Countering Adversarial Images Using Input Transformations

The code for this paper is available here[1]

Motivation

As the use of machine intelligence has increased, robustness has become a critical feature to guarantee the reliability of deployed machine-learning systems. However, recent research has shown that existing models are not robust to small, adversarially designed perturbations of the input. Adversarial examples are inputs to Machine Learning models that an attacker has intentionally designed to cause the model to make a mistake. Adversarially perturbed examples have been deployed to attack image classification services (Liu et al., 2016)[11], speech recognition systems (Cisse et al., 2017a)[12], and robot vision (Melis et al., 2017)[13]. The existence of these adversarial examples has motivated proposals for approaches that increase the robustness of learning systems to such examples. In the example below (Goodfellow et. al) [17], a small perturbation is applied to the original image of a panda, changing the prediction to a gibbon.

Introduction

The paper studies strategies that defend against adversarial-example attacks on image-classification systems by transforming the images before feeding them to a Convolutional Network Classifier. Generally, defenses against adversarial examples fall into two main categories -

1. Model-Specific – They enforce model properties such as smoothness and in-variance via the learning algorithm.
2. Model-Agnostic – They try to remove adversarial perturbations from the input.

Model-specific defense strategies make strong assumptions about expected adversarial attacks. As a result, they violate the Kerchkoffs principle, which states that adversaries can circumvent model-specific defenses by simply changing how an attack is executed. This paper focuses on increasing the effectiveness of Model Agnostic defense strategies. Specifically, they investigate the following image transformations as a means for protecting against adversarial images:

1. Image Cropping and Re-scaling (Graese et al, 2016).
2. Bit Depth Reduction (Xu et. al, 2017)
3. JPEG Compression (Dziugaite et al, 2016)
4. Total Variance Minimization (Rudin et al, 1992)
5. Image Quilting (Efros & Freeman, 2001).

These image transformations have been studied against Adversarial attacks such as the fast gradient sign method (Goodfelow et. al., 2015), its iterative extension (Kurakin et al., 2016a), Deepfool (Moosavi-Dezfooli et al., 2016), and the Carlini & Wagner (2017) [math]\displaystyle{ L_2 }[/math]attack.

From their experiments, the strongest defenses are based on Total Variance Minimization and Image Quilting. These defenses are non-differentiable and inherently random which makes it difficult for an adversary to get around them.

Previous Work

Recently, a lot of research has focused on countering adversarial threats. Wang et al [4], proposed a new adversary resistant technique that obstructs attackers from constructing impactful adversarial images. This is done by randomly nullifying features within images. Tramer et al [2], showed the state-of-the-art Ensemble Adversarial Training Method, which augments the training process but not only included adversarial images constructed from their model but also including adversarial images generated from an ensemble of other models. Their method implemented on an Inception V2 classifier finished 1st among 70 submissions of NIPS 2017 competition on Defenses against Adversarial Attacks. Graese, et al. [3], showed how input transformation such as shifting, blurring and noise can render the majority of the adversarial examples as non-adversarial. Xu et al.[5] demonstrated, how feature squeezing methods, such as reducing the color bit depth of each pixel and spatial smoothing, defends against attacks. Dziugaite et al [6], studied the effect of JPG compression on adversarial images.

Terminology

Gray Box Attack : Model Architecture and parameters are Public

Black Box Attack: Adversary does not have access to the model.

Non Targeted Adversarial Attack: The goal of the attack is to modify a source image in a way such that the image will be classified incorrectly by the network.

Targeted Adversarial Attack: The goal of the attack is to modify a source image in way such that image will be classified as a target class by the network.

Defense: A defense is a strategy that aims make the prediction on an adversarial example h(x') equal to the prediction on the corresponding clean example h(x).

Problem Definition

The paper discusses non-targeted adversarial attacks for image recognition systems. Given image space [math]\displaystyle{ \mathcal{X} = [0,1]^{H \times W \times C} }[/math], a source image [math]\displaystyle{ x \in \mathcal{X} }[/math], and a classifier [math]\displaystyle{ h(.) }[/math], a non-targeted adversarial example of [math]\displaystyle{ x }[/math] is a perturbed image [math]\displaystyle{ x' }[/math], such that [math]\displaystyle{ h(x) \neq h(x') }[/math] and [math]\displaystyle{ d(x, x') \leq \rho }[/math] for some dissimilarity function [math]\displaystyle{ d(·, ·) }[/math] and [math]\displaystyle{ \rho \geq 0 }[/math]. In the best case scenario, [math]\displaystyle{ d(·, ·) }[/math] measures the perceptual difference between the original image [math]\displaystyle{ x }[/math] and the perturbed image [math]\displaystyle{ x' }[/math], but usually, Euclidean distance ([math]\displaystyle{ ||x - x'||_2 }[/math]) or the Chebyshov distance ([math]\displaystyle{ ||x - x'||_{\infty} }[/math]) are used.

