# ensemble_adversarial_training_attacks_and_defenses__d182c672.pdf

Published as a conference paper at ICLR 2018

ENSEMBLE ADVERSARIAL TRAINING: ATTACKS AND DEFENSES

Florian Tram er Stanford University tramer@cs.stanford.edu

Alexey Kurakin Google Brain kurakin@google.com

Nicolas Papernot

Pennsylvania State University ngp5056@cse.psu.edu

Ian Goodfellow Google Brain goodfellow@google.com

Dan Boneh Stanford University dabo@cs.stanford.edu

Patrick Mc Daniel Pennsylvania State University mcdaniel@cse.psu.edu

Adversarial examples are perturbed inputs designed to fool machine learning models. Adversarial training injects such examples into training data to increase robustness. To scale this technique to large datasets, perturbations are crafted using fast single-step methods that maximize a linear approximation of the model s loss. We show that this form of adversarial training converges to a degenerate global minimum, wherein small curvature artifacts near the data points obfuscate a linear approximation of the loss. The model thus learns to generate weak perturbations, rather than defend against strong ones. As a result, we ﬁnd that adversarial training remains vulnerable to black-box attacks, where we transfer perturbations computed on undefended models, as well as to a powerful novel single-step attack that escapes the non-smooth vicinity of the input data via a small random step. We further introduce Ensemble Adversarial Training, a technique that augments training data with perturbations transferred from other models. On Image Net, Ensemble Adversarial Training yields models with strong robustness to black-box attacks. In particular, our most robust model won the ﬁrst round of the NIPS 2017 competition on Defenses against Adversarial Attacks (Kurakin et al., 2017c).

1 INTRODUCTION

Machine learning (ML) models are often vulnerable to adversarial examples, maliciously perturbed inputs designed to mislead a model at test time (Biggio et al., 2013; Szegedy et al., 2013; Goodfellow et al., 2014b; Papernot et al., 2016a). Furthermore, Szegedy et al. (2013) showed that these inputs transfer across models: the same adversarial example is often misclassiﬁed by different models, thus enabling simple black-box attacks on deployed models (Papernot et al., 2017; Liu et al., 2017).

Adversarial training (Szegedy et al., 2013) increases robustness by augmenting training data with adversarial examples. Madry et al. (2017) showed that adversarially trained models can be made robust to white-box attacks (i.e., with knowledge of the model parameters) if the perturbations computed during training closely maximize the model s loss. However, prior attempts at scaling this approach to Image Net-scale tasks (Deng et al., 2009) have proven unsuccessful (Kurakin et al., 2017b).

It is thus natural to ask whether it is possible, at scale, to achieve robustness against the class of black-box adversaries Towards this goal, Kurakin et al. (2017b) adversarially trained an Inception v3 model (Szegedy et al., 2016b) on Image Net using a single-step attack based on a linearization of the model s loss (Goodfellow et al., 2014b). Their trained model is robust to single-step perturbations but remains vulnerable to more costly multi-step attacks. Yet, Kurakin et al. (2017b) found that these attacks fail to reliably transfer between models, and thus concluded that the robustness of their model should extend to black-box adversaries. Surprisingly, we show that this is not the case.

Part of the work was done while the author was at Google Brain.

Published as a conference paper at ICLR 2018

We demonstrate, formally and empirically, that adversarial training with single-step methods admits a degenerate global minimum, wherein the model s loss can not be reliably approximated by a linear function. Speciﬁcally, we ﬁnd that the model s decision surface exhibits sharp curvature near the data points, thus degrading attacks based on a single gradient computation. In addition to the model of Kurakin et al. (2017b), we reveal similar overﬁtting in an adversarially trained Inception Res Net v2 model (Szegedy et al., 2016a), and a variety of models trained on MNIST (Le Cun et al., 1998).

We harness this result in two ways. First, we show that adversarially trained models using single-step methods remain vulnerable to simple attacks. For black-box adversaries, we ﬁnd that perturbations crafted on an undefended model often transfer to an adversarially trained one. We also introduce a simple yet powerful single-step attack that applies a small random perturbation to escape the nonsmooth vicinity of the data point before linearizing the model s loss. While seemingly weaker than the Fast Gradient Sign Method of Goodfellow et al. (2014b), our attack signiﬁcantly outperforms it for a same perturbation norm, for models trained with or without adversarial training.

Second, we propose Ensemble Adversarial Training, a training methodology that incorporates perturbed inputs transferred from other pre-trained models. Our approach decouples adversarial example generation from the parameters of the trained model, and increases the diversity of perturbations seen during training. We train Inception v3 and Inception Res Net v2 models on Image Net that exhibit increased robustness to adversarial examples transferred from other holdout models, using various single-step and multi-step attacks (Goodfellow et al., 2014b; Carlini & Wagner, 2017a; Kurakin et al., 2017a; Madry et al., 2017). We also show that our methods globally reduce the dimensionality of the space of adversarial examples (Tram er et al., 2017). Our Inception Res Net v2 model won the ﬁrst round of the NIPS 2017 competition on Defenses Against Adversarial Attacks (Kurakin et al., 2017c), where it was evaluated on other competitors attacks in a black-box setting.1

2 RELATED WORK

Various defensive techniques against adversarial examples in deep neural networks have been proposed (Gu & Rigazio, 2014; Luo et al., 2015; Papernot et al., 2016c; Nayebi & Ganguli, 2017; Cisse et al., 2017) and many remain vulnerable to adaptive attackers (Carlini & Wagner, 2017a;b; Baluja & Fischer, 2017). Adversarial training (Szegedy et al., 2013; Goodfellow et al., 2014b; Kurakin et al., 2017b; Madry et al., 2017) appears to hold the greatest promise for learning robust models.

Madry et al. (2017) show that adversarial training on MNIST yields models that are robust to whitebox attacks, if the adversarial examples used in training closely maximize the model s loss. Moreover, recent works by Sinha et al. (2018), Raghunathan et al. (2018) and Kolter & Wong (2017) even succeed in providing certiﬁable robustness for small perturbations on MNIST. As we argue in Appendix C, the MNIST dataset is peculiar in that there exists a simple closed-form denoising procedure (namely feature binarization) which leads to similarly robust models without adversarial training. This may explain why robustness to white-box attacks is hard to scale to tasks such as Image Net (Kurakin et al., 2017b). We believe that the existence of a simple robust baseline for MNIST can be useful for understanding some limitations of adversarial training techniques.

Szegedy et al. (2013) found that adversarial examples transfer between models, thus enabling blackbox attacks on deployed models. Papernot et al. (2017) showed that black-box attacks could succeed with no access to training data, by exploiting the target model s predictions to extract (Tram er et al., 2016) a surrogate model. Some prior works have hinted that adversarially trained models may remain vulnerable to black-box attacks: Goodfellow et al. (2014b) found that an adversarial maxout network on MNIST has slightly higher error on transferred examples than on white-box examples. Papernot et al. (2017) further showed that a model trained on small perturbations can be evaded by transferring perturbations of larger magnitude. Our ﬁnding that adversarial training degrades the accuracy of linear approximations of the model s loss is as an instance of a gradient-masking phenomenon (Papernot et al., 2016b), which affects other defensive techniques (Papernot et al., 2016c; Carlini & Wagner, 2017a; Nayebi & Ganguli, 2017; Brendel & Bethge, 2017; Athalye et al., 2018).

1We publicly released our model after the ﬁrst round, and it could thereafter be targeted using white-box attacks. Nevertheless, a majority of the top submissions in the ﬁnal round, e.g. (Xie et al., 2018) built upon our released model.

Published as a conference paper at ICLR 2018

3 THE ADVERSARIAL TRAINING FRAMEWORK

We consider a classiﬁcation task with data x [0, 1]d and labels ytrue Zk sampled from a distribution D. We identify a model with an hypothesis h from a space H. On input x, the model outputs class scores h(x) Rk. The loss function used to train the model, e.g., cross-entropy, is L(h(x), y).

3.1 THREAT MODEL

For some target model h H and inputs (x, ytrue) the adversary s goal is to ﬁnd an adversarial example xadv such that xadv and x are close yet the model misclassiﬁes xadv. We consider the wellstudied class of ℓ bounded adversaries (Goodfellow et al., 2014b; Madry et al., 2017) that, given some budget ϵ, output examples xadv where xadv x ϵ. As we comment in Appendix C.1, ℓ robustness is of course not an end-goal for secure ML. We use this standard model to showcase limitations of prior adversarial training methods, and evaluate our proposed improvements.

