A DVERSARIAL E XAMPLE D ETECTION AND CLASSIFICATION WITH ASYMMETRICAL A DVERSARIAL T RAINING
A BSTRACT
The vulnerabilities of deep neural networks against adversarial examples have become a signiﬁcant concern for deploying these models in sensitive domains.
Devising a deﬁnitive defense against such attacks is proven to be challenging, and the methods relying on detecting adversarial samples are only valid when the attacker is oblivious to the detection mechanism.
In this paper we ﬁrst present an adversarial example detection method that provides performance guarantee to norm constrained adversaries.
The method is based on the idea of training adversarial robust subspace detectors using asymmetrical adversarial training (AAT).
The novel AAT objective presents a minimax problem similar to that of GANs; it has the same convergence property, and consequently supports the learning of class conditional distributions.
We ﬁrst demonstrate that the minimax problem could be reasonably solved by PGD attack, and then use the learned class conditional generative models to deﬁne generative detection/classiﬁcation models that are both robust and more interpretable.
We provide comprehensive evaluations of the above methods, and demonstrate their competitive performances and compelling properties on adversarial detection and robust classiﬁcation problems.
I NTRODUCTION
Deep neural networks have become the staple of modern machine learning pipelines, achieving stateof-the-art performance on extremely difﬁcult tasks in various applications such as computer vision (He et al., 2016),
speech recognition (Amodei et al., 2016), machine translation (Vaswani et al., 2017), robotics (Levine et al., 2016), and biomedical image analysis (Shen et al., 2017).
Despite their outstanding performance, these networks are shown to be vulnerable against various types of adversarial attacks, including evasion attacks (aka, inference or perturbation attacks) (Szegedy et al., 2013; Goodfellow et al., 2014b; Carlini & Wagner, 2017b;
Su et al., 2019) and poisoning attacks (Liu et al., 2017; Shafahi et al., 2018).
These vulnerabilities in deep neural networks hinder their deployment in sensitive domains including, but not limited to, health care, ﬁnances, autonomous driving, and defense-related applications and have become a major security concern.
Due to the mentioned vulnerabilities, there has been a recent surge toward designing defense mechanisms against adversarial attacks (Gu & Rigazio, 2014; Jin et al., 2015; Papernot et al., 2016b;
Bastani et al., 2016; Madry et al., 2017; Sinha et al., 2018), which has in turn motivated the design of stronger attacks that defeat the proposed defenses (Goodfellow et al., 2014b; Kurakin et al., 2016b;a; Carlini & Wagner, 2017b; Xiao et al., 2018; Athalye et al., 2018; Chen et al., 2018;
He et al., 2018).
Besides, the proposed defenses have been shown to be limited and often not effective and easy to overcome (Athalye et al., 2018).
Alternatively, a large body of work has focused on detection of adversarial examples (Bhagoji et al., 2017; Feinman et al., 2017;
Gong et al., 2017;
Grosse et al., 2017; Metzen et al., 2017; Hendrycks & Gimpel, 2017; Li & Li, 2017; Xu et al., 2017;
Pang et al., 2018; Roth et al., 2019;
Bahat et al., 2019; Ma et al., 2018; Zheng & Hong, 2018; Tian et al., 2018).
While training robust classiﬁers focuses on maintaining performance in presence of adversarial examples, adversarial detection only cares for detecting such examples.
The majority of the current detection mechanisms focus on non-adaptive threats, for which the attacks are not speciﬁcally tuned/tailored to bypass the detection mechanism, and the attacker is oblivious to the detection mechanism.
In fact, Carlini & Wagner (2017a) and Athalye et al.
(2018) showed that the detection methods presented in (Bhagoji et al., 2017; Feinman et al., 2017;
Gong et al., 2017; Grosse et al., 2017; Metzen et al., 2017; Hendrycks & Gimpel, 2017; Li & Li, 2017;
Ma et al., 2018), are signiﬁcantly less effective than their claims under adaptive attacks.
Overall, current solutions are mostly heuristic approaches that cannot provide performance guarantees to the adversary they considered.
[Figure]
In this paper we present a detection mechanism that can withstand adaptive attacks.
The idea is to partition the input space into subspaces based on the classiﬁcation system’s output and perform adversarial/natural sample classiﬁcation in these subspaces.
The partition allows us to drop the adversarial constrain and employ a novel asymmetrical adversarial training (AAT) objective to train robust binary classiﬁers in subspaces.
The robustness could be further explained by the fact that AAT objective presents a minimax problem that supports the learning of class conditional distributions.
Our speciﬁc contributions are:
• We develop adversarial example detection techniques that provide performance guarantees to norm constrained adversaries.