From a set of N clean images [math]\displaystyle{ [{x_{1}, …, x_{n}}] }[/math], an adversarial attack aims to generate [math]\displaystyle{ [{x'_{1}, …, x'_{n}}] }[/math] images, such that ([math]\displaystyle{ x'_{n} }[/math]) is an adversary of ([math]\displaystyle{ x_{n} }[/math]).

The success rate of an attack is given as:

,

which is the proportions of predictions that were altered by an attack.

The success rate is generally measured as a function of the magnitude of perturbations performed by the attack. In this paper, L2 perturbations are used and are quantified using the normalized L2-dissimilarity metric: [math]\displaystyle{ \frac{1}{N} \sum_{n=1}^N{\frac{\vert \vert x_n - x'_n \vert \vert_2}{\vert \vert x_n \vert \vert_2}} }[/math]

A strong adversarial attack has a high rate, while its normalized L2-dissimilarity given by the above equation is less.

Adversarial Attacks

For the experimental purposes, below 4 attacks have been studied in the paper:

1. Fast Gradient Sign Method (FGSM; Goodfellow et al. (2015)) [17]: Given a source input [math]\displaystyle{ x }[/math], and true label [math]\displaystyle{ y }[/math], and let [math]\displaystyle{ l(.,.) }[/math] be the differentiable loss function used to train the classifier [math]\displaystyle{ h(.) }[/math]. Then the corresponding adversarial example is given by:

[math]\displaystyle{ x' = x + \epsilon \cdot sign(\nabla_x l(x, y)) }[/math]

for some [math]\displaystyle{ \epsilon \gt 0 }[/math] which controls the perturbation magnitude.

2. Iterative FGSM ((I-FGSM; Kurakin et al. (2016b)) [14]: iteratively applies the FGSM update, where M is the number of iterations. It is given as:

[math]\displaystyle{ x^{(m)} = x^{(m-1)} + \epsilon \cdot sign(\nabla_{x^{m-1}} l(x^{m-1}, y)) }[/math]

where [math]\displaystyle{ m = 1,...,M; x^{(0)} = x; }[/math] and [math]\displaystyle{ x' = x^{(M)} }[/math]. M is set such that [math]\displaystyle{ h(x) \neq h(x') }[/math].

Both FGSM and I-FGSM work by minimizing the Chebyshov distance between the inputs and the generated adversarial examples.

3. DeepFool ((Moosavi-Dezfooliet al., 2016) [15]: projects x onto a linearization of the decision boundary defined by binary classifier h(.) for M iterations. This can be particularly effictive when a network uses ReLU activation functions. It is given as:

4. Carlini-Wagner's L2 attack (CW-L2; Carlini & Wagner (2017)) [16]: propose an optimization-based attack that combines a differentiable surrogate for the model’s classification accuracy with an L2-penalty term which encourages the adversary image to be close to the original image. Let [math]\displaystyle{ Z(x) }[/math] be the operation that computes the logit vector (i.e., the output before the softmax layer) for an input [math]\displaystyle{ x }[/math], and [math]\displaystyle{ Z(x)_k }[/math] be the logit value corresponding to class [math]\displaystyle{ k }[/math]. The untargeted variant of CW-L2 finds a solution to the unconstrained optimization problem. It is given as:

As mentioned earlier, the first two attacks minimize the Chebyshov distance whereas the last two attacks minimize the Euclidean distance between the inputs and the adversarial examples.

All the methods described above maintain [math]\displaystyle{ x' \in \mathcal{X} }[/math] by performing value clipping.

Below figure shows adversarial images and corresponding perturbations at five levels of normalized L2-dissimilarity for all four attacks, mentioned above.

Defenses

Defense is a strategy that aims make the prediction on an adversarial example equal to the prediction on the corresponding clean example. Five image transformations that alter the structure of these perturbations have been studied:

1. Image Cropping and Re-scaling,
2. Bit Depth Reduction,
3. JPEG Compression,
4. Total Variance Minimization,
5. Image Quilting.

Image cropping and Rescaling has the effect of altering the spatial positioning of the adversarial perturbation. In this study, images are cropped and re-scaled during training time. At test time, the predictions of randomly cropped are averaged.

Bit Depth Reduction (Xu et. al) [5] performs a simple type of quantization that can remove small (adversarial) variations in pixel values from an image. Images are reduced to 3 bits in the experiment.