We distinguish between white-box adversaries that have access to the target model s parameters (i.e., h), and black-box adversaries with only partial information about the model s inner workings. Formal deﬁnitions for these adversaries are in Appendix A. Although security against white-box attacks is the stronger notion (and the one we ideally want ML models to achieve), black-box security is a reasonable and more tractable goal for deployed ML models.

3.2 ADVERSARIAL TRAINING

Following Madry et al. (2017), we consider an adversarial variant of standard Empirical Risk Minimization (ERM), where our aim is to minimize the risk over adversarial examples:

h = arg min h H E (x,ytrue) D

h max xadv x ϵ L(h(xadv), ytrue) i . (1)

Madry et al. (2017) argue that adversarial training has a natural interpretation in this context, where a given attack (see below) is used to approximate solutions to the inner maximization problem, and the outer minimization problem corresponds to training over these examples. Note that the original formulation of adversarial training (Szegedy et al., 2013; Goodfellow et al., 2014b), which we use in our experiments, trains on both the clean examples x and adversarial examples xadv.

We consider three algorithms to generate adversarial examples with bounded ℓ norm. The ﬁrst two are single-step (i.e., they require a single gradient computation); the third is iterative it computes multiple gradient updates. We enforce xadv [0, 1]d by clipping all components of xadv.

Fast Gradient Sign Method (FGSM). This method (Goodfellow et al., 2014b) linearizes the inner maximization problem in (1):

xadv FGSM := x + ε sign ( x L(h(x), ytrue)) . (2)

Single-Step Least-Likely Class Method (Step-LL). This variant of FGSM introduced by Kurakin et al. (2017a;b) targets the least-likely class, y LL = arg min{h(x)}:

xadv LL := x ε sign ( x L(h(x), y LL)) . (3)

Although this attack only indirectly tackles the inner maximization in (1), Kurakin et al. (2017b) ﬁnd it to be the most effective for adversarial training on Image Net.

Iterative Attack (I-FGSM or Iter-LL). This method iteratively applies the FGSM or Step-LL k times with step-size α ϵ/k and projects each step onto the ℓ ball of norm ϵ around x. It uses projected gradient descent to solve the maximization in (1). For ﬁxed ϵ, iterative attacks induce higher error rates than single-step attacks, but transfer at lower rates (Kurakin et al., 2017a;b).

3.3 A DEGENERATE GLOBAL MINIMUM FOR SINGLE-STEP ADVERSARIAL TRAINING

When performing adversarial training with a single-step attack (e.g., the FGSM or Step-LL methods above), we approximate Equation (1) by replacing the solution to the inner maximization problem

Published as a conference paper at ICLR 2018

in with the output of the single-step attack (e.g., xadv FGSM in (2)). That is, we solve

h = arg min h H E (x,ytrue) D

h L(h(xadv FGSM), ytrue) i . (4)

For model families H with high expressive power, this alternative optimization problem admits at least two substantially different global minima h :

For an input x from D, there is no xadv close to x (in ℓ norm) that induces a high loss. That is,

L(h (xadv FGSM), ytrue) max xadv x ϵ L(h (xadv), ytrue)] 0 . (5)

In other words, h is robust to all ℓ bounded perturbations. The minimizer h is a model for which the approximation method underlying the attack (i.e., linearization in our case) poorly ﬁts the model s loss function. That is,

L(h (xadv FGSM), ytrue) max xadv x ϵ L(h (xadv), ytrue)] . (6)

Thus the attack when applied to h produces samples xadv that are far from optimal.

Note that this second degenerate minimum can be more subtle than a simple case of overﬁtting to samples produced from single-step attacks. Indeed, we show in Section 4.1 that single-step attacks applied to adversarially trained models create adversarial examples that are easy to classify even for undefended models. Thus, adversarial training does not simply learn to resist the particular attack used during training, but actually to make that attack perform worse overall. This phenomenon relates to the notion of Reward Hacking (Amodei et al., 2016) wherein an agent maximizes its formal objective function via unintended behavior that fails to captures the designer s true intent.

3.4 ENSEMBLE ADVERSARIAL TRAINING

The degenerate minimum described in Section 3.3 is attainable because the learned model s parameters inﬂuence the quality of both the minimization and maximization in (1). One solution is to use a stronger adversarial example generation process, at a high performance cost (Madry et al., 2017). Alternatively, Baluja & Fischer (2017) suggest training an adversarial generator model as in the GAN framework (Goodfellow et al., 2014a). The power of this generator is likely to require careful tuning, to avoid similar degenerate minima (where the generator or classiﬁer overpowers the other).

We propose a conceptually simpler approach to decouple the generation of adversarial examples from the model being trained, while simultaneously drawing an explicit connection with robustness to black-box adversaries. Our method, which we call Ensemble Adversarial Training, augments a model s training data with adversarial examples crafted on other static pre-trained models. Intuitively, as adversarial examples transfer between models, perturbations crafted on an external model are good approximations for the maximization problem in (1). Moreover, the learned model can not inﬂuence the strength of these adversarial examples. As a result, minimizing the training loss implies increased robustness to black-box attacks from some set of models.

Domain Adaptation with multiple sources. We can draw a connection between Ensemble Adversarial Training and multiple-source Domain Adaptation (Mansour et al., 2009; Zhang et al., 2012). In Domain Adaptation, a model trained on data sampled from one or more source distributions S1, . . . , Sk is evaluated on samples x from a different target distribution T .

Let Ai be an adversarial distribution obtained by sampling (x, ytrue) from D, computing an adversarial example xadv for some model such that xadv x ϵ, and outputting (xadv, ytrue). In Ensemble Adversarial Training, the source distributions are D (the clean data) and A1, . . . , Ak (the attacks overs the currently trained model and the static pre-trained models). The target distribution takes the form of an unseen black-box adversary A . Standard generalization bounds for Domain Adaptation (Mansour et al., 2009; Zhang et al., 2012) yield the following result. Theorem 1 (informal). Let h H be a model learned with Ensemble Adversarial Training and static black-box adversaries A1, . . . , Ak. Then, if h is robust against the black-box adversaries A1, . . . Ak used at training time, then h has bounded error on attacks from a future black-box adversary A , if A is not much stronger , on average, than the static adversaries A1, . . . , Ak.

Published as a conference paper at ICLR 2018

(a) Loss of model v3adv.

(b) Zoom in for small ϵ1, ϵ2.

Figure 1: Gradient masking in single-step adversarial training. We plot the loss of model v3adv on points x = x+ϵ1 g+ϵ2 g , where g is the signed gradient and g is an orthogonal adversarial direction. Plot (b) is a zoom of (a) near x. The gradient poorly approximates the global loss.

We give a formal statement of this result and of the assumptions on A in Appendix B. Of course, ideally we would like guarantees against arbitrary future adversaries. For very low-dimensional tasks (e.g., MNIST), stronger guarantees are within reach for speciﬁc classes of adversaries (e.g., ℓ bounded perturbations (Madry et al., 2017; Sinha et al., 2018; Raghunathan et al., 2018; Kolter & Wong, 2017)), yet they also fail to extend to other adversaries not considered at training time (see Appendix C.1 for a discussion). For Image Net-scale tasks, stronger formal guarantees appear out of reach, and we thus resort to an experimental assessment of the robustness of Ensemble Adversarially Trained models to various non-interactive black-box adversaries in Section 4.2.

4 EXPERIMENTS

We show the existence of a degenerate minimum, as described in Section 3.3, for the adversarially trained Inception v3 model of Kurakin et al. (2017b). Their model (denoted v3adv) was trained on a Step-LL attack with ϵ 16/256. We also adversarially train an Inception Res Net v2 model (Szegedy et al., 2016a) using the same setup. We denote this model by IRv2adv. We refer the reader to (Kurakin et al., 2017b) for details on the adversarial training procedure.

We ﬁrst measure the approximation-ratio of the Step-LL attack for the inner maximization in (1). As we do not know the true maximum, we lower-bound it using an iterative attack. For 1,000 random test points, we ﬁnd that for a standard Inception v3 model, step-LL gets within 19% of the optimum loss on average. This attack is thus a good candidate for adversarial training. Yet, for the v3adv model, the approximation ratio drops to 7%, conﬁrming that the learned model is less amenable to linearization. We obtain similar results for Inception Res Net v2 models. The ratio is 17% for a standard model, and 8% for IRv2adv. Similarly, we look at the cosine similarity between the perturbations given by a single-step and multi-step attack. The more linear the model, the more similar we expect both perturbations to be. The average similarity drops from 0.13 for Inception v3 to 0.02 for v3adv. This effect is not due to the decision surface of v3adv being too ﬂat near the data points: the average gradient norm is larger for v3adv (0.17) than for the standard v3 model (0.10).