Empirically, our best models improve previous state-ofthe-art mean L 2 distortion from 3.68 to 5.65 on MNIST dataset, and from 1.1 to 1.5 on CIFAR10 dataset.
• We study powerful and versatile generative classiﬁcation models derived from our detection framework and demonstrate their competitive performances over discriminative robust classiﬁers.
While defense mechanisms based on ordinary adversarial training are vulnerable to rubbish examples, inputs that cause conﬁdent predictions of our models have humanunderstandable semantic meanings.
• We propose the AAT objective, that presents a minimax problem similar to that of GANs, and supports the learning of class conditional distributions.
We used PGD attack (Madry et al., 2017) to solve the optimization problem, and obtained competitive generative modeling results on CIFAR10, ImageNet, and synthetic datasets.
R ELATED WORKS
Adversarial attacks.
Since the pioneering work of Szegedy et al.
(2013), a large body of work has focused on designing algorithms that achieve successful attacks on neural networks (Goodfellow et al., 2014b; Moosavi-Dezfooli et al., 2016;
Kurakin et al., 2016b; Chen et al., 2018; Papernot et al., 2016a; Carlini & Wagner, 2017b).
More recently, iterative projected gradient descent (PGD), has been empirically identiﬁed as the most effective approach for performing norm constrained attacks, and the attack reasonably approximates the optimal attack (Madry et al., 2017).
Adversarial detection techniques.
The majority of the methods developed for detecting adversarial attacks are based on the following core idea: given a trained K -class classiﬁer, f : R d → { 1 ...K } , and its corresponding natural training samples, D = { x i ∈ R d } Ni =1 , generate a set of adversarially attacked samples D (cid:48)
= { x (cid:48)
j ∈ R d } Mj =1 , and devise a mechanism to discriminate D from D (cid:48) .
For instance, Gong et al.
(2017) use this exact idea and learn a binary classiﬁer to distinguish the natural and adversarially perturbed sets.
Similarly, Grosse et al.
(2017) append a new “attacked” class to the classiﬁer, f , and re-train a secured network that classiﬁes natural images, x ∈ D , into the K classes and all attacked images, x (cid:48) ∈ D (cid:48) , to the ( K + 1) -th class.
In contrast to Gong et al.
; Grosse et al.
(2017), which aim at detecting adversarial examples directly from the image content, Metzen et al.
(2017) trained a binary classiﬁer that receives as input the intermediate layer features extracted from the classiﬁer network f , and distinguished D from D (cid:48) based on such input features.
More importantly, Metzen et al.
(2017) considered the so-called case of adaptive / dynamic adversary and proposed to harden the detector against such attacks using a similar adversarial training approach as in Goodfellow et al.
(2014b).
Unfortunately, the mentioned detection methods are signiﬁcantly less effective under an adaptive adversary equipped with a strong attack (Carlini & Wagner, 2017a;
Athalye et al., 2018).
A DVERSARIAL D ETECTION M ETHODS 3. I NTEGRATED ADVERSARIAL DETECTION
For a K ( K ≥ 2 ) class classiﬁcation problem, given a dataset of natural samples D = { x i } Ni =1 , x i ∈ R d , along with labels { y i } Ni =1 , y i ∈ { 1 ...K } , let f : R d → { 1 ...K } be a classiﬁer that is used to do classiﬁcation on D .
With the labels and predicted labels the dataset respectively forms the partition
D =
(cid:83) D k and D f = (cid:83) D fk , where D k = { x : y
=
k, x ∈ D} , and D fk
= { x : f ( x ) =
k, x ∈ D} .
Let H =
{ h k } Kk =1 , h k : R d → { 0 , 1 } be a set of binary classiﬁers (detectors), in which h k is trained to discriminate natural samples classiﬁed as k , from adversarial samples that fool to be classiﬁed as k .
Also, let D (cid:48) be a set of L p norm bounded adversarial examples crafted from D :
D (cid:48)
= { x + δ : f ( x + δ ) (cid:54) =
y, f ( x ) =
y, x ∈ D , δ ∈ S} , S = {
δ ∈ R d | (cid:107) δ (cid:107) p ≤ (cid:15) } .
Consider the following procedure to determine whether a sample x in D ∪ D (cid:48) is an adversarial example:
First obtain the estimated class label k
= f ( x ) , then use the k -th detector to predict: if h k ( x ) = 1 then categorize x as a natural sample, otherwise categorize it as an adversarial example.
The detection accuracy of the algorithm is given by
[Equation]
where D (cid:48)
fk =
{ x : f ( x ) =
k, x ∈ D (cid:48) } .
Hence minimizing the algorithm’s classiﬁcation error is equivalent to minimizing classiﬁcation error of individual detectors.
