M AKING S UBSTITUTE M ODELS M ORE B AYESIAN C AN E NHANCE T RANSFERABILITY OF A DVERSARIAL E XAMPLES
Anonymous authors
Paper under double-blind review
A BSTRACT
The transferability of adversarial examples across deep neural networks (DNNs) is the crux of many black box attacks
Many prior efforts have been devoted to im proving the transferability via increasing the diversity in inputs of some substitute models
In this paper
, by contrast
, we opt for the diversity in substitute models and advocate to attack a Bayesian model for achieving desirable transferability.
Deriv- ing from the Bayesian formulation, we develop a principled strategy for possible finetuning, which can be combined with many off-the-shelf Gaussian posterior approximations over DNN parameters.
Extensive experiments have been conducted to verify the effectiveness of our method, on common benchmark datasets and the results demonstrate that our method outperforms recent state-of-the-arts by large margins (roughly 16 .
8% absolute increase in average attack success rate on ImageNet) and, by combining with these recent methods, further performance gain can be obtained.
Our code will be publicly available.
I NTRODUCTION
The adversarial vulnerability of deep neural networks (DNNs) has attracted great attention over the past decade (Szegedy et al., 2014; Goodfellow et al., 2015; Papernot et al., 2016; Carlini & Wagner, 2017; Madry et al., 2018; Athalye et al., 2018).
It has been demonstrated that the prediction of stateof-the-art DNNs can be arbitrarily altered by adding designated perturbations, even imperceptible to human eyes, to their inputs.
Threat models involving the adversarial examples can be divided into white-box and black-box settings according to the amount of information (of victim models) being exposed to the adversary.
In black-box attacks, where the adversary can hardly get access to the architecture and parameters of the victim model, the transferability of adversarial examples is often relied on, given the fact that adversarial examples crafted on a substitute model can sometimes fool other models as well.
However, such methods also suffer from considerable failure rate when the perturbation budge is small.
Thus, much recent effort has been devoted to improve the black-box transferability of adversarial examples, and a variety of transfer-based attacks have been proposed.
Assuming that the substitute model was pre-trained and given, most of the recent research focused only on improving the backpropagation process when issuing attacks, yet little attention has been paid to possible training or finetuning of the substitute model.
In this paper, we shall focus more on the training process, and for which we advocate to perform in a Bayesian manner, in order to issue more powerful transfer-based attacks.
By introducing probability measures to weights and biases of the substitute model, all these parameters are represented under assumptions of some distributions to be learned.
In this way, an ensemble of infinitely many DNNs (that are jointly trained in our view) can be obtained from a single run of training.
Adversarial examples are then crafted by maximizing average prediction loss over such a distribution of models, which is a referred to as posterior learned in the Bayesian manner.
Extensive experiments on attacking a variety of CIFAR-10 (Krizhevsky &
Hinton, 2009) and ImageNet (Russakovsky et al., 2015) victim models have been performed, and we show that the proposed method outperforms previous state-of-the-arts considerably.
Moreover, our method can be conjugated with existing methods easily and reliably for further improving the attack performance.
B ACKGROUND AND R ELATED W ORK 2. A DVERSARIAL A TTACKS
White-box attacks.
Given full knowledge of the architecture and parameters of a victim model, white-box attacks are typically performed via utilizing some loss gradient with respect to the model inputs.
For instance, given a normal sample ( x , y ) and a source model f w : R n → R c that is trained to classify x into y ∈ R c , it is a popular choice to craft the adversarial example x + ∆x within an ℓ p bounded small region of x , by maximizing the prediction loss, i.e. , max ∥
∆x ∥ p ≤ ϵ L ( x + ∆x , y, w ) , where ϵ is the perturbation budget.
FGSM proposed to calculate ϵ · sgn( ∇ x L ( x , y, w )) for ∆x in the p = ∞ setting (Goodfellow et al., 2015), and the iterative variants of FGSM, e.g. , I-FGSM (Kurakin et al., 2017) and PGD (Madry et al., 2018) can be more powerful.
Black-box attacks.
Black-box attacks are more challenging compared to the white-box attacks.
Many existing methods largely rely on the transferability of adversarial examples, i.e. , adversarial examples crafted on one classification model can generally succeed in attacking some other victim models as well.
It is normally assumed to be able to query the victim model to annotate training samples, or be possible to collect a pre-trained source model that is trained to accomplish the same task as the victim models.
