply scale and shift each pixel value. Suppose the input image $x\in[-0.5,0.5]^{d}$, we scale and shift each test data example $x$ element-wise to form a new example $x^{\prime}$:

where $\alpha$ is a constant close to 1 and $\beta$ is a constant close to 0. We make sure that the selection of $\alpha$ and $\beta$ will result in a $x^{\prime}$ that is still in the valid input range $[-0.5,0.5]^{d}$. This transformation effectively adjusts the contrast of the image, and/or adds a gray background to the image. We then perform Carlini & Wagner’s attacks on these transformed images $x^{\prime}$ to find their adversarial examples $x^{\prime}_{\text{adv}}$. It is important that the blind-spot images $x^{\prime}$ are still undoubtedly valid images; for example, a digit that is slightly darker than the one in test set is still considered as a valid digit and can be well recognized by humans. Also, we found that with appropriate $\alpha$ and $\beta$ the accuracy of MNIST and Fashion-MNIST models barely decreases; the model has enough generalization capability for this set of slightly transformed images, yet their adversarial examples can be easily found.

Experiments

In this section we present our experimental results on adversarially trained models by Madry et al. (2018). Results on certified defense models by Wong & Kolter (2018); Wong et al. (2018) and Sinha et al. (2018) are very similar and are demonstrated in Section 6.4 in the Appendix.

We conduct experiments on adversarially trained models by Madry et al. (2018) on four datasets: MNIST, Fashion MNIST, and CIFAR-10. For MNIST, we use the “secret” model release for the MNIST attack challengehttps://github.com/MadryLab/mnist_challenge. For CIFAR-10, we use the public “adversarially trained” modelhttps://github.com/MadryLab/cifar10_challenge. For Fashion MNIST, we train our own model with the same model structure and parameters as the robust MNIST model, except that the iterative adversary is allowed to perturb each pixel by at most $\epsilon=0.1$ as a larger $\epsilon$ will significantly reduce model accuracy.

2 Effectiveness of adversarial training and the distance to training set

In this set of experiments, we build a connection between attack success rate on adversarially trained models and the distance between a test example and the whole training set. We use the metric defined in Section 3.1 to measure this distance. For MNIST and Fashion-MNIST, the outputs of the first fully connected layer (after all convolutional layers) are used as the neural feature extractor $h(x)$; for CIFAR, we use the outputs of the last average pooling layer. We consider both naturally and adversarially trained networks as the neural feature extractor, with $p=2$ and $k=5$. The results are shown in Figure 1, 2 and 3. For each test set, after obtaining the distance of each test point, we bin the test data points based on their distances to the training set and show them in the histogram at the bottom half of each figure (red). The top half of each figure (blue) represents the attack success rates for the test images in the corresponding bins. Some bars on the right are missing because there are too few points in the corresponding bins. We only attack correctly classified images and only calculate success rate on those images. Note that we should not compare the distances shown between the left and right columns of Figures 1, 2 and 3 because they are obtained using different embeddings, however the overall trends are very similar.

As we can observe in all three figures, most successful attacks in test sets for adversarially trained networks concentrate on the right hand side of the distance distribution, and the success rates tend to grow when the distance is increasing. The trend is independent of the feature extractor being used (naturally or adversarially trained). The strong correlation between attack success rates and the distance from a test point to the training dataset supports our hypothesis that adversarial training tends to fail on test points that are far enough from the training data distribution.

3 K-L Divergence between training and test sets vs attack success rate

Clearly, Fashion-MNIST is the dataset with the strongest defense as measured by the attack success rates on test set, and its K-L divergence is also the smallest. For CIFAR, the divergence between training and test sets is significantly larger, and adversarial training only has limited success. The hardness of training a robust model for MNIST is in between Fashion-MNIST and CIFAR. Another important observation is that the effectiveness of adversarial training does not depend on the accuracy; for Fashion-MNIST, classification is harder as the data is more complicated than MNIST, but training a robust Fashion-MNIST model is easier as the data distribution is more concentrated and adversarial training has less “blind-spots”.