Employing empirical risk minimization, detector k , parameterized by θ k , is trained by
[Equation]
where L is a loss function that measures the distance between h k ’s output and the supplied label (e.g., the binary cross-entropy loss).
In the case of adaptive attacks, when the adversary aims to fool both the classiﬁer and detectors, the accuracy of a naively trained detector could be signiﬁcantly reduced.
In order to be robust to adaptive attacks, by taking a similar approach as Madry et al.
(2017) we incorporate the attack into the training objective:
[Equation]
where D f \ k = { x : f ( x ) (cid:54) =
k, y (cid:54)
=
k, x ∈ D} , and we assume that perturbation budget is large enough such that ∀ x ∈ D f \ k , ∃ δ ∈ S , s.t. f
( x + δ ) =
k .
Now by dropping the f ( x + δ )
=
k constrain we could derive an upper bound for the ﬁrst loss term:
[Equation]
The detector could instead be trained by minimizing this upper bound using the following unconstrained objective,
[Equation] [Equation]
Further, we use the fact that when D is used as the training set, f could overﬁt on D such that
D \ k = { x i : y i (cid:54) =
k } and D k are respectively good approximations of D f \ k and D fk .
This leads to our proposed asymmetrical adversarial training (AAT) objective: min
[Equation]
, where
[Equation]
In a nutshell, each detector is trained using in-class natural samples and detector-adversarial examples crafted from out-of-class samples.
We use iterative PGD attack (Madry et al., 2017) to solve the inner maximization.
The outer minimization is solved using regular gradient descent based optimization methods.
3. G ENERATIVE ADVERSARIAL DETECTION / CLASSIFICATION
If we assume L to be the binary cross-entropy loss, the minimax problem presented in objectiveis then similar to that of GANs, and it is straightforward to use GANs’ convergence theorem to show convergence (if no perturbation limit) of the out-of-class distribution to the target class distribution.
The perturbation limit provides a mechanism for constraining the optimization such that the discriminator (detector) could retain density information and not converge to a degenerate uniform solution as in the case of GANs (Goodfellow et al., 2014a; Dai et al., 2017).
In Appendix F we provide experimental results on 1D and 2D benchmark datasets to demonstrate AAT’s capability to do density estimation.
Unfortunately, the detectors don’t deﬁne explicit density functions.
Under the energy-based learning framework (LeCun et al., 2006), we could, however, obtain the joint probability of the input and a class category using the Gibbs distribution: p ( x, k ) =
exp( − E θk ( x ))
Z Θ , where E θ k
( x ) = − z
( h k ( x )) , and Z Θ = (cid:80) k (cid:82) exp( − E θ k
( x )) dx is an intractable normalizing constant known as the partition function.
We could then apply the Bayes classiﬁcation rule to obtain a generative classiﬁer :
H ( x ) =
arg max k p ( x, k ) =
arg max k z ( h k ( x )) .
In addition, we could base on p ( x, k ) to reject low probability inputs.
We implement the reject option by thresholding ˆ k -th detector’s logit output, where ˆ k is the predicted class.
In the context of adversarial example detection, rejected samples are considered as adversarial examples.
E VALUATION METHODOLOGY 4. R OBUSTNESS TEST
We ﬁrst test the robustness of individual detectors.
We show that, once a detector is trained with an adequately conﬁgured PGD attack, its performance cannot be signiﬁcantly reduced by an adversary with much stronger conﬁgurations.
We note that although the PGD attack has been shown (Madry et al., 2017) to be able to induce robustness in ordinary adversarial training, the test is necessary as it is not clear whether the optimization landscape of the asymmetrical objective is the same as its symmetrical counterparts.
For instance, we found that the step size used by Madry et al.
(2017) to train their CIFAR10 robust classiﬁer would not induce robustness to our detectors (see Appendix C.2.2).
Furthermore, we use adversarial ﬁnetuning on CIFAR10 and ImageNet to speedup detector training.
With the robustness test, we show that PGD attack also introduces robustness within this new training paradigm.
We use AUC (area under the ROC Curve) to measure detection performances.
The metric could be interpreted as the probability that the detector assigns a higher score to a random positive sample than to a random negative example.
While true positive rate and false positive rate are the commonly used metrics for measuring detection performances, they require a detection threshold to be speciﬁed.
AUC, however, is an aggregated measurement of detection performance across a range of thresholds, and we found it to be a more stable and reliable metric.
For the k -th detector h k , its AUC is computed on the set { ( x, 0) : x ∈ D (cid:48)
f \ k } ∪ { ( x, 1) : x ∈ D fk } , where D (cid:48)
f \ k = { arg max x + δ L
h k ( x + δ ; θ k ) , 0) : x ∈ D f \ k ) } (refer to loss 4).