JPEG Compression and Decompression (Dziugaite etal., 2016) removes small perturbations by performing simple quantization. The authors use a quality level of 75/100 in their experiments

Total Variance Minimization (Rudin et. al) [9] : This combines pixel dropout with total variance minimization. This approach randomly selects a small set of pixels, and reconstructs the “simplest” image that is consistent with the selected pixels. The reconstructed image does not contain the adversarial perturbations because these perturbations tend to be small and localized.Specifically, we first select a random set of pixels by sampling a Bernoulli random variable [math]\displaystyle{ X(i; j; k) }[/math] for each pixel location [math]\displaystyle{ (i; j; k) }[/math];we maintain a pixel when [math]\displaystyle{ (i; j; k) }[/math]= 1. Next, we use total variation, minimization to constructs an image z that is similar to the (perturbed) input image x for the selected set of pixels, whilst also being “simple” in terms of total variation by solving:

,

where [math]\displaystyle{ TV_{p}(z) }[/math] represents [math]\displaystyle{ L_{p} }[/math] total variation of z :

The total variation (TV) measures the amount of fine-scale variation in the image z, as a result of which TV minimization encourages removal of small (adversarial) perturbations in the image.

Image Quilting (Efros & Freeman, 2001) [8] Image Quilting is a non-parametric technique that synthesizes images by piecing together small patches that are taken from a database of image patches. The algorithm places appropriate patches in the database for a predefined set of grid points and computes minimum graph cuts in all overlapping boundary regions to remove edge artifacts. Image Quilting can be used to remove adversarial perturbations by constructing a patch database that only contains patches from "clean" images ( without adversarial perturbations); the patches used to create the synthesized image are selected by finding the K nearest neighbors ( in pixel space) of the corresponding patch from the adversarial image in the patch database, and picking one of these neighbors uniformly at random. The motivation for this defense is that resulting image only contains pixels that were not modified by the adversary - the database of real patches is unlikely to contain the structures that appear in adversarial images.

Experiments

Five experiments were performed to test the efficacy of defenses. The first four experiments consider gray and black box attacks, while the final experiment compares the authors' defenses with prior work.

Set up: Experiments are performed on the ImageNet image classification dataset. The dataset comprises 1.2 million training images and 50,000 test images that correspond to one of 1000 classes. The adversarial images are produced by attacking a ResNet-50 model, with different kinds of attacks mentioned in Section5. The strength of an adversary is measured in terms of its normalized L2-dissimilarity. To produce the adversarial images, L2 dissimilarity for each of the attack was set as below:

- FGSM. Increasing the step size [math]\displaystyle{ \epsilon }[/math], increases the normalized L2-dissimilarity.

- I-FGSM. We fix M=10, and increase [math]\displaystyle{ \epsilon }[/math] to increase the normalized L2-dissimilarity.

- DeepFool. We fix M=5, and increase [math]\displaystyle{ \epsilon }[/math] to increase the normalized L2-dissimilarity.

- CW-L2. We fix [math]\displaystyle{ k }[/math]=0 and [math]\displaystyle{ \lambda_{f} }[/math] =10, and multiply the resulting perturbation

The hyperparameters of the defenses have been fixed in all the experiments. Specifically the pixel dropout probability was set to [math]\displaystyle{ p }[/math]=0.5 and regularization parameter of total variation minimizer [math]\displaystyle{ \lambda_{TV} }[/math]=0.03.

Below figure shows the difference between the set up in different experiments below. The network is either trained on a) regular images or b) transformed images. The different settings are marked by 8.1, 8.2 and 8.3

GrayBox- Image Transformation at Test Time

This experiment applies a transformation on adversarial images at test time before feeding them to a ResNet -50 which was trained to classify clean images. Below figure shows the results for five different transformations applied and their corresponding Top-1 accuracy. Few of the interesting observations from the plot are: All of the image transformations partly eliminate the effects of the attack, Crop ensemble gives the best accuracy around 40-60 percent, with an ensemble size of 30. The accuracy of Image Quilting Defense hardly deteriorates as the strength of the adversary increases. However, it does impact accuracy on non-adversarial examples.

BlackBox - Image Transformation at Training and Test Time

ResNet-50 model was trained on transformed ImageNet Training images. Before feeding the images to the network for training, standard data augmentation (from He et al) along with bit depth reduction, JPEG Compression, TV Minimization, or Image Quilting were applied on the images. The classification accuracy on the same adversarial images as in the previous case is shown Figure below. (Adversary cannot get this trained model to generate new images - Hence this is assumed as a Black Box setting!). Below figure concludes that training Convolutional Neural Networks on images that are transformed in the same way at test time, dramatically improves the effectiveness of all transformation defenses. Nearly 80 -90 % of the attacks are defended successfully, even when the L2- dissimilarity is high.