Aiming at enhancing the adversarial transferability, methods have been proposed to modify the backpropagation computation, see for example the skip gradient method (SGM) (Wu et al., 2020), the linear backpropagation (LinBP) method (Guo et al., 2020), the intermediate-level attack (ILA) (Huang et al., 2019), and ILA++ (Li et al., 2020a; Guo et al., 2022).
It is also widely adopted to increase the diversity in inputs (Xie et al., 2019;
Dong et al., 2019; Lin et al., 2019; Wang et al., 2021).
In this paper, we consider the diversity from another perspective, the substitute model(s), and we introduce a Bayesian approximation for achieving this.
Ensemble-based attacks.
Our method can be equivalent to utilizing an ensemble of infinitely many substitute models with different parameters for performing attacks.
There exists prior work that also took advantage of multiple substitute models.
For instance, Liu et al.
(2017) proposed to generate adversarial examples on an ensemble of multiple models that differ in their architectures.
Li et al.
(2020b) proposed ghost network for gaining transferability, using dropout and skip connection erosion to obtain multiple models.
Following the spirit of stochastic variance reduced gradient (Johnson & Zhang, 2013), Xiong et al.
(2022) proposed stochastic variance reduced ensemble (SVRE) to reduce the variance of gradients of different substitute models.
From a geometric perspective, Gubri et al.
(2022b) suggested finetuning with a constant and high learning rate for collecting multiple models along the training trajectory, on which the ensemble attack was performed.
Another method collected substitute models by using cSGLD (Gubri et al., 2022a), which is more related to our work, but being different in the sense of posterior approximation and sampling strategy.
We will provide detailed comparison in Section 4.2.
2. B AYESIAN DNN S
If a DNN is viewed as a probabilistic model, then the training of its parameters w can be regarded as maximum likelihood estimation or maximum a posterior estimation (with regularization).
Bayesian deep learning opts for estimating a posterior of the parameter given data at the same time.
Prediction of any new input instance is given by taking expectation over such a posterior.
Since DNNs normally involves a huge number of parameters, making the optimization of Bayesian model more challenging than in shallow models, a series of studies have been conducted and many scalabble approximations have been developed.
Effective methods utilize variational inference (Graves, 2011; Blundell et al., 2015; Kingma et al., 2015; Khan et al., 2018; Zhang et al., 2018; Wu et al., 2018; Osawa et al., 2019; Dusenberry et al., 2020) dropout inference (Gal & Ghahramani, 2016; Kendall & Gal, 2017;
Gal et al., 2017), Laplace approximation (Kirkpatrick et al., 2017; Ritter et al., 2018; Li, 2000), or SGD-based approximation (Mandt et al., 2017; Maddox et al., 2019; 2021; Wilson & Izmailov, 2020).
Taking SWAG (Maddox et al., 2019) as an example, which is an SGD-based approximation, it approximates the posterior using a Gaussian distribution with the stochastic weight averaging (SWA) solution as its first raw moment and the composition of a low rank matrix and a diagonal matrix as its second central moment.
Our method is develop in a Bayesian spirit and we shall discuss SWAG thoroughly in this paper.
Due to the space limit of this paper, we omit detailed introduction of these methods and encourage readers to check references if needed.
The robustness of Bayesian DNNs has also been studied over the last few years.
In addition to the probabilistic robustness/safety measures of such models (Cardelli et al., 2019; Wicker et al., 2020), attacks have been adapted (Liu et al., 2018b; Yuan et al., 2020) to testing the robustness in Bayesian settings.
Theoretical studies have also been made (Gal & Smith, 2018).
Although Bayesian models are suggested to be more robust (Carbone et al., 2020; Li et al., 2019), adversarial training has also been proposed for them, as in Liu et al. (2018b)’s work. Yet, to the best of our knowledge, these studies did not pay attention to adversarial transferability.  T RANSFER - BASED A TTACK AND B AYESIAN S UBSTITUTE M ODELS
An intuition for improving the transferability of adversarial examples suggests improving the diversity during backpropagation.
Prior work has tried increasing input diversity (Xie et al., 2019; Dong et al., 2019; Lin et al., 2019; Wang et al., 2021) and has indeed achieved remarkable improvements.
In this paper we consider model diversity.
A straightforward idea seems to train a set of models with diverse architecture or from different initialization.