4. D ETECTION P ERFORMANCE
Having validated the robustness of individual detectors, we evaluate the overall performance of our integrated detection system.
Recalling our detection rule, we ﬁrst obtain the estimated class label k
= f ( x ) , then use the k -th detector’s logit output z ( h k ( x )) to predict: if z ( h k ( x )) ≥ T k , then x is a natural sample, otherwise it is an adversarially perturbed sample.
For the sake of this evaluation, we use a universal threshold for all the detectors: ∀ k ∈ { 1 ...K } T k = T , and report detection performance at a range of universal thresholds.
In practice, however, the optimal value of each detector’s detection threshold T k should be determined by optimizing a utility function.
We use D = { ( x i , y i ) } Ni =1 to denote the test set that contains natural samples, and D (cid:48)
= { ( x i
+ δ i , y i ) }
Ni =1 to denote the corresponding perturbed test set.
For a given threshold T , we compute the true positive rate (TPR) on D and false positive rate (FPR) on D (cid:48) .
These two metrics are respectively deﬁned as
[Equation]
6) and
[Equation]
In the FPR deﬁnition we use f ( x ) (cid:54) =
y to constrain that only true adversarial examples are counted as false positives.
This constraint is necessary, as we found that for the norm ball constraint we considered in the experiments, not all perturbed samples are adversarial examples that cause misclassiﬁcation on f .
In order to craft the perturbed dataset D (cid:48) , we consider three attacking scenarios.
Classiﬁer attack.
This attack corresponds to the scenario where the adversary is oblivious to the detection mechanism.
For a given natural sample x and its label y , the perturbed sample x (cid:48) is computed by minimizing the loss,
[Equation]
where z ( f ( x (cid:48) )) is the classiﬁer’s logit outputs.
This objective is derived from the CW attack (Carlini & Wagner, 2017b) and used in MadryLab (b;a) to perform untargeted attacks.
Detectors attack.
In this scenario adversarial examples are produced by attacking only the detectors.
We construct a single detection function H by using the i -th detector’s logit output as its i -th logit output: z ( H ( x ))
i =
z ( h i ( x )) .
H is then recognized as a single network, and the perturbed sample x (cid:48) for a given input ( x, y ) is computed by minimizing the loss
[Equation]
According to our detection rule, a low value of the detector’s logit output indicates detection of an adversarial example, thus by minimizing the negative of logit output we make the perturbed example harder to detect.
H could also be fed directly to the CW loss 8 or to the cross-entropy loss, but we found the attack based on the loss in 9 to be signiﬁcantly more effective.
Combined attack.
With the goal of fooling both the classiﬁer and detectors, perturbed samples are produced by attacking the integrated detection system.
We consider two loss functions for realizing the combined attack.
The ﬁrst is based on the combined loss function (Carlini & Wagner, 2017a) that has been shown to be effective against existing detection methods.
Given a natural example x and its label y , same as the detectors-attack scenario, we ﬁrst construct a single detection function
H by aggregating the logit outputs of individual detectors: z ( H ( x ))
i =
z ( h i ( x )) .
We then use the aggregated detector’s largest logit output max k (cid:54) =
y z ( H ( x ))
k (low value of this quantity indicates detection of an adversarial example) and the classiﬁer logit outputs z ( f ( x )) to construct a surrogate classiﬁer g , with its logit outputs being
[Equation]
A perturbed example x (cid:48) is then computed by minimizing the loss function
[Equation] [Equation]
In practice we observe that the optimization of this loss tends to stuck at the point where max i (cid:54) =
y z ( f ( x (cid:48) ))
i keeps changing signs while max j (cid:54) =
y z ( H ( x ))
j stays as a large negative number (which indicates detection).
To derive a more effective attack we consider a simple combination of loss 8 and loss 9:
[Equation]
The objective is straightforward: if x (cid:48) is not yet an adversarial example on f , optimize it for that goal; otherwise optimize it for fooling the aggregated detector.
We mention brieﬂy here that we perform the same performance analysis of the generative detection method as detailed in Section 3.2, by computing TPR on D and FPR on D (cid:48) , where D (cid:48) is computed using loss 9, loss 8 and the cross-entropy loss.
4. R OBUST C LASSIFICATION P ERFORMANCE
Integrated classiﬁcation.
In addition to the generative classiﬁer proposed in Section 3.2, we introduce another classiﬁcation scheme that comes up with a reject option.
The scheme is based on a combination of the naive classiﬁer f and the detectors: for a given input x and its prediction label k
= f ( x ) , if z ( h k ( x )) <
T , then x is rejected, otherwise it’s classiﬁed as k .