We visualize this gradient-masking effect (Papernot et al., 2016b) by plotting the loss of v3adv on examples x = x + ϵ1 g + ϵ2 g , where g is the signed gradient of model v3adv and g is a signed vector orthogonal to g. Looking forward to Section 4.1, we actually chose g to be the signed gradient of another Inception model, from which adversarial examples transfer to v3adv. Figure 1 shows that the loss is highly curved in the vicinity of the data point x, and that the gradient poorly reﬂects the global loss landscape. Similar plots for additional data points are in Figure 4.

We show similar results for adversarially trained MNIST models in Appendix C.2. On this task, input dropout (Srivastava et al., 2014) mitigates adversarial training s overﬁtting problem, in some cases. Presumably, the random input mask diversiﬁes the perturbations seen during training (dropout at intermediate layers does not mitigate the overﬁtting effect). Mishkin et al. (2017) ﬁnd that input dropout signiﬁcantly degrades accuracy on Image Net, so we did not include it in our experiments.

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Table 1: Error rates (in %) of adversarial examples transferred between models. We use Step LL with ϵ = 16/256 for 10,000 random test inputs. Diagonal elements represent a white-box attack. The best attack for each target appears in bold. Similar results for MNIST models appear in Table 7.

Source Target v4 v3 v3adv IRv2 IRv2adv v4 60.2 39.2 31.1 36.6 30.9 v3 43.8 69.6 36.4 42.1 35.1 v3adv 36.3 35.6 26.6 35.2 35.9 IRv2 38.0 38.0 30.8 50.7 31.9 IRv2adv 31.0 30.3 25.7 30.6 21.4

Source Target v4 v3 v3adv IRv2 IRv2adv v4 31.0 14.9 10.2 13.6 9.9 v3 18.7 42.7 13.0 17.8 12.8 v3adv 13.6 13.5 9.0 13.0 14.5 IRv2 14.1 14.8 9.9 24.0 10.6 IRv2adv 10.3 10.5 7.7 10.4 5.8

4.1 ATTACKS AGAINST ADVERSARIALLY TRAINED NETWORKS

Kurakin et al. (2017b) found their adversarially trained model to be robust to various single-step attacks. They conclude that this robustness should translate to attacks transferred from other models. As we have shown, the robustness to single-step attacks is actually misleading, as the model has learned to degrade the information contained in the model s gradient. As a consequence, we ﬁnd that the v3adv model is substantially more vulnerable to single-step attacks than Kurakin et al. (2017b) predicted, both in a white-box and black-box setting. The same holds for the IRv2adv model.

In addition to the v3adv and IRv2adv models, we consider standard Inception v3, Inception v4 and Inception Res Net v2 models. These models are available in the Tensor Flow-Slim library (Abadi et al., 2015). We describe similar results for a variety of models trained on MNIST in Appendix C.2.

Black-box attacks. Table 1 shows error rates for single-step attacks transferred between models. We compute perturbations on one model (the source) and transfer them to all others (the targets). When the source and target are the same, the attack is white-box. Adversarial training greatly increases robustness to white-box single-step attacks, but incurs a higher error rate in a black-box setting. Thus, the robustness gain observed when evaluating defended models in isolation is misleading. Given the ubiquity of this pitfall among proposed defenses against adversarial examples (Carlini & Wagner, 2017a; Brendel & Bethge, 2017; Papernot et al., 2016b), we advise researchers to always consider both white-box and black-box adversaries when evaluating defensive strategies. Notably, a similar discrepancy between white-box and black-box attacks was recently observed in Buckman et al. (2018).

Attacks crafted on adversarial models are found to be weaker even against undefended models (i.e., when using v3adv or IRv2adv as source, the attack transfers with lower probability). This conﬁrms our intuition from Section 3.3: adversarial training does not just overﬁt to perturbations that affect standard models, but actively degrades the linear approximation underlying the single-step attack.

A new randomized single-step attack. The loss function visualization in Figure 1 shows that sharp curvature artifacts localized near the data points can mask the true direction of steepest ascent. We thus suggest to prepend single-step attacks by a small random step, in order to escape the non-smooth vicinity of the data point before linearizing the model s loss. Our new attack, called R+FGSM (alternatively, R+Step-LL), is deﬁned as follows, for parameters ϵ and α (where α < ϵ):

xadv = x + (ε α) sign x J(x , ytrue) , where x = x + α sign(N(0d, Id)) . (7)

Note that the attack requires a single gradient computation. The R+FGSM is a computationally efﬁcient alternative to iterative methods that have high success rates in a white-box setting. Our attack can be seen as a single-step variant of the general PGD method from (Madry et al., 2017).

Table 2 compares error rates for the Step-LL and R+Step-LL methods (with ϵ = 16/256 and α = ϵ/2). The extra random step yields a stronger attack for all models, even those without adversarial training. This suggests that a model s loss function is generally less smooth near the data points. We further compared the R+Step-LL attack to a two-step Iter-LL attack, which computes two gradient steps. Surprisingly, we ﬁnd that for the adversarially trained Inception v3 model, the R+Step-LL attack is stronger than the two-step Iter-LL attack. That is, the local gradients learned by the adversarially trained model are worse than random directions for ﬁnding adversarial examples!

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Table 2: Error rates (in %) for Step-LL, R+Step-LL and a two-step Iter-LL on Image Net. We use ϵ = 16/256, α = ϵ/2 on 10,000 random test inputs. R+FGSM results on MNIST are in Table 7.

Step-LL R+Step-LL Iter-LL(2)

v4 v3 v3adv IRv2 IRv2adv 60.2 69.6 26.6 50.7 21.4 70.5 80.0 64.8 56.3 37.5 78.5 86.3 58.3 69.9 41.6

v4 v3 v3adv IRv2 IRv2adv 31.0 42.7 9.0 24.0 5.8 42.8 57.1 37.1 29.3 15.0 56.2 70.2 29.6 45.4 16.5

Table 3: Models used for Ensemble Adversarial Training on Image Net. The Res Nets (He et al., 2016) use either 50 or 101 layers. Inc Res stands for Inception Res Net (Szegedy et al., 2016a).

Trained Model Pre-trained Models Holdout Models Inception v3 (v3adv-ens3) Inception v3, Res Net v2 (50) ( Inception v4 Res Net v1 (50) Res Net v2 (101)

Inception v3 (v3adv-ens4) Inception v3, Res Net v2 (50), Inc Res v2 Inc Res v2 (IRv2adv-ens) Inception v3, Inc Res v2

We ﬁnd that the addition of this random step hinders transferability (see Table 9). We also tried adversarial training using R+FGSM on MNIST, using a similar approach as (Madry et al., 2017). We adversarially train a CNN (model A in Table 5) for 100 epochs, and attain > 90.0% accuracy on R+FGSM samples. However, training on R+FGSM provides only little robustness to iterative attacks. For the PGD attack of (Madry et al., 2017) with 20 steps, the model attains 18.0% accuracy.

4.2 ENSEMBLE ADVERSARIAL TRAINING

We now evaluate our Ensemble Adversarial Training strategy described in Section 3.4. We recall our intuition: by augmenting training data with adversarial examples crafted from static pre-trained models, we decouple the generation of adversarial examples from the model being trained, so as to avoid the degenerate minimum described in Section 3.3. Moreover, our hope is that robustness to attacks transferred from some ﬁxed set of models will generalize to other black-box adversaries.

We train Inception v3 and Inception Res Net v2 models (Szegedy et al., 2016a) on Image Net, using the pre-trained models shown in Table 3. In each training batch, we rotate the source of adversarial examples between the currently trained model and one of the pre-trained models. We select the source model at random in each batch, to diversify examples across epochs. The pre-trained models gradients can be precomputed for the full training set. The per-batch cost of Ensemble Adversarial Training is thus lower than that of standard adversarial training: using our method with n 1 pre-trained models, only every nth batch requires a forward-backward pass to compute adversarial gradients. We use synchronous distributed training on 50 machines, with minibatches of size 16 (we did not pre-compute gradients, and thus lower the batch size to ﬁt all models in memory). Half of the examples in a minibatch are replaced by Step-LL examples. As in Kurakin et al. (2017b), we use RMSProp with a learning rate of 0.045, decayed by a factor of 0.94 every two epochs.

To evaluate how robustness to black-box attacks generalizes across models, we transfer various attacks crafted on three different holdout models (see Table 3), as well as on an ensemble of these models (as in Liu et al. (2017)). We use the Step-LL, R+Step-LL, FGSM, I-FGSM and the PGD attack from Madry et al. (2017) using the hinge-loss function from Carlini & Wagner (2017a). Our results are in Table 4. For each model, we report the worst-case error rate over all black-box attacks transfered from each of the holdout models (20 attacks in total). Results for MNIST are in Table 8.