If all models (including the victim models) that are trained to accomplish the same classification task follow a common distribution, then training multiple source models seems to perform multiple point estimates with maximum likelihood estimation.
The power of performing attacks on the ensemble of these models may increase along with the number of source models.
However, the time complexity of such a straightforward method is high, and it scales with the number of source models that could be trained.
Here we opt for an Bayesian approach to address this issue and the method resembles performing transfer-based attack on an ensemble of infinitely many DNNs.
3. G ENERATE A DVERSARIAL E XAMPLES ON A B AYESIAN M ODEL
Bayesian learning aims to discover a distribution of likely models instead of a single one.
Given a posterior distribution over parameters p ( w |D ) ∝ p ( D| w ) p ( w ) , we can predict the label of a new input x by Bayesian model averaging, i.e. ,
[Equation]
where p ( y | x , w ) is the likelihood, sometimes also referred to as the predictive distribution (Izmailov et al., 2021; Lakshminarayanan et al., 2017) for a given w , which is obtained from the DNN output with the assistance of the softmax function.
To perform attack on such a Bayesian model, a straightforward idea is to solve the following optimization problem (Liu et al., 2018b; Carbone et al., 2020):
[Equation]
Obviously, it is intractable to perform exact inference on DNNs using Eq.
(2), since there are a very large number of parameters.
A series of methods aim to address this, and, as in prior work, we adopt the Monte Carlo sampling method to approximate the integral, i.e. , p ( y | x , D )
≈ 1 M (cid:80) i p ( y | x , w i ) , where a set of M models each of which parameterized by w i are sampled from the posterior p ( w |D ) .
One can then solve Eq.
(2) by performing attacks on the ensemble of these models,
[Equation]
where L ( · , · , w i ) is a function evaluating prediction loss of a DNN model parameterized by w i .
With iterative optimization methods, e.g. , I-FGSM and PGD, different sets of models can be sampled at different iterations, as if there exist infinitely many substitute models.
3. T HE B AYESIAN F ORMULATION AND P OSSIBLE F INETUNING
In this subsection, we discuss the way of obtaining the Bayesian posterior.
Following prior work, we consider a threat model in which finetuning a source model is sometimes possible on benchmark datasets collected for the same task as that of the victim models, though it is feasible to approximate the posterior without taking special care of the training process (Gal & Ghahramani, 2016). [Figure]
To get started, we simply assume that the posterior is an isotropic Gaussian N ( ˆ w , σ 2 I ) , where ˆ w is the parameter to be trained and σ is a positive constant for controlling the diversity of distribution.
The rationality of such an assumption of isotropic posterior comes from the fact that the distribution of victim models is unknown and nearly infeasible to estimate in practice, and higher probability of the victim parameters in the posterior may imply stronger transferability.
We thus encourage exploration towards all directions (departed from ˆ w ) of equal importance in the first place.
Discussions with a more practical assumption of the posterior will be given in the next section.
Optimization of the Bayesian model can be formulated as
[Equation]
We can further reformulate Eq.
(4) into
[Equation]
by adopting Monte Carlo sampling.
The computational complexity of the objective in Eq.
(5) seems still high, thus we focus on the worst-case parameters from the posterior, whose loss in fact bounds the objective from below.
The optimization problem then becomes:
[Equation]
I ) and p ( ∆w )
[Equation]
where ε controls the confidence region of the Gaussian posterior.
With Taylor’s theorem, we further approximate Eq.
(6) with
[Equation]
As ∆w is sampled from a zero-mean isotropic Gaussian distribution, the inner maximization problem has an analytic solution, which is ∆w ∗ =
λ ε,σ ∇ ˆ w L ( x , y, ˆ w ) / ∥∇ ˆ w L ( x , y, ˆ w ) ∥ .
λ ε,σ is computed with the probability density of the zero-mean isotropic Gaussian distribution.
Thereafter, the outer gradient for solving Eq.
(7) is ∇ ˆ w L ( x , y, ˆ w ) + H∆w ∗ , which involves second-order partial derivatives in the Hessian matrix H which can be approximately calculated using the finite difference method.
More specifically, the gradient is estimated with ∇ ˆ w L ( x , y, ˆ w ) + (1 /γ )
( ∇ ˆ w L ( ˆ w + γ ∆w ∗ ) − ∇ ˆ w L ( w )) , where γ is a small positive constant.