4 Blind-Spot Attack on MNIST and Fashion MNIST

One might think that we can generally detect blind-spot attacks by observing their distances to the training dataset, using a metric similar to Eq. (2). Thus, we plot histograms for the distances between tests points and training dataset, for both original test images and those slightly transformed ones in Figure 5. We set $\alpha=0.7,\beta=0$ for MNIST and $\alpha=0.9,\beta=0$ for Fashion-MNIST. Unfortunately, the differences in distance histograms for these blind-spot images are so tiny that we cannot reliably detect the change, yet the robustness property drastically changes on these transformed images.

Conclusion

In this paper, we observe that the effectiveness of adversarial training is highly correlated with the characteristics of the dataset, and data points that are far enough from the distribution of training data are prone to adversarial attacks despite adversarial training. Following this observation, we defined a new class of attacks called “blind-spot attack” and proposed a simple scale-and-shift scheme for conducting blind-spot attacks on adversarially trained MNIST and Fashion MNIST datasets with high success rates. Our findings suggest that adversarial training can be challenging due to the prevalence of blind-spots in high dimensional datasets.

Acknowledgment

We thank Zeyuan Allen-Zhu, Lijie Chen, Sébastien Bubeck, Rasmus Kyng, Yin Tat Lee, Aleksander Mądry, Jelani Nelson, Eric Price, Ilya Razenshteyn, Aviad Rubinstein, Ludwig Schmidt and Pengchuan Zhang for fruitful discussions. We also thank Eric Wong for kindly providing us with pre-trained models to perform our experiments.

References

Appendix

As discussed in Section 3.1, we use $k$-nearest neighbour in embedding space to measure the distance between a test example and the training set. In Section 4.2 we use $k=5$. In this section we show that the choice of $k$ does not have much influence on our results. We use the adversarially trained model on the CIFAR dataset as an example. In Figures 6, 7 and 8 we choose $k=10,100,1000$, respectively. The results are similar to those we have shown in Figure 3: a strong correlation between attack success rates and the distance from a test point to the training dataset.

2 German traffic sign (GTS) dataset

We also studied the German Traffic Sign (GTS) (Houben et al., 2013) dataset. For GTS, we train our own model with the same model structure and parameters as the adversarially trained CIFAR model (Madry et al., 2018). We set $\epsilon=8/255$ for adversarial training with PGD, and also use the same $\epsilon$ as the threshold of success. The results are shown in Figure 9. The GTS model behaves similarly to the CIFAR model: attack success rates are much higher when the distances between the test example and the training dataset are larger.

3 More visualization results

We demonstrate more MNIST and Fashion-MNIST visualizations in Figure 10.

4 Results on other robust training methods

In this section we demonstrate our experimental results on two other state-of-the-art certified defense methods, including convex adversarial polytope by Wong et al. (2018) and Wong & Kolter (2018), and distributional robust optimization based adversarial training by Sinha et al. (2018). Different from the adversarial training by Madry et al. (2018), these two methods can provide a formal certification on the robustness of the model and provably improve robustness on the training dataset. However, they cannot practically guarantee non-trivial robustness on test data. We did not include other certified defenses like Raghunathan et al. (2018) and Hein & Andriushchenko (2017) because they are not applicable to multi-layer networks. For all defenses, we use their official implementations and pretrained models (if available). Figure 11 shows the results on CIFAR using the small CIFAR model in Wong et al. (2018). Tables 4 and 5 show the blind-spot attack results on MNIST and Fashion-MNIST for robust models in Wong & Kolter (2018) and Sinha et al. (2018), respectively. Figure 12 shows the blind-spot attack examples on Madry et al. (2018), Wong & Kolter (2018) and Sinha et al. (2018).