Convergence speed. Convergence of Ensemble Adversarial Training is slower than for standard adversarial training, a result of training on hard adversarial examples and lowering the batch size. Kurakin et al. (2017b) report that after 187 epochs (150k iterations with minibatches of size 32), the v3adv model achieves 78% accuracy. Ensemble Adversarial Training for models v3adv-ens3 and v3adv-ens4 converges after 280 epochs (450k iterations with minibatches of size 16). The Inception Res Net v2 model is trained for 175 epochs, where a baseline model converges at around 160 epochs.

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Table 4: Error rates (in %) for Ensemble Adversarial Training on Image Net. Error rates on clean data are computed over the full test set. For 10,000 random test set inputs, and ϵ = 16/256, we report error rates on white-box Step-LL and the worst-case error over a series of black-box attacks (Step-LL, R+Step-LL, FGSM, I-FGSM, PGD) transferred from the holdout models in Table 3. For both architectures, we mark methods tied for best in bold (based on 95% conﬁdence).

Top 1 Top 5

Model Clean Step-LL Max. Black-Box Clean Step-LL Max. Black-Box v3 22.0 69.6 51.2 6.1 42.7 24.5 v3adv 22.0 26.6 40.8 6.1 9.0 17.4 v3adv-ens3 23.6 30.0 34.0 7.6 10.1 11.2 v3adv-ens4 24.2 43.3 33.4 7.8 19.4 10.7

IRv2 19.6 50.7 44.4 4.8 24.0 17.8 IRv2adv 19.8 21.4 34.5 4.9 5.8 11.7 IRv2adv-ens 20.2 26.0 27.0 5.1 7.6 7.9

White-box attacks. For both architectures, the models trained with Ensemble Adversarial Training are slightly less accurate on clean data, compared to standard adversarial training. Our models are also more vulnerable to white-box single-step attacks, as they were only partially trained on such perturbations. Note that for v3adv-ens4, the proportion of white-box Step-LL samples seen during training is 1/4 (instead of 1/3 for model v3adv-ens3). The negative impact on the robustness to white-box attacks is large, for only a minor gain in robustness to transferred samples. Thus it appears that while increasing the diversity of adversarial examples seen during training can provide some marginal improvement, the main beneﬁt of Ensemble Adversarial Training is in decoupling the attacks from the model being trained, which was the goal we stated in Section 3.4.

Ensemble Adversarial Training is not robust to white-box Iter-LL and R+Step-LL samples: the error rates are similar to those for the v3adv model, and omitted for brevity (see Kurakin et al. (2017b) for Iter-LL attacks and Table 2 for R+Step-LL attacks). Kurakin et al. (2017b) conjecture that larger models are needed to attain robustness to such attacks. Yet, against black-box adversaries, these attacks are only a concern insofar as they reliably transfer between models.

Black-box attacks. Ensemble Adversarial Training signiﬁcantly boosts robustness to all attacks transferred from the holdout models. For the IRv2adv-ens model, the accuracy loss (compared to IRv2 s accuracy on clean data) is 7.4% (top 1) and 3.1% (top 5). We ﬁnd that the strongest attacks in our test suite (i.e., with highest transfer rates) are the FGSM attacks. Black-box R+Step-LL or iterative attacks are less effective, as they do not transfer with high probability (see Kurakin et al. (2017b) and Table 9). Attacking an ensemble of all three holdout models, as in Liu et al. (2017), did not lead to stronger black-box attacks than when attacking the holdout models individually.

Our results have little variance with respect to the attack parameters (e.g., smaller ϵ) or to the use of other holdout models for black-box attacks (e.g., we obtain similar results by attacking the v3adv-ens3 and v3adv-ens4 models with the IRv2 model). We also ﬁnd that v3adv-ens3 is not vulnerable to perturbations transferred from v3adv-ens4. We obtain similar results on MNIST (see Appendix C.2), thus demonstrating the applicability of our approach to different datasets and model architectures.

The NIPS 2017 competition on adversarial examples. Our Inception Res Net v2 model was included as a baseline defense in the NIPS 2017 competition on Adversarial Examples (Kurakin et al., 2017c). Participants of the attack track submitted non-interactive black-box attacks that produce adversarial examples with bounded ℓ norm. Models submitted to the defense track were evaluated on all attacks over a subset of the Image Net test set. The score of a defense was deﬁned as the average accuracy of the model over all adversarial examples produced by all attacks.

Our IRv2adv-ens model ﬁnished 1st among 70 submissions in the ﬁrst development round, with a score of 95.3% (the second placed defense scored 89.9%). The test data was intentionally chosen as an easy subset of Image Net. Our model achieved 97.9% accuracy on the clean test data.

After the ﬁrst round, we released our model publicly, which enabled other users to launch white-box attacks against it. Nevertheless, a majority of the ﬁnal submissions built upon our released model. The winning submission (team liaofz with a score of 95.3%) made use of a novel adversarial

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Probability

v3 v3adv v3adv ens3

Figure 2: The dimensionality of the adversarial cone. For 500 correctly classiﬁed points x, and for ϵ {4, 10, 16}, we plot the probability that we ﬁnd at least k orthogonal vectors ri such that ri = ϵ and x + ri is misclassiﬁed. For ϵ 10, model v3adv shows a bimodal phenomenon: most points x either have 0 adversarial directions or more than 90.

denoising technique. The second placed defense (team cihangxie with a score of 92.4%) prepends our IRv2adv-ens model with random padding and resizing of the input image (Xie et al., 2018).

It is noteworthy that the defenses that incorporated Ensemble Adversarial Training faired better against the worst-case black-box adversary. Indeed, although very robust on average, the winning defense achieved as low as 11.8% accuracy on some attacks. The best defense under this metric (team rafaelmm which randomly perturbed images before feeding them to our IRv2adv-ens model) achieved at least 53.6% accuracy against all submitted attacks, including the attacks that explicitly targeted our released model in a white-box setting.

Decreasing gradient masking. Ensemble Adversarial Training decreases the magnitude of the gradient masking effect described previously. For the v3adv-ens3 and v3adv-ens4 models, we ﬁnd that the loss incurred on a Step-LL attack gets within respectively 13% and 18% of the optimum loss (we recall that for models v3 and v3adv, the approximation ratio was respectively 19% and 7%). Similarly, for the IRv2adv-ens model, the ratio improves from 8% (for IRv2adv) to 14%. As expected, not solely training on a white-box single-step attack reduces gradient masking. We also verify that after Ensemble Adversarial Training, a two-step iterative attack outperforms the R+Step-LL attack from Section 4.1, thus providing further evidence that these models have meaningful gradients.

Finally, we revisit the Gradient-Aligned Adversarial Subspace (GAAS) method of Tram er et al. (2017). Their method estimates the size of the space of adversarial examples in the vicinity of a point, by ﬁnding a set of orthogonal perturbations of norm ϵ that are all adversarial. We note that adversarial perturbations do not technically form a subspace (e.g., the 0 vector is not adversarial). Rather, they may form a cone , the dimension of which varies as we increase ϵ. By linearizing the loss function, estimating the dimensionality of this cone reduces to ﬁnding vectors ri that are strongly aligned with the model s gradient g = x L(h(x), ytrue). Tram er et al. (2017) give a method that ﬁnds k orthogonal vectors ri that satisfy g ri ϵ g 2 1

k (this bound is tight). We extend this result to the ℓ norm, an open question in Tram er et al. (2017). In Section E, we give a randomized combinatorial construction (Colbourn, 2010), that ﬁnds k orthogonal vectors ri satisfying ri = ϵ and E g ri ϵ g 1 1

k. We show that this result is tight as well.

For models v3, v3adv and v3adv-ens3, we select 500 correctly classiﬁed test points. For each x, we search for a maximal number of orthogonal adversarial perturbations ri with ri = ϵ. We limit our search to k 100 directions per point. The results are in Figure 2. For ϵ {4, 10, 16}, we plot the proportion of points that have at least k orthogonal adversarial perturbations. For a ﬁxed ϵ, the value of k can be interpreted as the dimension of a slice of the cone of adversarial examples near a data point. For the standard Inception v3 model, we ﬁnd over 50 orthogonal adversarial directions for 30% of the points. The v3adv model shows a curious bimodal phenomenon for ϵ 10: for most points ( 80%), we ﬁnd no adversarial direction aligned with the gradient, which is consistent with the gradient masking effect. Yet, for most of the remaining points, the adversarial space is very high-dimensional (k 90). Ensemble Adversarial Training yields a more robust model, with only a small fraction of points near a large adversarial space.