A quick experiment on CIFAR-10 (Krizhevsky & Hinton, 2009) using ResNet-18 (He et al., 2016) was performed to evaluate the effectiveness of such finetuning.
Introduction to victim models and detailed experimental settings are deferred to Section 4.1.
In Figure 1, we compare attack transferability using the conventional deterministic formulation and the Bayesian formulation, by applying FGSM and I-FGSM for single-step and multi-step attacks, respectively.
We evaluated with different choices of the posterior covariance by varying the value of σ .
Notice that, as has been mentioned, the posterior can also be given even without any finetuning, and we achieve this via directly applying the pre-trained parameters to ˆ w in N ( ˆ w , σ 2 I ) . Attacking such a “Bayesian” model can be equivalent to attacking the deterministic model when σ → 0 .
With σ being strictly greater than 0 , one [Figure] can perform attack in a way as described in Section 3.1.
Apparently, more significant adversarial transferability can be achieved by introducing the Bayesian formulation (along with σ > 0 ), while the substitute model itself is suggested to be more robust with such a Bayesian formulation (Li et al., 2019; Carbone et al., 2020).
In Figure 1, see the blue curves that are obtained without any finetuning, the best performance is achieved with σ = 0 .
012 , showing +5 .
71% and +14 . 30% absolute gain in attack success rate caused by attacks with FGSM and I-FGSM, respectively. It can further be observed that obvious improvement (over +21% absolute increase in the I-FGSM success rate) can be achieved by introducing the additional finetuning, though attacking the pretrained parameters in a Bayesian manner already outperforms the baseline considerably.
Detailed performance on each victim model is reported in Table 1.
Conclusions on ImageNet is consistent with that on CIFAR-10, see Appendix A.
The (possible) finetuning improves the performance of unsatisfactory models that could be sampled.
For instance, with σ = 0 . 009 , the worst performance in 100 model samples from the posterior shows an test-set prediction accuracy of 90 .
59% while the model at population mean shows an accuracy of 91 .
After finetuning, both results increase, to 91 .
91% and 92 .
65% , respectively, which means finetuning improves the test-set performance of Bayesian samples and this could be beneficial to adversarial transferability.
Bayesian substitute model (with and without additional finetuning) under the ℓ ∞ constraint with ϵ = 4 / 255 on CIFAR-10.
The architecture of the substitute models is ResNet-18, and “average” was calculated over all six victim models except for ResNet-
[Table] We performed 10 runs of the experiment and report the average performance in the table.
[Table]
3. F ORMULATION WITH I MPROVED A PPROXIMATION
In Section 3.2, we have demonstrated the power of adopting Bayesian substitute models for generating transferable adversarial examples, with a relatively strong assumption of the posterior with a presumed isotropic covariance matrix though.
Taking one step further, we try to learn the covariance matrix from data in this subsection.
There exist dozens of methods for achieving the goal, here we choose SWAG (Maddox et al., 2019) which is a simple and scalable one.
It introduces an improved approximation to the posterior over [Table] parameters.
Gaussian approximation is still considered [Table] and more specifically, the SWA solution (Izmailov et al., 2018) is adopted as its mean and the covariance is decomposed into a low rank matrix and a diagonal matrix, i
[Table] e. , w ∼ N ( w SWA , Σ SWAG ) , where Σ SWAG = 12 ( Σ diag + Σ low − rank ) . In SWAG
[Table]
both the mean and the covariance are calculated using temporary models during training, and thus the posterior is estimated from the training dynamics.
Recall that the posterior concerned in the previous section is constructed using only a single model sampling, thus it can be readily combined with SWAG for improving the diversity and flexibility.
Specifically, since w SWA is unknown before training terminates, we still optimize w i during training
[Table]
and the same learning objective as in Section 3.2 can be derived. On this point, the dispersion of w in the final Bayesian model comes from two independent Gaussian distribution and the covariance matrices are added together, i.e. , w ∼ N ( w SWA , Σ SWAG + β I ) , where β = σ 2 .
Figure 2 illustrates the attack performance when SWAG is further incorporated for approximating the posterior.
It can be observed that our method still works favourably well, and further improvements can be achieved comparing to the results in Figure 1.
Detailed attack success rates on each victim model are provided in Table 1.
Conclusions on ImageNet are the same (see Appendix A).
