R OBUSTNESS M AY BE AT O DDS WITH A CCURACY
Anonymous authors
Paper under double-blind review
A BSTRACT
We show that there exists an inherent tension between the goal of adversarial robustness and that of standard generalization.
Speciﬁcally, training robust models may not only be more resource-consuming, but also lead to a reduction of standard accuracy.
We demonstrate that this trade-off between the standard accuracy of a model and its robustness to adversarial perturbations provably exists even in a fairly simple and natural setting.
These ﬁndings also corroborate a similar phenomenon observed in practice.
Further, we argue that this phenomenon is a consequence of robust classiﬁers learning fundamentally different feature representations than standard classiﬁers.
These differences, in particular, seem to result in unexpected beneﬁts: the representations learned by robust models tend to align better with salient data characteristics and human perception.
I NTRODUCTION
Deep learning models have achieved impressive performance on a number of challenging benchmarks in computer vision, speech recognition and competitive game playing (Krizhevsky et al., 2012; Graves et al., 2013; Mnih et al., 2015; Silver et al., 2016;
He et al., 2015a).
However, it turns out that these models are actually quite brittle.
In particular, one can often synthesize small, imperceptible perturbations of the input data and cause the model to make highly-conﬁdent but erroneous predictions (Dalvi et al., 2004; Biggio & Roli, 2017; Szegedy et al., 2013).
This problem of so-called adversarial examples has garnered signiﬁcant attention recently and resulted in a number of approaches both to ﬁnding these perturbations, and to training models that are robust to them (Goodfellow et al., 2014b; Nguyen et al., 2015; Moosavi-Dezfooli et al., 2016; Carlini & Wagner, 2016; Sharif et al., 2016; Kurakin et al., 2016a; Evtimov et al., 2017; Athalye et al., 2017).
However, building such adversarially robust models has proved to be quite challenging.
In particular, many of the proposed robust training methods were subsequently shown to be ineffective (Carlini & Wagner, 2017; Athalye et al., 2018;
Uesato et al., 2018).
Only recently, has there been progress towards models that achieve robustness that can be demonstrated empirically and, in some cases, even formally veriﬁed (Madry et al., 2017; Kolter & Wong, 2017; Sinha et al., 2017; Tjeng & Tedrake, 2017; Raghunathan et al., 2018; Dvijotham et al., 2018a; Xiao et al., 2018b).
The vulnerability of models trained using standard methods to adversarial perturbations makes it clear that the paradigm of adversarially robust learning is different from the classic learning setting.
In particular, we already know that robustness comes at a cost.
This cost takes the form of computationally expensive training methods (more training time), but also, as shown recently in
Schmidt et al.
(2018), the potential need for more training data.
It is natural then to wonder:
Are these the only costs of adversarial robustness?
And, if so, once we choose to pay these costs, would it always be preferable to have a robust model instead of a standard one?
The goal of this work is to explore these questions and thus, in turn, to bring us closer to understanding the phenomenon of adversarial robustness.
Our contributions It might be natural to expect that training models to be adversarially robust, albeit more resource-consuming, can only improve performance in the standard classiﬁcation setting.
In this work, we show, however, that the picture here is much more nuanced: these two goals might be fundamentally at odds.
Speciﬁcally, even though applying adversarial training, the leading method for training robust models, can be beneﬁcial in some regimes of training data size, in general, there is a trade-off between the standard accuracy and adversarially robust accuracy of a model.
In fact, we show that this trade-off provably exists even in a fairly simple and natural setting.
At the root of this trade-off is the fact that representations learned by the optimal standard and optimal robust classiﬁers are fundamentally different and, interestingly, this phenomenon persists even in the limit of inﬁnite data.
This thus also goes against the natural expectation that given sufﬁcient data, classic machine learning tools would be sufﬁcient to learn robust models and emphasizes the need for techniques speciﬁcally tailored to training robust models.
Our exploration also uncovers certain unexpected beneﬁt of adversarially robust models.
In particular, adversarially robust learning tends to equip the resulting models with invariances that we would expect to be also present in human vision.
This, in turn, leads to feature representations that align better with human perception, and could also pave the way towards building models that are easier to understand.