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5 CONCLUSION AND FUTURE WORK

Previous work on adversarial training at scale has produced encouraging results, showing strong robustness to (single-step) adversarial examples (Goodfellow et al., 2014b; Kurakin et al., 2017b). Yet, these results are misleading, as the adversarially trained models remain vulnerable to simple black-box and white-box attacks. Our results, generic with respect to the application domain, suggest that adversarial training can be improved by decoupling the generation of adversarial examples from the model being trained. Our experiments with Ensemble Adversarial Training show that the robustness attained to attacks from some models transfers to attacks from other models.

We did not consider black-box adversaries that attack a model via other means than by transferring examples from a local model. For instance, generative techniques (Baluja & Fischer, 2017) might provide an avenue for stronger attacks. Yet, a recent work by Xiao et al. (2018) found Ensemble Adversarial Training to be resilient to such attacks on MNIST and CIFAR10, and often attaining higher robustness than models that were adversarially trained on iterative attacks.

Moreover, interactive adversaries (see Appendix A) could try to exploit queries to the target model s prediction function in their attack, as demonstrated in Papernot et al. (2017). If queries to the target model yield prediction conﬁdences, an adversary can estimate the target s gradient at a given point (e.g., using ﬁnite-differences as in Chen et al. (2017)) and fool the target with our R+FGSM attack. Note that if queries only return the predicted label, the attack does not apply. Exploring the impact of these classes of black-box attacks and evaluating their scalability to complex tasks is an interesting avenue for future work.

ACKNOWLEDGMENTS

We thank Ben Poole and Jacob Steinhardt for feedback on early versions of this work. Nicolas Papernot is supported by a Google Ph D Fellowship in Security. Research was supported in part by the Army Research Laboratory, under Cooperative Agreement Number W911NF-13-2-0045 (ARL Cyber Security CRA), and the Army Research Ofﬁce under grant W911NF-13-1-0421. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the ofﬁcial policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for government purposes notwithstanding any copyright notation hereon.

Mart ın Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro, Greg S. Corrado, Andy Davis, Jeffrey Dean, Matthieu Devin, Sanjay Ghemawat, Ian Goodfellow, Andrew Harp, Geoffrey Irving, Michael Isard, Yangqing Jia, Rafal Jozefowicz, Lukasz Kaiser, Manjunath Kudlur, Josh Levenberg, Dan Man e, Rajat Monga, Sherry Moore, Derek Murray, Chris Olah, Mike Schuster, Jonathon Shlens, Benoit Steiner, Ilya Sutskever, Kunal Talwar, Paul Tucker, Vincent Vanhoucke, Vijay Vasudevan, Fernanda Vi egas, Oriol Vinyals, Pete Warden, Martin Wattenberg, Martin Wicke, Yuan Yu, and Xiaoqiang Zheng. Tensor Flow: Large-scale machine learning on heterogeneous systems, 2015. URL https://www.tensorflow.org/. Software available from tensorﬂow.org.

Dario Amodei, Chris Olah, Jacob Steinhardt, Paul Christiano, John Schulman, and Dan Man e. Concrete problems in ai safety. ar Xiv preprint ar Xiv:1606.06565, 2016.

Anish Athalye, Nicholas Carlini, and David Wagner. Obfuscated gradients give a false sense of security: Circumventing defenses to adversarial examples. ar Xiv preprint ar Xiv:1802.00420, 2018.

Shumeet Baluja and Ian Fischer. Adversarial transformation networks: Learning to generate adversarial examples. ar Xiv preprint ar Xiv:1703.09387, 2017.

Battista Biggio, Igino Corona, Davide Maiorca, Blaine Nelson, Nedim ˇSrndi c, Pavel Laskov, Giorgio Giacinto, and Fabio Roli. Evasion attacks against machine learning at test time. In ECMLKDD, pp. 387 402. Springer, 2013.

Published as a conference paper at ICLR 2018

Wieland Brendel and Matthias Bethge. Comment on biologically inspired protection of deep networks from adversarial attacks . ar Xiv preprint ar Xiv:1704.01547, 2017.

Jacob Buckman, Aurko Roy, Colin Raffel, and Ian Goodfellow. Thermometer encoding: One hot way to resist adversarial examples. In International Conference on Learning Representations, 2018. URL https://openreview.net/forum?id=S18Su--CW.

Nicholas Carlini and David Wagner. Towards evaluating the robustness of neural networks. In IEEE Symposium on Security and Privacy, 2017a.

Nicholas Carlini and David Wagner. Adversarial examples are not easily detected: Bypassing ten detection methods. ar Xiv preprint ar Xiv:1705.07263, 2017b.

Pin-Yu Chen, Huan Zhang, Yash Sharma, Jinfeng Yi, and Cho-Jui Hsieh. Zoo: Zeroth order optimization based black-box attacks to deep neural networks without training substitute models. In Proceedings of the 10th ACM Workshop on Artiﬁcial Intelligence and Security, pp. 15 26. ACM, 2017.

Moustapha Cisse, Bojanowski Piotr, Grave Edouard, Dauphin Yann, and Usunier Nicolas. Parseval networks: Improving robustness to adversarial examples. ar Xiv preprint ar Xiv:1704.08847, 2017.

Charles J Colbourn. CRC handbook of combinatorial designs. CRC press, 2010.

J. Deng, W. Dong, R. Socher, L.-J. Li, K. Li, and L. Fei-Fei. Image Net: A Large-Scale Hierarchical Image Database. In CVPR09, 2009.

Logan Engstrom, Dimitris Tsipras, Ludwig Schmidt, and Aleksander Madry. A rotation and a translation sufﬁce: Fooling cnns with simple transformations. ar Xiv preprint ar Xiv:1712.02779, 2017.

Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In Advances in neural information processing systems, pp. 2672 2680, 2014a.

Ian J Goodfellow, Jonathon Shlens, and Christian Szegedy. Explaining and harnessing adversarial examples. ar Xiv preprint ar Xiv:1412.6572, 2014b.

Shixiang Gu and Luca Rigazio. Towards deep neural network architectures robust to adversarial examples. ar Xiv preprint ar Xiv:1412.5068, 2014.

Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 770 778, 2016.

J Zico Kolter and Eric Wong. Provable defenses against adversarial examples via the convex outer adversarial polytope. ar Xiv preprint ar Xiv:1711.00851, 2017.

Alexey Kurakin, Ian Goodfellow, and Samy Bengio. Adversarial examples in the physical world. In ICLR, 2017a.

Alexey Kurakin, Ian Goodfellow, and Samy Bengio. Adversarial machine learning at scale. In ICLR, 2017b.

Alexey Kurakin, Ian J Goodfellow, and Samy Bengio. Nips 2017: Defense against adversarial attack, 2017c. URL https://www.kaggle.com/c/ nips-2017-defense-against-adversarial-attack.

Yann Le Cun, L eon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278 2324, 1998.

Yanpei Liu, Xinyun Chen, Chang Liu, and Dawn Song. Delving into transferable adversarial examples and black-box attacks. In ICLR, 2017.

Yan Luo, Xavier Boix, Gemma Roig, Tomaso Poggio, and Qi Zhao. Foveation-based mechanisms alleviate adversarial examples. ar Xiv preprint ar Xiv:1511.06292, 2015.

Published as a conference paper at ICLR 2018

Aleksander Madry, Aleksandar Makelov, Ludwig Schmidt, Dimitris Tsipras, and Adrian Vladu. Towards deep learning models resistant to adversarial attacks. ar Xiv preprint ar Xiv:1706.06083, 2017.

Yishay Mansour, Mehryar Mohri, and Afshin Rostamizadeh. Domain adaptation: Learning bounds and algorithms. ar Xiv preprint ar Xiv:0902.3430, 2009.

Dmytro Mishkin, Nikolay Sergievskiy, and Jiri Matas. Systematic evaluation of convolution neural network advances on the imagenet. Computer Vision and Image Understanding, 2017.

Aran Nayebi and Surya Ganguli. Biologically inspired protection of deep networks from adversarial attacks. ar Xiv preprint ar Xiv:1703.09202, 2017.

Nicolas Papernot, Patrick Mc Daniel, Somesh Jha, Matt Fredrikson, Z Berkay Celik, and Ananthram Swami. The limitations of deep learning in adversarial settings. In Security and Privacy (Euro S&P), 2016 IEEE European Symposium on, pp. 372 387. IEEE, 2016a.