Such empirical improvements indicate that the assumption of a more general Gaussian posterior may still align with the distribution of victim parameters in practice, and SWAG help estimate the posterior appropriately.
Therefore, without further clarification, we shall incorporate SWAG into our method in the following experiments considering the superior performance.
Note that for SWAG without our possible finetuning, we estimate the mean and covariance in SWAG by finetuning using crossentropy loss from the pre-trained model.
E XPERIMENTS
In this section, we evaluate the effectiveness of our method by comparing it to a bunch of recent state-of-the-arts.
4. E XPERIMENTAL S ETTINGS
We tested untargeted ℓ ∞ attacks in the black-box setting, just like prior work (Dong & Yang, 2019;
Lin et al., 2019; Wang et al., 2021; Guo et al., 2020; 2022).
A bunch of methods were considered for comparison on CIFAR-10 (Krizhevsky & Hinton, 2009), and ImageNet (Russakovsky et al., 2015), using I-FGSM as the back-end method.
On CIFAR-10, we set the perturbation budget to ϵ = 4 /and used ResNet-18 (He et al., 2016) as source model.
While on ImageNet, the perturbation bud- [Table] get was set to ϵ = 8 / 255 and used ResNet-50 (He et al., 2016) as the source model.
For victim models on CIFAR-10, we chose a ResNet-18 (He et al., 2016), a VGG-19 with batch normalization (Simonyan & Zisserman, 2015), a PyramidNet (Han et al., 2017), GDAS (Dong & Yang, 2019), a WRN-28-10 (Zagoruyko & Komodakis, 2016), a ResNeXt-29 (Xie et al., 2017), and a DenseNetBC (Huang et al
[Table]
, 2017) 1 , following Guo et al.
[Table] 2022)’s work.
On ImageNet, ResNet-50 (He et al., 2016), VGG-19 (Simonyan & Zisserman, 2015), ResNet-152 (He et al., 2016), Inception v3 (Szegedy et al., 2016), DenseNet-121 (Huang et al., 2017), MobileNet v2 (Sandler et al., 2018),
SENet-154 (Hu et al., 2018), ResNeXt-101 (Xie et al., 2017), WRN-101 (Zagoruyko & Komodakis, 2016), PNASNet (Liu et al., 2018a), and MNASNet (Tan et al., 2019) were adopted as victim models 2 .
Since these victim models may require different sizes of inputs, we strictly followed their official pre-processing pipeline to obtain inputs of specific sizes.
For CIFAR-10 tests, we performed attacks on all test data.
For ImageNet, we randomly sampled 5000 test images from a set of the validation data that could be classified correctly by these victim models, and we learned perturbations to these images, following prior work (Huang & Kong, 2022; Guo et al., 2020; 2022).
For comparison, we evaluated attack success rates of adversarial examples crafted utilizing different methods on all victim models.
Inputs to all models were re-scaled to [0 . 0 , 1 . 0] .
Temporary results after each attack iteration were all clipped to this range to guarantee the inputs to DNNs were close to valid images.
When establishing a multi-step baseline using I-FGSM we run it for 20 iterations on CIFAR-10 data and 50 iterations on ImageNet data with a step size of 1 / 255 .
In possible finetuning, we set γ = 0 .
1 / ∥
∆w ∗ ∥ 2 and a finetuning learning rate of 0 .
if SWAG was incorporated.
We set a smaller finetuning learning rate of 0 .
001 if it was not.
On CIFAR-10, finetune without and with SWAG, we set λ ε,σ = 2 and λ ε,σ = 0 . 2 , respectively.
When performing attacks, we set σ = 0 .
009 and σ = 0 .
002 for source models finetuned without and with SWAG, respectively.
On ImageNet, we always set λ ε,σ = 1 , and σ = 0 .
005 and σ = 0 .
002 for source models finetuned without and with SWAG, respectively.
Since we found little difference in success rate between using diagonal matrix and diagonal matrix plus low rank matrix as the covariance for SWAG posterior, we always use the diagonal matrix for simplicity.
For compared competitors, we followed their official implementations and evaluated on the same test images on CIFAR-10 and ImageNet for generating adversarial examples fairly.
Implementation details about these methods are deferred to Appendix D
All experiments are performed on an NVIDIA V100 GPU.