We respectively use loss 12 and 9 to attack the integrated classiﬁer and the generative classiﬁer.
Performance metric.
In the context of robust classiﬁcation, the performance of a robust classiﬁer is measured using standard accuracy and robust accuracy — accuracies respectively computed on the natural dataset and perturbed dataset.
On the natural dataset D = { ( x i , y i ) }
Ni =1 , we compute the accuracy as the fraction of samples that are correctly classiﬁed ( f ( x ) =
y ) and at the same time not rejected ( z ( h k ( x )) ≥ T ):
[Equation]
On the perturbed dataset D (cid:48) = { ( x i + δ i , y i ) }
Ni =
we compute the error as the fraction of samples that are misclassiﬁed ( f ( x ) (cid:54) =
y ) and at the same time not rejected:
[Equation]
Note that in this case the error is no longer a complement of the accuracy.
For a classiﬁcation system with a reject option, any perturbed samples that are rejected should be considered as properly handled, regardless of whether they are misclassiﬁed.
Thus on the perturbed dataset, the error, which is the fraction of misclassiﬁed and not rejected samples, is a more proper notion of such system’s performance.
For a standard softmax robust classiﬁer that comes without an reject option, its perturbed set error is computed as the complement of its accuracy on the perturbed set.
E XPERIMENTS 5. MNIST
Using different p -norm and maximum perturbation (cid:15) constrains we trained four detection systems (each has 10 base detectors), with training and validation adversarial examples optimized using PGD attacks of different steps and step sizes (see Table 6).
At each step of PGD attack we use the Adam optimizer to perform gradient descent, both for L 2 and L ∞ constrained scenarios.
Appendix A. provides more training details.
Robustness results.
The robustness test results in Table 1 conﬁrm that the base detectors trained with objective 5 are able to withstand much stronger PGD attacks, for both L 2 and L ∞ scenarios.
In Table 8 we present the robustness test result using normalized steepest descent based attack.
We also performed other types of tests, including cross-norm test, cross-perturbation test, and random restart test.
The results of these tests, along with the complete robustness test results on L ∞ (cid:15) = 0 .and
L ∞ (cid:15) = 0 .
5 trained detectors, are presented in Appendix C.1.
[Table] [Table]
Detection results.
Figure 2a shows that generative detection outperforms integrated detection (compare their performances under their respective most effective attacks).
Combined attack, as expected, is the most effective attack against integrated detection.
Attack based on loss 9 is the most effective attack against generative detection (see Figure 7).
Notably, the red curve that overlaps the y-axis indicates that integrated detection is able to perfectly detect adversarial examples crafted by attacking only the classiﬁer (using objective 8).
In Table 2 we list the performances of the generative detection method along with the state-of-the-art detection method as identiﬁed by Carlini & Wagner (2017a).
Our method, apart from being able to provide performance guarantee, outperforms the state-of-the-art method by large margins.
(Please refer to Appendix B for a description of how we compute the mean L 2 distortions.)
Classiﬁcation results.
In Figure 2b we plot the robust classiﬁcation performance of our classiﬁcation methods and a state-of-the-art discriminative robust classiﬁer (Madry et al., 2017).
Our methods provide reject options that allow the user to ﬁnd a balance between standard accuracy and robust error by adjusting the rejection threshold.
We observe that a stronger attack ( (cid:15) = 0 . 4 ) breaks the robust classiﬁer (as indicated by the right red cross), while the generative classiﬁer still exhibits robustness, even though both systems are trained with L ∞ (cid:15)
= 0 .
3 constrain.
[Figure]
(discussed in Section 4.3)
Figure 3 shows perturbed samples produced by performing targeted attacks against the generative classiﬁer and robust classiﬁer.
The generative classiﬁer’s perturbation samples have distinguishable visible features of the target class, indicating that the base detectors, from which the generative classiﬁer is built, have learned the class conditional distributions, and the perturbations have to change the semantics for a successful attack.
In contrast, perturbations introduced by attacking the robust classiﬁer are not interpretable, even though they could cause high logit output of the target classes (see Figure 8 for the logit outputs distribution).
Following this path and with a larger perturbation limit, it is straightforward to generate completely unrecognizable inputs that fool the discriminative robust classiﬁer.
[Figure]
5. CIFAR
On CIFAR10 we train the base detectors using L ∞ (cid:15) = 8 constrain PGD attack of steps 40 and step size 0.5.
Note that the scale of (cid:15) and step size here is 0-255 (rather than 0-1 as in the case of
MNIST).