Nicolas Papernot, Patrick Mc Daniel, Arunesh Sinha, and Michael Wellman. Towards the science of security and privacy in machine learning. ar Xiv preprint ar Xiv:1611.03814, 2016b.

Nicolas Papernot, Patrick Mc Daniel, Xi Wu, Somesh Jha, and Ananthram Swami. Distillation as a defense to adversarial perturbations against deep neural networks. In Security and Privacy (SP), 2016 IEEE Symposium on, pp. 582 597. IEEE, 2016c.

Nicolas Papernot, Patrick Mc Daniel, Ian Goodfellow, Somesh Jha, Z Berkay Celik, and Ananthram Swami. Practical black-box attacks against machine learning. In Asia Conference on Computer and Communications Security (ASIACCS), pp. 506 519. ACM, 2017.

Aditi Raghunathan, Jacob Steinhardt, and Percy Liang. Certiﬁed defenses against adversarial examples. In International Conference on Learning Representations, 2018. URL https: //openreview.net/forum?id=Bys4ob-Rb.

Aman Sinha, Hongseok Namkoong, and John Duchi. Certiﬁable distributional robustness with principled adversarial training. In International Conference on Learning Representations, 2018. URL https://openreview.net/forum?id=Hk6k Pg ZA-.

Nitish Srivastava, Geoffrey E Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov. Dropout: a simple way to prevent neural networks from overﬁtting. Journal of machine learning research, 15(1):1929 1958, 2014.

Christian Szegedy, Wojciech Zaremba, Ilya Sutskever, Joan Bruna, Dumitru Erhan, Ian Goodfellow, and Rob Fergus. Intriguing properties of neural networks. ar Xiv preprint ar Xiv:1312.6199, 2013.

Christian Szegedy, Sergey Ioffe, Vincent Vanhoucke, and Alex Alemi. Inception-v4, inceptionresnet and the impact of residual connections on learning. ar Xiv preprint ar Xiv:1602.07261, 2016a.

Christian Szegedy, Vincent Vanhoucke, Sergey Ioffe, Jon Shlens, and Zbigniew Wojna. Rethinking the inception architecture for computer vision. In CVPR, pp. 2818 2826, 2016b.

Florian Tram er, Fan Zhang, Ari Juels, Michael K Reiter, and Thomas Ristenpart. Stealing machine learning models via prediction apis. In Usenix Security, 2016.

Florian Tram er, Nicolas Papernot, Ian Goodfellow, Dan Boneh, and Patrick Mc Daniel. The space of transferable adversarial examples. ar Xiv preprint ar Xiv:1704.03453, 2017.

Chaowei Xiao, Bo Li, Jun-Yan Zhu, Warren He, Mingyan Liu, and Dawn Song. Generating adversarial examples with adversarial networks, 2018. URL https://openreview.net/ forum?id=Hknby Qb C-.

Cihang Xie, Jianyu Wang, Zhishuai Zhang, Zhou Ren, and Alan Yuille. Mitigating adversarial effects through randomization. In International Conference on Learning Representations, 2018. URL https://openreview.net/forum?id=Sk9yuql0Z.

Chao Zhang, Lei Zhang, and Jieping Ye. Generalization bounds for domain adaptation. In Advances in neural information processing systems, pp. 3320 3328, 2012.

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A THREAT MODEL: FORMAL DEFINITIONS

We provide formal deﬁnitions for the threat model introduced in Section 3.1. In the following, we explicitly identify the hypothesis space H that a model belongs to as describing the model s architecture. We consider a target model h H trained over inputs (x, ytrue) sampled from a data distribution D. More precisely, we write h train(H, Xtrain, Ytrain, r) , where train is a randomized training procedure that takes in a description of the model architecture H, a training set Xtrain, Ytrain sampled from D, and randomness r.

Given a set of test inputs X, Y = {(x1, y1), . . . , (xm, ym)} from D and a budget ϵ > 0, an adversary A produces adversarial examples Xadv = {xadv 1 , . . . , xadv m }, such that xi xadv i ϵ for all i [1, m]. We evaluate success of the attack as the error rate of the target model over Xadv:

i=1 1(arg max h(xadv i ) = yi) .

We assume A can sample inputs according to the data distribution D. We deﬁne three adversaries. Deﬁnition 2 (White-Box Adversary). For a target model h H, a white-box adversary is given access to all elements of the training procedure, that is train (the training algorithm), H (the model architecture), the training data Xtrain, Ytrain, the randomness r and the parameters h. The adversary can use any attack (e.g., those in Section 3.2) to ﬁnd adversarial inputs.

White-box access to the internal model weights corresponds to a very strong adversarial model. We thus also consider the following relaxed and arguably more realistic notion of a black-box adversary. Deﬁnition 3 (Non-Interactive Black-Box Adversary). For a target model h H, a non-interactive black-box adversary only gets access to train (the target model s training procedure) and H (the model architecture). The adversary can sample from the data distribution D, and uses a local algorithm to craft adversarial examples Xadv.

Attacks based on transferability (Szegedy et al., 2013) fall in this category, wherein the adversary selects a procedure train and model architecture H , trains a local model h over D, and computes adversarial examples on its local model h using white-box attack strategies.

Most importantly, a black-box adversary does not learn the randomness r used to train the target, nor the target s parameters h. The black-box adversaries in our paper are actually slightly stronger than the ones deﬁned above, in that they use the same training data Xtrain, Ytrain as the target model.

We provide A with the target s training procedure train to capture knowledge of defensive strategies applied at training time, e.g., adversarial training (Szegedy et al., 2013; Goodfellow et al., 2014b) or ensemble adversarial training (see Section 4.2). For ensemble adversarial training, A also knows the architectures of all pre-trained models. In this work, we always mount black-box attacks that train a local model with a different architecture than the target model. We actually ﬁnd that black-box attacks on adversarially trained models are stronger in this case (see Table 1).

The main focus of our paper is on non-interactive black-box adversaries as deﬁned above. For completeness, we also formalize a stronger notion of interactive black-box adversaries that additionally issue prediction queries to the target model (Papernot et al., 2017). We note that in cases where ML models are deployed as part of a larger system (e.g., a self driving car), an adversary may not have direct access to the model s query interface. Deﬁnition 4 (Interactive Black-Box Adversary). For a target model h H, an interactive blackbox adversary only gets access to train (the target model s training procedure) and H (the model architecture). The adversary issues (adaptive) oracle queries to the target model. That is, for arbitrary inputs x [0, 1]d, the adversary obtains y = arg max h(x) and uses a local algorithm to craft adversarial examples (given knowledge of H, train, and tuples (x, y)).

Papernot et al. (2017) show that such attacks are possible even if the adversary only gets access to a small number of samples from D. Note that if the target model s prediction interface additionally returns class scores h(x), interactive black-box adversaries could use queries to the target model to estimate the model s gradient (e.g., using ﬁnite differences) (Chen et al., 2017), and then apply the attacks in Section 3.2. We further discuss interactive black-box attack strategies in Section 5.

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B GENERALIZATION BOUND FOR ENSEMBLE ADVERSARIAL TRAINING

We provide a formal statement of Theorem 1 in Section 3.4, regarding the generalization guarantees of Ensemble Adversarial Training. For simplicity, we assume that the model is trained solely on adversarial examples computed on the pre-trained models (i.e., we ignore the clean training data and the adversarial examples computed on the model being trained). Our results are easily extended to also consider these data points.

Let D be the data distribution and A1, . . . , Ak, A be adversarial distributions where a sample (x, y) is obtained by sampling (x, ytrue) from D, computing an xadv such that xadv x ϵ and returning (xadv, ytrue). We assume the model is trained on N data points Ztrain, where N

k data points are sampled from each distribution Ai, for 1 i k. We denote Atrain = {A1, . . . , Ak}. At test time, the model is evaluated on adversarial examples from A .

For a model h H we deﬁne the empirical risk

ˆR(h, Atrain) := 1

(xadv,ytrue) Ztrain

L(h(xadv), ytrue) , (8)

and the risk over the target distribution (or future adversary)

R(h, A ) := E (xadv,ytrue) A [L(h(xadv), ytrue)] . (9)

We further deﬁne the average discrepancy distance (Mansour et al., 2009) between distributions Ai and A with respect to a hypothesis space H as

disc H(Atrain, A ) := 1

i=1 sup h1,h2 H

E Ai [1{h1(xadv)=h2(xadv)}] E A [1{h1(xadv)=h2(xadv)}] . (10)

This quantity characterizes how different the future adversary is from the train-time adversaries. Intuitively, the distance disc(Atrain, A ) is small if the difference in robustness between two models to the target attack A is somewhat similar to the difference in robustness between these two models to the attacks used for training (e.g., if the static black-box attacks Ai induce much higher error on some model h1 than on another model h2, then the same should hold for the target attack A ). In other words, the ranking of the robustness of models h H should be similar for the attacks in Atrain as for A .