4. C OMPARISON TO S TATE - OF - THE - ARTS
We compare our method to recent state-of-the-arts in Table 2 and 3 on attacking 10 victim models on ImageNet and 6 victim models on CIFAR-10.
Methods that increase input diversity, i.e. , and
TIM (Dong et al., 2019), SIM (Lin et al., 2019), Admix (Wang et al., 2021), that modify backpropagation, i.e. , LinBP (Guo et al., 2020) and ILA++ (Guo et al., 2022), and very recent methods including TAIG (Huang & Kong, 2022) and LGV (Gubri et al., 2022b) are compared.
It can be observed in the tables that our method outperforms all these methods significantly.
Specifically, our method achieves an average success rate of nearly 89 .
15% on ImageNet, which outperforms the second best by 16 .
When attacking Inception v3 and PyramidNet, which are the most challenging victim models on ImageNet and CIFAR-10, respectively, out method outperforms the second best by 23 .
04% and 7 .
43% while outperforms the I-FGSM baseline by 53 .
20% and 36 .
Comparison to (more) ensemble attacks.
From certain perspectives, our method shows a similar spirit to that of ensemble attacks, thus here we compare more such attacks (Liu et al., 2017; Xiong et al., 2022) empirically.
Table 4 reports results of all methods under the ℓ ∞ constraint and ϵ = 8 / 255 .
Our method taking ResNet-50 as the substitute architecture achieves an average success rate of 90 .
03% on ImageNet, while the two prior methods taking ResNet-50, Inception v3, MobileNet, and MNASNet altogether as substitute architectures achieve 46 .
58% and 57 .
One may also utilize more than one architecture for our method, as a natural combination of our method and prior methods, see also Table 4 for results.
We adopt ResNet-50 and MobileNet as substitute architectures when combine with the ensemble methods [Table]
[Table]
Comparison to Gubri et al.
(2022a)’s method.
Gubri et al.
(2022a)’s method suggested to collect multiple substitute models along a single run of training using cSGLD (Zhang et al., 2019).
It is related to Bayesian learning in the sense that cSGLD is a Bayesian posterior sampling method.
The difference between our method and Gubri et al.
’s lie in three main aspect.
First, our method is motivated by the belief that scaling the number of substitute models, to infinity if possible , improves the transferability of adversarial examples; thus, by establishing a proper posterior with or without finetuning, our method is capable of producing different sets of substitute models at different iterations of I-FGSM, as if there exist infinitely many models (see Appendix C for the benefit in scaling the number of models). While, Gubri et al.
’s method utilizes a fixed finite set of models collected during a single run of finetuning. Second, as have been demonstrated in Section 3.2, our method is capable of achieving superior transferability even without finetuning, while finetuning is requisite for Gubri et al. ’s method.
Third, the derivation in Section 3.2 leads to a principled finetuning objective for our method, which is also strikingly different from that of Gubri et al.
’s method. Table 5 compares the two methods empirically and shows the superiority of our method in experiments.
We followed their experimental settings to evaluate the methods and copy their results in the paper.
More specifically, the back-end I-FGSM was perform with ϵ = 4 / 255 and a step size of ϵ/ 10 for 50 optimization steps in total.
Victim models include ResNeXt-50, DenseNet-121, EfficientNet-B0 (Tan & Le, 2019), etc.
[Table]
Attack robust models.
It is also of interest to evaluate the transferability of adversarial examples to robust models, and we compare the performance of competitive methods in this setting in Table 6.
The victim models here include a robust Inception v3 and a robust EfficientNet-B0 collected from the public timm (Wightman, 2019) repository, together with a robust ResNet-50 and a robust DeiTS (Touvron et al., 2021a) provided by Bai et al.
All these models were trained using some sorts of advanced adversarial training (Madry et al., 2018; Wong et al., 2020; Xie et al., 2020) on ImageNet. We still used the ResNet-50 source model which was trained just as normal and not robust to adversarial examples.
Obviously, in Table 6, our method outperforms the others consistently in attacking these models.
In addition to adversarial training, we also tested on robust models obtained via randomized smoothing (Cohen et al., 2019), which is one of the most popular methods for achieving certified robustness.
In particular, our method achieves an success rate of 29 . 08 % in attacking a smoothed ResNet-50 on ImageNet, following the same settings as in Table 6, while I-FGSM, ILA++, and LGV achieve 9 . 06% , and, 17 .