The robust classiﬁer (Madry et al., 2017) that we compare with is trained with the same
L ∞ (cid:15) = 8 constraint but with a different step size (see Appendix C.2.2 for a discussion of the effects of step size).
Appendix A.2 provides the training details.
Robustness results.
Table 3 shows that the base detector models can withstand attacks that are signiﬁcantly stronger than the training attack.
In Appendix C.2.1 we report random restart test results, cross-norm and cross-perturbation test results, and robustness test result for L 2 based models.
[Table]
Detection results.
Consistent with the MNIST results, in Figure 4a combined attack is the most effective method against integrated detection.
Generative detection outperforms integrated detection when the detection threshold is low (i
[Table]
e., when TPR is high).
In this ﬁgure we use loss 9 to attack generative detection
[Table]
and in Figure 9 we show that this loss is more effective than cross-entropy loss and CW loss.
In Table 4 our method outperforms the state-of-the-art adversarial detection method
[Table] [Table]
[Figure]
u
[Figure]
i
[Figure]
i v
[Figure]
t
[Figure]
Classiﬁcation results.
Contrary to MNIST’s result we did not observe a dramatic decrease in the robust classiﬁer’s performance when we increase the perturbation limit to (cid:15)
=
12 (Figure 4b).
Integrated classiﬁcation can reach the standard accuracy of a regular classiﬁer, but at the cost of signiﬁcantly increased error on the perturbed set.
Figure 5 shows some perturbed samples produced by attacking the generative classiﬁer and robust classiﬁer.
While these two classiﬁers have similar errors on the perturbed set, samples produced by attacking the generative classiﬁer have more visible features of the targets, suggesting that the adversary has to change more semantic to cause the same error.
Figures 6 and 10 demonstrate that hard to recognize images are able to cause high logit outputs of the robust classiﬁer.
Such examples highlight a major defect of the defense mechanisms based on ordinary adversary training: they could be easily fooled by unrecognizable inputs (Nguyen et al., 2015; Goodfellow et al., 2014b; Schott et al., 2018).
In contrast, samples that cause high logit outputs of the generative classiﬁer all have clear semantic meaning.
In Figure 13 we present image synthesis results using L ∞ (cid:15) = 16 constrained detectors.
These results conform to the theory that
AAT supports the learning of class conditional distributions.
(See Appendix D for Gaussian noise attack results and a discussion about the interpretability of the generative classiﬁcation approach.)
[Figure]
5. I MAGE N ET
On ImageNet we show AAT induces detection robustness and supports the learning of class conditional distributions.
Our experiment is based on Restricted ImageNet (Tsipras et al., 2018), a subset of ImageNet that has its samples reorganized into customized categories.
The dog category consists of images of different breeds collected from ImageNet class from 151 to 268.
We trained a dog class detector by ﬁnetuning a pre-trained ResNet50 (He et al., 2016) model.
The dog category covers a range of ImageNet classes, with each one having its logit output.
We use the subnetwork deﬁned by the logit output of class 151 as the detector (in principle logit output of other classes in the range should also work).
Due to computational resource constraints, we only validated the robustness of a L ∞ (cid:15) = 0 .
02 trained detector (trained with PGD attack of steps 40 and step size 0.001), and we present the result in Table 5.
(On Restricted ImageNet in the case of L ∞ scenario Tsipras et al.
(2018) only demonstrates the robustness of a (cid:15)
= 0 .
005 constrained model).
Please refer to
Appendix C.3 for more results of adversarial examples, image synthesis, and image enhancement.
[Table]
C ONCLUSION
We studied the problem of adversarial example detection under the robust optimization framework and proposed a novel detection method based on input space partitioning.
Our formulation leads to a new generative modeling technique which we called asymmetrical adversarial training (AAT).
AAT’s capability to learn class conditional distributions further gives rise to generative detection/classiﬁcation approaches that show competitive performance and improved interpretability.
In particular, our generative classiﬁcation method is more resistant to “rubbish examples”, a major threat to even the most successful defense mechanisms.
High computational cost (see Appendix E for more discussion) is a major drawback of our methods, and in the future, we will explore the idea of shared computation between detectors.
A T RAINING D ETAILS A. MNIST TRAINING
We use 50K samples from the original training set for training and the rest 10K samples for validation and report test performances based on the epoch-saved checkpoint that gives the best validation performance.
All base detectors are trained using a network consisting of two max-pooled convolutional layers each with 32 and 64 ﬁlters and a fully connected layer of size 1024: same as the one used in Madry et al.
At each iteration we sample a batch of 320 samples, from which in-class samples are used as positive samples and out-of-class samples are used for crafting adversarial examples that will be used as negative samples.