Finally, let RN(H) be the average Rademacher complexity of the distributions A1, . . . , Ak (Zhang et al., 2012). Note that RN(H) 0 as N . The following theorem is a corollary of Zhang et al. (2012, Theorem 5.2): Theorem 5. Assume that H is a function class consisting of bounded functions. Then, with probability at least 1 ϵ,

sup h H | ˆR(h, Atrain) R(h, A )| disc H(Atrain, A ) + 2RN(H) + O

Compared to the standard generalization bound for supervised learning, the generalization bound for Domain Adaptation incorporates the extra term disc H(Atrain, A ) to capture the divergence between the target and source distributions. In our context, this means that the model h learned by Ensemble Adversarial Training has guaranteed generalization bounds with respect to future adversaries that are not too different from the ones used during training. Note that A need not restrict itself to perturbation with bounded ℓ norm for this result to hold.

C EXPERIMENTS ON MNIST

We re-iterate our Image Net experiments on MNIST. For this simpler task, Madry et al. (2017) show that training on iterative attacks conveys robustness to white-box attacks with bounded ℓ norm. Our goal is not to attain similarly strong white-box robustness on MNIST, but to show that our observations on limitations of single-step adversarial training, extend to other datasets than Image Net.

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Figure 3: Adversarial Examples on MNIST. (top) clean examples. (middle) inputs are rotated by 20 and 5 random pixels are ﬂipped. (bottom) The I-FGSM with ϵ = 0.3 is applied.

C.1 A NOTE ON ℓ ROBUSTNESS ON MNIST

The MNIST dataset is a simple baseline for assessing the potential of a defense, but the obtained results do not always generalize to harder tasks. We suggest that this is because achieving robustness to ℓ perturbations admits a simple closed-form solution, given the near-binary nature of the data. Indeed, for an average MNIST image, over 80% of the pixels are in {0, 1} and only 6% are in the range [0.2, 0.8]. Thus, for a perturbation with ϵ 0.3, binarized versions of x and xadv can differ in at most 6% of the input dimensions. By binarizing the inputs of a standard CNN trained without adversarial training, we obtain a model that enjoys robustness similar to the model trained by Madry et al. (2017). Concretely, for a white-box I-FGSM attack, we get at most 11.4% error.

The existence of such a simple robust representation begs the question of why learning a robust model with adversarial training takes so much effort. Finding techniques to improve the performance of adversarial training, even on simple tasks, could provide useful insights for more complex tasks such as Image Net, where we do not know of a similarly simple denoising procedure.

These positive results on MNIST for the ℓ norm also leave open the question of deﬁning a general norm for adversarial examples. Let us motivate the need for such a deﬁnition: we ﬁnd that if we ﬁrst rotate an MNIST digit by 20 , and then use the I-FGSM, our rounding model and the model from Madry et al. (2017) achieve only 65% accuracy (on clean rotated inputs, the error is < 5%). If we further randomly ﬂip 5 pixels per image, the accuracy of both models drops to under 50%. Thus, we successfully evade the model by slightly extending the threat model (see Figure 3).

Of course, we could augment the training set with such perturbations (see Engstrom et al. (2017)). An open question is whether we can enumerate all types of adversarial perturbations. In this work, we focus on the ℓ norm to illustrate our ﬁndings on the limitations of single-step adversarial training on Image Net and MNIST, and to showcase the beneﬁts of our Ensemble Adversarial Training variant. Our approach can easily be extended to consider multiple perturbation metrics. We leave such an evaluation to future work.

C.2 RESULTS

We repeat experiments from Section 4 on MNIST. We use the architectures in Table 5. We train a standard model for 6 epochs, and an adversarial model with the FGSM (ϵ = 0.3) for 12 epochs.

During adversarial training, we avoid the label leaking effect described by Kurakin et al. (2017b) by using the model s predicted class arg max h(x) instead of the true label ytrue in the FGSM,

We ﬁrst analyze the degenerate minimum of adversarial training, described in Section 3.3. For each trained model, we compute the approximation-ratio of the FGSM for the inner maximization problem in equation (1). That is, we compare the loss produced by the FGSM with the loss of a

Table 5: Neural network architectures used in this work for the MNIST dataset. Conv: convolutional layer, FC: fully connected layer.

Conv(64, 5, 5) + Relu Dropout(0.2) Conv(128, 3, 3) + Tanh h FC(300) + Relu Dropout(0.5)

i 4 Conv(64, 5, 5) + Relu Conv(64, 8, 8) + Relu Max Pool(2,2) Dropout(0.25) Conv(128, 6, 6) + Relu Conv(64, 3, 3) + Tanh FC + Softmax FC(128) + Relu Conv(128, 5, 5) + Relu Max Pool(2,2) Dropout(0.5) Dropout(0.5) FC(128) + Relu FC + Softmax FC + Softmax FC + Softmax

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Table 6: Approximation ratio between optimal loss and loss induced by single-step attack on MNIST. Architecture B is the same as B without the input dropout layer.

A Aadv B Badv B B adv C Cadv D Dadv 17% 0% 25% 8% 23% 1% 25% 0% 49% 16%

strong iterative attack. The results appear in Table 6. As we can see, for all model architectures, adversarial training degraded the quality of a linear approximation to the model s loss.

We ﬁnd that input dropout (Srivastava et al., 2014) (i.e., randomly dropping a fraction of input features during training) as used in architecture B limits this unwarranted effect of adversarial training.2

If we omit the input dropout (we call this architecture B ) the single-step attack degrades signiﬁcantly. We discuss this effect in more detail below. For the fully connected architecture D, we ﬁnd that the learned model is very close to linear and thus also less prone to the degenerate solution to the min-max problem, as we postulated in Section 3.3.

Attacks. Table 7 compares error rates of undefended and adversarially trained models on whitebox and black-box attacks, as in Section 4.1. Again, model B presents an anomaly. For all other models, we corroborate our ﬁndings on Image Net for adversarial training: (1) black-box attacks trump white-box single-step attacks; (2) white-box single-step attacks are signiﬁcantly stronger if prepended by a random step. For model Badv, the opposite holds true. We believe this is because input dropout increases diversity of attack samples similarly to Ensemble Adversarial Training.

Table 7: White-box and black-box attacks against standard and adversarially trained models. For each model, the strongest single-step white-box and black box attacks are marked in bold.

white-box black-box

FGSM R+FGSM FGSMA FGSMB FGSMB* FGSMC FGSMD A 64.7 69.7 - 61.5 53.2 46.8 41.5 Aadv 2.2 14.8 6.6 10.7 8.8 6.5 8.3

B 85.0 86.0 45.7 - 69.9 59.9 85.9 Badv 11.6 11.1 6.4 8.9 8.5 4.9 6.1

B 75.7 74.1 44.3 72.8 - 46.0 62.6 B adv 4.3 40.6 16.1 14.7 15.0 17.9 9.1

C 81.8 81.8 40.2 55.8 49.5 - 59.4 Cadv 3.7 17.1 9.8 29.3 21.5 11.9 21.9

D 92.4 95.4 61.3 74.1 68.9 65.1 - Dadv 25.5 47.5 32.1 30.5 29.3 28.2 21.8

While training with input dropout helps avoid the degradation of the single-step attack, it also signiﬁcantly delays convergence of the model. Indeed, model Badv retains relatively high error on white-box FGSM examples. Adversarial training with input dropout can be seen as comparable to training with a randomized single-step attack, as discussed in Section 4.1.

The positive effect of input dropout is architecture and dataset speciﬁc: Adding an input dropout layer to models A, C and D confers only marginal beneﬁt, and is outperformed by Ensemble Adversarial Training, discussed below. Moreover, Mishkin et al. (2017) ﬁnd that input dropout signiﬁcantly degrades accuracy on Image Net. We thus did not incorporate it into our models on Image Net.

Ensemble Adversarial Training. To evaluate Ensemble Adversarial Training 3.4, we train two models per architecture. The ﬁrst, denoted [A-D]adv-ens, uses a single pre-trained model of the same type (i.e., Aadv-ens is trained on perturbations from another model A). The second model, denoted [AD]adv-ens3, uses 3 pre-trained models ({A, C, D} or {B, C, D}). We train all models for 12 epochs.

We evaluate our models on black-box attacks crafted on models A,B,C,D (for a fair comparison, we do not use the same pre-trained models for evaluation, but retrain them with different random seeds).

2We thank Arjun Bhagoji, Bo Li and Dawn Song for this observation.