64% , respectively. [Table] Attack vision transformers. The community has witnessed a surge of transformers in computer vision and machine learning applications.
Here we also test the transferability of adversarial examples (generated on convolutional substitute architectures) to some vision transformers.
Specifically, we tested with a ViT-B (Dosovitskiy et al., 2020), a DeiT-B (Touvron et al., 2021b) [Table] a Swin-B (Liu et al., 2021) [Table]
and a BEiT(Bao et al., 2021).
These models were all collected from timm (Wightman, 2019).
Results in Table 6 show that, though it is indeed more challenging to transfer to vision transformers, our method can also gain considerable improvements.
4. C OMBINATION WITH O THER M ETHODS
We would also like to mention that it is also possible to combine our method with some methods in
Table 2 to further enhance the transferability, since our method focuses on model diversity and does not consider improvement in input diversity and backpropagation.
In Table 7, we report the attack success rate of our method, in combination with Admix, LinBP, and ILA++.
It can be seen that the transferability to all victim models are further enhanced.
The best performance is obtained when combined with ILA++, leading to an average success rate achieve of 92 .
90% (which is roughly 30% higher than the original performance of ILA++) and a worst success rate of only 81 .
64% (when attacking PNASNet).
[Table]
C ONCLUSION
In this paper, we have considered diversity in substitute models for performing transfer-based attacks.
Specifically, we have developed a Bayesian formulation for performing attacks and advocated possible finetuning for improving the Bayesian model.
By simply assuming the posterior to be an isotropic Gaussian distribution or, one step further, a more general Gaussian distribution, our attack can be equivalently regarded as generating adversarial examples on a set of infinitely many substitute models while the time complexity of possible finetuning is just as normal.
Extensive experiments have been conducted to demonstrate the effectiveness of our method on ImageNet and CIFAR-10.
It has been shown our method outperforms recent state-of-the-arts by large margins in attacking more than 10 convolutional DNNs and 4 vision transformers.
The transferability to robust models has also been evaluated.
Moreover, we have also shown that the proposed method can further be combined with prior methods to achieve even more powerful adversarial transferability.
A P ERFORMANCE OF P OSSIBLE F INETUNING ON I MAGE N ET
[Table]
B S ENSITIVITY OF λ ε,σ
When finetuning is possible, we have λ ε,σ as a hyper-parameter. An empirical study was performed to show how the performance of our method varies along with the value of such a hyper-parameter. We tested with λ ε,σ ∈ { 0 , 0 . 5 , 1 , 1 . 5 , 2 } on ImageNet and show the average attack success rates of attacking ten victim models in Figure 3.
It can be seen by increasing the value of λ ε,σ , the adversarial transferability is improved, while it drastically drop when it is too large.
Tuning of such a hyper-parameter can be done coarsely in a logarithmic scale on a validation set on different datasets.
C B ENEFIT OF S CALING THE N UMBER OF S UBSTITUTE M ODELS
Our method is developed based on the belief that utilizing more substitute models should improve the transferability of adversarial examples, and the Bayesian formulation is considered since infinitely many models can be sampled from it in principle.
In this section, we evaluate the transferability of adversarial examples crafted on different numbers of substitute models, which are sampled before issuing attacks.
Figure B illustrates the results, and it can be observed that using more substitute models can indeed improve the transferability of adversarial examples.
Average success rates calculated on the same victim models as in Table 2 are reported.
in Table
D D ETAILED E XPERIMENTAL S ETTINGS OF C OMPARED M ETHODS
Here we report some detailed experimental settings of compared methods.
For TIM (Dong et al., 2019), we adopted the maximal shift of 3 × 3 and 7 × 7 on CIFAR-10 and ImageNet, respectively.
For SIM (Lin et al., 2019), the number of scale copies is 5 .
For Admix (Wang et al., 2021), we set the number of scale copies and mix copies to 5 and 3 , respectively, and let the mix coefficient be 0 . 2 .
For LinBP (Guo et al., 2020), the last six building blocks in ResNet-50 and the last four building blocks in Resnet-18 were modified to be more linear during backpropagation.
For TAIG (Huang &
Kong, 2022), we set 30 as the number of tuning points.
For ILA++ (Li et al., 2020a; Guo et al., 2022), we chose the first block of the last meta-block of ResNet-18 and the first block of the third block of ResNet-50.