To balance positive and negative examples at each batch, we resample the out-of-class set to have same number of samples as in-class set
[Table]
A. CIFAR10 TRAINING
We train CIFAR10 base detectors using a ResNet model (same as the one used by Madry et al.
(2017); MadryLab (a)).
To speedup training, we take advantage of a natural trained classiﬁer: the subnetwork of f that deﬁnes the output logit
z ( f ( · ))
k is essentially a “detector”, that would output high values for samples of class k , and low values for others.
The base detector is then trained by ﬁnetuning the subnetwork using objective 5.
The pretrained classiﬁer has a test accuracy of 95.01% (fetched from MadryLab (a)).
At each iteration of training we sample a batch of 300 samples, from which in-class samples are used as positive samples, while an equal number of out-of-class samples are used for crafting adversarial examples.
Adversarial examples for training L 2 and L ∞ models are both optimized using normalized steepest descent based PGD attacks (MadryLab, b).
We report results based on the best performances on the CIFAR10 test set (thus don’t claim generalization performance of the proposed method).
B C OMPUTING MEAN L 2 DISTANCE
We ﬁrst ﬁnd the detection threshold T with which the detection system has 0.95 TPR.
We construct a new loss function by adding a weighted loss term that measures perturbation size to objective
[Equation]
We then use unconstrained PGD attack to optimize L ( x (cid:48) ) .
We use binary search to ﬁnd the optimal c , where in each bsearch attempt if x (cid:48) is a false positive ( max i z
( H ( x (cid:48) ))
i (cid:54) =
y and max i (cid:54) =
y z ( H ( x (cid:48) ))
i > T ) we consider the current c as effective and continue with a larger c .
The conﬁgurations for performing binary search and PGD attack are detailed in Table 7.
The c upper bound is established such that with this upper bound, no samples except those that are inherently misclassiﬁed by the generative classiﬁer, could be perturbed as a false positive.
With these settings, our MNIST L ∞ (cid:15) = 0 . 3 and L ∞ (cid:15) = 0 .
5 generative detection models respective reached 1.0 FPR and 0.9455 FPR, and CIFAR10 generative model reached 0.9995 FPR.
Due to the technical and computational difﬁculties of ﬁnding the optimal c in loss 15 using binary search, performances based on mean L 2 distortion are not precise, and we encourage future work to measure detection performances based on norm constrained attacks.
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C M ORE EXPERIMENTAL RESULTS C. M ORE MNIST RESULTS
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RESULTS
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2. M ORE ROBUSTNESS TEST RESULTS
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C.2. T RAINING STEP SIZE AND ROBUSTNESS
We found training with adversarial examples optimized with a sufﬁciently small step size to be essential for detection robustness.
In table 18 we tested two L ∞ (cid:15) = 2 . 0 base detectors respectively trained with 0.5 and 1.0 step size.
The step size 1.0 model is not robust when tested with a much smaller step size.
We observe that when training the step size 1.0 model, training set adv AUC reached 1.0 in less than one hundred iterations, but test set natural AUC plummeted to around 0.
and couldn’t recover thereafter.
(Please refer to Figure 11 for the deﬁnitions of adv AUC and nat
AUC.)
This suggests that naturally occurring data, and perturbed data produced using a large step size, which are essentially noise images, live in two easily separable spaces, and training on such data is not beneﬁcial for model performance on natural data or adversarial examples.
[Table]
E FFECTS OF PERTURBATION LIMIT
To study the effects of perturbation limit on asymmetrical adversarial training, we analysis the training dynamics of one L ∞ (cid:15) = 2 . 0 constrained and one L ∞ (cid:15) = 8 . 0 constrained base detector.
In
Figure 11 we show the training and testing history of these two models.
The (cid:15) = 2 . 0 model history shows that by adversarial ﬁnetuning the model reaches robustness in just a few thousands of iterations, and the performance on natural samples is preserved (test natural AUC begins at 0.9971, and ends at 0.9981).
Adversarial ﬁnetuning on the (cid:15) = 8 . 0 model didn’t converge after an extended 20K iterations of training.
The gap between train adv AUC and test adv AUC of the (cid:15) = 8 . 0 model is more pronounced, and we observed a decrease of test natural AUC from 0.9971 to 0.9909.
These results suggest that training with larger perturbation limit is more time and resource consuming, and could lead to performance decrease on natural samples.
The beneﬁt is that the detector is pushed to a better approximation of the target data distribution.
As an illustration, in Figure 12, perturbations generated by attacking the naturally trained classiﬁer (corresponds to 0 perturbation limit) don’t have clear semantics, while perturbed samples of the L ∞ (cid:15) = 8 . 0 model are completely recognizable.