Published as a conference paper at ICLR 2018

Table 8: Ensemble Adversarial Training on MNIST. For black-box robustness, we report the maximum and average error rate over a suite of 12 attacks, comprised of the FGSM, I-FGSM and PGD (Madry et al., 2017) attacks applied to models A,B,C and D. We use ϵ = 16 in all cases. For each model architecture, we mark the models tied for best (at a 95% conﬁdence level) in bold.

Clean FGSM Max. Black Box Avg. Black Box Aadv 0.8 2.2 10.8 7.7 Aadv-ens 0.8 7.0 6.6 5.2 Aadv-ens3 0.7 5.4 6.5 4.3

Badv 0.8 11.6 8.9 5.5 Badv-ens 0.7 10.5 6.8 5.3 Badv-ens3 0.8 14.0 8.8 5.1

Cadv 1.0 3.7 29.3 18.7 Cadv-ens 1.3 1.9 17.2 10.7 Cadv-ens3 1.4 3.6 14.5 8.4

Dadv 2.6 25.5 32.5 23.5 Dadv-ens 2.6 21.5 38.6 28.0 Dadv-ens3 2.6 29.4 29.8 15.6

The attacks we consider are the FGSM, I-FGSM and the PGD attack from Madry et al. (2017) with the loss function from Carlini & Wagner (2017a)), all with ϵ = 0.3. The results appear in Table 8. For each model, we report the worst-case and average-case error rate over all black-box attacks.

Ensemble Adversarial Training signiﬁcantly increases robustness to black-box attacks, except for architecture B, which we previously found to not suffer from the same overﬁtting phenomenon that affects the other adversarially trained networks. Nevertheless, model Badv-ens achieves slightly better robustness to white-box and black-box attacks than Badv. In the majority of cases, we ﬁnd that using a single pre-trained model produces good results, but that the extra diversity of including three pre-trained models can sometimes increase robustness even further. Our experiments conﬁrm our conjecture that robustness to black-box attacks generalizes across models. Indeed, we ﬁnd that when training with three external models, we attain very good robustness against attacks initiated from models with the same architecture (as evidenced by the average error on our attack suite), but also increased robustness to attacks initiated from the fourth holdout model

D TRANSFERABILITY OF RANDOMIZED SINGLE-STEP PERTURBATIONS.

In Section 4.1, we introduced the R+Step-LL attack, an extension of the Step-LL method that prepends the attack with a small random perturbation. In Table 9, we evaluate the transferability of R+Step-LL adversarial examples on Image Net. We ﬁnd that the randomized variant produces perturbations that transfer at a much lower rate (see Table 1 for the deterministic variant).

Table 9: Error rates (in %) of randomized single-step attacks transferred between models on Image Net. We use R+Step-LL with ϵ = 16/256, α = ϵ/2 for 10,000 random test set samples. The white-box attack always outperforms black-box attacks.

Source Target v4 v3 v3adv IRv2 IRv2adv v4 70.5 37.2 23.2 34.0 24.6 v3 42.6 80.0 26.7 38.5 27.6 v3adv 31.4 30.7 64.8 30.4 34.0 IRv2 36.2 35.7 23.0 56.3 24.6 IRv2adv 26.8 26.3 25.2 26.9 37.5

Source Target v4 v3 v3adv IRv2 IRv2adv v4 42.8 14.3 6.3 11.9 6.9 v3 18.0 57.1 8.0 15.6 8.6 v3adv 10.7 10.4 37.1 10.1 12.9 IRv2 12.8 13.6 6.1 29.3 7.0 IRv2adv 8.0 8.0 7.7 8.3 15.0

Published as a conference paper at ICLR 2018

E GRADIENT ALIGNED ADVERSARIAL SUBSPACES FOR THE ℓ NORM

Tram er et al. (2017) consider the following task for a given model h: for a (correctly classiﬁed) point x, ﬁnd k orthogonal vectors {r1, . . . , rk} such that ri 2 ϵ and all the x + ri are adversarial (i.e., arg max h(x + ri) = ytrue). By linearizing the model s loss function, this reduces to ﬁnding k orthogonal vectors ri that are maximally aligned with the model s gradient g = x L(h(x), ytrue). Tram er et al. (2017) left a construction for the ℓ norm as an open problem.

We provide an optimal construction for the ℓ norm, based on Regular Hadamard Matrices (Colbourn, 2010). Given the ℓ constraint, we ﬁnd orthogonal vectors ri that are maximally aligned with the signed gradient, sign(g). We ﬁrst prove an analog of (Tram er et al., 2017, Lemma 1).

Lemma 6. Let v { 1, 1}d and α (0, 1). Suppose there are k orthogonal vectors r1, . . . rn { 1, 1}d satisfying v ri α d. Then α k 1

Proof. Let ˆri = ri ri 2 = ri

d. Then, we have

i=1 |v ˆri|2 = d 1 k X

i=1 |v ri|2 d 1 k (α d)2 = k α2 d , (12)

from which we obtain α k 1

This result bounds the number of orthogonal perturbations we can expect to ﬁnd, for a given alignment with the signed gradient. As a warm-up consider the following trivial construction of k orthogonal vectors in { 1, 1}d that are somewhat aligned with sign(g). We split sign(g) into k chunks of size d

k and deﬁne ri to be the vector that is equal to sign(g) in the ith chunk and zero otherwise. We obtain sign(g) ri = d

k, a factor

k worse than the the bound in Lemma 6.

We now provide a construction that meets this upper bound. We make use of Regular Hadamard Matrices of order k (Colbourn, 2010). These are square matrices Hk such that: (1) all entries of Hk are in { 1, 1}k; (2) the rows of Hk are mutually orthogonal; (3) All row sums are equal to

The order of a Regular Hadamard Matrix is of the form 4u2 for an integer u. We use known constructions for k {4, 16, 36, 64, 100}.

Lemma 7. Let g Rd and k be an integer for which a Regular Hadamard Matrix of order k exists. Then, there is a randomized construction of k orthogonal vectors r1, . . . rn { 1, 1}d, such that sign(g) ri = d k 1/2. Moreover, E[g ri] = k 1/2 g 1.

Proof. We construct k orthogonal vectors r1, . . . , rk { 1, 1}d, where ri is obtained by repeating the ith row of Hk d/k times (for simplicity, we assume that k divides d. Otherwise we pad ri with zeros). We then multiply each ri component-wise with sign(g). By construction, the k vectors ri { 1, 1}d are mutually orthogonal, and we have sign(g) ri = d

k = d k 1/2, which is tight according to Lemma 6.

As the weight of the gradient g may not be uniformly distributed among its d components, we apply our construction to a random permutation of the signed gradient. We then obtain

E[g ri] = E h d X

j=1 |g(j)| sign(g(j)) r(j) i i (13)

j=1 |g(j)| E h sign(g(j)) r(j) i i = k 1/2 g 1 . (14)

It can be shown that the bound in Lemma 7 can be attained if and only if the ri are constructed from the rows of a Regular Hadamard Matrix (Colbourn, 2010). For general integers k for which no such matrix exists, other combinatorial designs may be useful for achieving looser bounds.

Published as a conference paper at ICLR 2018

F ILLUSTRATIONS OF GRADIENT MASKING IN ADVERSARIAL TRAINING

In Section 3.3, we show that adversarial training introduces spurious curvature artifacts in the model s loss function around data points. As a result, one-shot attack strategies based on ﬁrst-order approximations of the model loss produce perturbations that are non-adversarial. In Figures 4 and 5 we show further illustrations of this phenomenon for the Inception v3adv model trained on Image Net by Kurakin et al. (2017b) as well as for the model Aadv we trained on MNIST.

(a) Loss surface for model v3adv.

(b) Zoom in of the loss for small ϵ1, ϵ2.

Figure 4: Additional illustrations of the local curvature artifacts introduced by adversarial training on Image Net. We plot the loss of model v3adv on samples of the form x = x+ϵ1 g +ϵ2 g , where g is the signed gradient of v3adv and g is an orthogonal adversarial direction, obtained from an Inception v4 model. The right-side plots are zoomed in versions of the left-side plots.

Published as a conference paper at ICLR 2018

(a) Loss surface for model Aadv (log-scale).

(b) Zoom in of the loss for small ϵ1, ϵ2.

Figure 5: Illustrations of the local curvature artifacts introduced by adversarial training on MNIST. We plot the loss of model Aadv on samples of the form x = x + ϵ1 g + ϵ2 g , where g is the signed gradient of model Aadv and g is an orthogonal adversarial direction, obtained from model B. The right-side plots are zoomed in versions of the left-side plots.