Blackbox - Ensembling

Four networks ResNet-50, ResNet-10, DenseNet-169, and Inception-v4 along with an ensemble of defenses were studied, as shown in Table 1. The adversarial images are produced by attacking a ResNet-50 model. The results in the table conclude that Inception-v4 performs best. This could be due to that network having a higher accuracy even in non-adversarial settings. The best ensemble of defenses achieves an accuracy of about 71% against all the other attacks. The attacks deteriorate the accuracy of the best defenses (a combination of cropping, TVM, image quilting, and model transfer) by at most 6%. Gains of 1-2% in classification accuracy could be found from ensembling different defenses, while gains of 2-3% were found from transferring attacks to different network architectures.

Table 1. Top-1 classification accuracy of ensemble and model transfer defenses (columns) against four black-box attacks (rows). The four networks we use to classify images are ResNet-50 (RN50), ResNet-101 (RN101), DenseNet-169 (DN169), and Inception-v4 (Iv4). Adversarial images are generated by running attacks against the ResNet-50 model, aiming for an average normalized [math]\displaystyle{ L_2 }[/math]-dissimilarity of 0.06. Higher is better. The best defense against each attack is typeset in boldface.

GrayBox - Image Transformation at Training and Test Time

In this experiment, the adversary has access to the network and the related parameters (but does not have access to the input transformations applied at test time). From the network trained in-(BlackBox: Image Transformation at Training and Test Time), novel adversarial images were generated by the four attack methods. The results show that Bit-Depth Reduction and JPEG Compression are weak defenses in such a gray box setting. In contrast, image cropping, rescaling, variation minimization, and image quilting are more robust against adversarial images in this setting. The results for this experiment are shown in below figure. Networks using these defenses classify up to 50 % of images correctly.

Comparison With Ensemble Adversarial Training

The results of the experiment are compared with the state of the art ensemble adversarial training approach proposed by Tramer et al. [2] 2017. Ensemble Training fits the parameters of a Convolutional Neural Network on adversarial examples that were generated to attack an ensemble of pre-trained models. The model release by Tramer et al [2]: an Inception-Resnet-v2, trained on adversarial examples generated by FGSM against Inception-Resnet-v2 and Inception-v3 models. The results of ensemble training and the preprocessing techniques mentioned in this paper are shown in Table 2. The results show that ensemble adversarial training works better on FGSM attacks (which it uses at training time), but is outperformed by each of the transformation-based defenses all other attacks.

Table 2. Top-1 classification accuracy on images perturbed using attacks against ResNet-50 models trained on input-transformed images and an Inception-v4 model trained using ensemble adversarial. Adversarial images are generated by running attacks against the models, aiming for an average normalized [math]\displaystyle{ L_2 }[/math]-dissimilarity of 0.06. The best defense against each attack is typeset in boldface.

Discussion/Conclusions

The paper proposed reasonable approaches to countering adversarial images. The authors evaluated Total Variance Minimization and Image Quilting and compared it with already proposed ideas like Image Cropping- Rescaling, Bit Depth Reduction, JPEG Compression, and Decompression on the challenging ImageNet dataset. Previous work by (Wang et al) [10], shows that a strong input defense should, be nondifferentiable and randomized. Two of the defenses - namely Total Variation Minimization and Image Quilting, both possess this property. Future work suggests applying the same techniques to other domains such as speech recognition and image segmentation. For example, in speech recognition, total variance minimization can be used to remove perturbations from waveforms and "spectrogram quilting" techniques that reconstruct a spectrogram could be developed. The input transformations can also be studied with ensemble adversarial training by Tramèr et al.[2]

Critiques

1. The terminology of Black Box, White Box, and Grey Box attack is not exactly given and clear.

2. White Box attacks could have been considered where the adversary has a full access to the model as well as the pre-processing techniques.

3. Though the authors did a considerable work in showing the effect of four attacks on ImageNet database, much stronger attacks (Madry et al) [7], could have been evaluated.

4. Authors claim that the success rate is generally measured as a function of the magnitude of perturbations, performed by the attack using the L2- dissimilarity, but the claim is not supported by any references. None of the previous work has used these metrics.

References

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2. Florian Tramèr, Alexey Kurakin, Nicolas Papernot, Ian Goodfellow, Dan Boneh, Patrick McDaniel, Ensemble Adversarial Training: Attacks and defenses.

3. Abigail Graese, Andras Rozsa, and Terrance E. Boult. Assessing threat of adversarial examples of deep neural networks. CoRR, abs/1610.04256, 2016.

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