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C. M ORE I MAGE N ET RESULTS
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D G AUSSIAN NOISE ATTACK AND MODEL INTERPRETABILITY
In this section we use Gaussian noise attack experiment to motivate a comparative analysis of the interpretabilities of our generative classiﬁcation approach and discriminative robust classiﬁcation approach (Madry et al., 2017).
We ﬁrst discuss how these two approaches determine the posterior class probabilities.
For the discriminative classiﬁer, the posterior probabilities are computed from the logit outputs of the classiﬁer using the softmax function p ( k | x ) =
[Equation]
z
[Equation]
f ( x )) j ) .
For the generative classiﬁer, the posterior probabilities are computed in two steps: in the ﬁrst, we train the base detectors, which is the process of solving the inference problem of determining the joint probability p ( x, k ) , and in the second, we use Bayes rule to compute the posterior probability p ( k | x )
=
p ( x,k ) p ( x
[Equation]
=
[Equation]
Coincidentally, the formulas for computing the posterior probabilities take the same form.
But in our approach, the exponential of the logit output of a detector (i.e., exp( z ( h k ( x ))) ) has a clear probabilistic interpretation: it’s the unnormalized joint probability of the input and the corresponding class category.
We use Gaussian noise attack to demonstrate that this probabilistic interpretation is consistent with visual perception.
We start from a Gaussian noise image, and gradually perturb it to cause higher and higher logit outputs.
This is implemented by targeted PGD attack against logit outputs of these two classiﬁcation models.
The resulting images in Figure 19 show that, in our model, the logit output increase direction, i.e. the join probability increase direction, indicates the class semantic changing direction; while for the discriminative robust model, the perturbed image computed by increasing logit outputs are not as clearly interpretable.
In particular, the perturbed images that cause high logit outputs of the robust classiﬁers are not recognizable.
In summary, as a generative classiﬁcation approach that explicitly models class conditional distributions, our system offers a probabilistic view of the decision making process of the classiﬁcation problem; adversarial attacks that rely on imperceptible or uninterpretable noises are not effective against such a system.
E C OMPUTATIONAL COST ISSUE
In this section we provide an analysis of the computational cost of our generative classiﬁcation approach.
In terms of memory requirements, if we assume the softmax classiﬁer (i.e., the discriminative robust classiﬁer) and the detectors use the same architecture (i.e., only defer in the ﬁnal layer)
then the detector based generative classiﬁer is approximately K times more expensive than the Kclass softmax classiﬁer.
This also means that the computational graph of the generative classiﬁer is K times larger than the softmax classiﬁer.
Indeed, in the CIFAR10 task, on our Quadro M 24GB GPU (TensorFlow 1.13.1), the inference speed of the generative classiﬁer is roughly ten times slower than the softmax classiﬁer.
We next benchmark the training speed of these two types of classiﬁers.
The generative classiﬁer has K logit outputs, with each one deﬁned by the logit output of a detector.
Same with the softmax classiﬁer, except that the K outputs share the parameters in the convolutional base.
Now consider ordinary adversarial training on the softmax classiﬁer and asymmetrical adversarial training on the generative classiﬁer.
To train the softmax classiﬁer, we use batches of N samples.
For the generative classiﬁer, we train each detector with batches of 2 × M samples ( M positive samples and M negative samples).
At each iteration, we need to respectively compute N and M × K adversarial examples for these two classiﬁers.
Now we test the speed of the following two scenarios:
1) compute the gradient w.r.t.
to N samples on a single computational graph, and 2) compute the gradient w.r.t to M × K samples on K computational graphs, with each graph working on M samples.
We assume that in scenario 2 all the computational graphs are loaded to GPUs, and thus their computations are in parallel.
In our CIFAR10 experiment, we used batches consisting of 30 positive samples and 30 negative samples to train each ResNet50 based detectors.
In Madry et al.
(2017), the softmax classiﬁer was trained with batches of 128 samples.
In this case, K = 10 , M = 30 , and N = 128 .
On our GPU, scenario 1 took 683 ms ± 6.76 ms per loop, while scenario 2 took 1.85 s ± 42.7 ms per loop.
In this case, we could expect asymmetrical adversarial training to be about 2.7 times slower than ordinary adversarial training, if not considering parameter gradient computation.
In practice, large batch size is almost always preferred.
And our method won’t compare as favorably if we choose to use one.
F D ENSITY ESTIMATION ON SYNTHETIC DATASETS
While ordinary discriminative training only learns a good decision boundary, AAT is able to learn the underlying density functions that generate the training data.
Results on 1D (Figure 20) and 2D benchmark datasets (Figure 21) show that through properly conﬁgured asymmetrical adversarial training, detectors’ output recover target density functions.
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