Consequently, the feature embeddings learnt by robust models yield also clean inter-class interpolations, similar to those found by generative adversarial networks (GANs) (Goodfellow et al., 2014b) and other generative models.
This hints at the existence of a stronger connection between
GANs and adversarial robustness.
O N THE P RICE OF A DVERSARIAL R OBUSTNESS
Recall that in the canonical classiﬁcation setting, the primary focus is on maximizing standard accuracy, i.e. the performance on (yet) unseen samples from the underlying distribution.
Speciﬁcally, the goal is to train models that have low expected loss (also known as population risk):
[Equation]
Adversarial robustness
The existence of adversarial examples largely changed this picture.
In particular, there has been a lot of interest in developing models that are resistant to them, or, in other words, models that are adversarially robust .
In this context, the goal is to train models that have low expected adversarial loss :
[Equation]
Here, ∆ represents the set of perturbations that the adversary can apply to induce misclassiﬁcation.
In this work, we focus on the case when ∆ is the set of (cid:96) p -bounded perturbations, i.e. ∆ =
{ δ ∈
R d | (cid:107) δ (cid:107) p ≤ ε } .
This choice is the most common one in the context of adversarial examples and serves as a standard benchmark.
It is worth noting though that several other notions of adversarial perturbations have been studied.
These include rotations and translations (Fawzi & Frossard, 2015;
Engstrom et al., 2017), and smooth spatial deformations (Xiao et al., 2018a).
In general, determining the “right” ∆ to use is a domain speciﬁc question.
Adversarial training The most successful approach to building adversarially robust models so far (Madry et al., 2017; Kolter & Wong, 2017; Sinha et al., 2017; Raghunathan et al., 2018) was socalled adversarial training (Goodfellow et al., 2014b).
Adversarial training is motivated by viewing (2) as a statistical learning question, for which we need to solve the corresponding (adversarial) empirical risk minimization problem:
[Equation]
The resulting saddle point problem can be hard to solve in general.
However, it turns out to be often tractable in practice, at least in the context of (cid:96) p -bounded perturbations (Madry et al., 2017).
Speciﬁcally, adversarial training corresponds to a natural robust optimization approach to solving this problem (Ben-Tal et al., 2009).
In this approach, we repeatedly ﬁnd the worst-case input perturbations δ (solving the inner maximization problem), and then update the model parameters to reduce the loss on these perturbed inputs.
Though adversarial training is effective, this success comes with certain drawbacks.
The most obvious one is an increase in the training time (we need to compute new perturbations each parameter update step).
Another one is the potential need for more training data as shown recently in (Schmidt et al., 2018).
These costs make training more demanding, but is that the whole price of being adversarially robust?
In particular, if we are willing to pay these costs: Are robust classiﬁers better than standard ones in every other aspect?
This is the key question that motivates our work.
[Figure]
Adversarial Training as a Form of Data Augmentation Our point of start is a popular view of adversarial training as the “ultimate” form of data augmentation.
According to this view, the adversarial perturbation set ∆ is seen as the set of invariants that a good model should satisfy (regardless of the adversarial robustness considerations).
Thus, ﬁnding the worst-case δ corresponds to augmenting the training data in the “most confusing” and thus also “most helpful” manner.
A key implication of this view is that adversarial training should be beneﬁcial for the standard accuracy of a model (Torkamani & Lowd, 2013; 2014; Goodfellow et al., 2014b; Miyato et al., 2018).
Indeed, in Figure 1, we see this effect, when classiﬁers are trained with relatively few samples (particularly on MNIST).
In this setting, the amount of training data available is insufﬁcient to learn a good standard classiﬁer and the set of adversarial perturbations used is “compatible” with the learning task.
(That is, good standard models for this task need to be also somewhat invariant to these perturbations.)
In such regime, robust training does indeed act as data augmentation, regularizing the model and leading to a better solution (from standard accuracy point of view).
(Note that this effect seems less pronounced for CIFAR-10, possibly because (cid:96) p -invariance is not as important for a good standard CIFAR-10 classiﬁer.)
Surprisingly however, in Figure 6 we see that as we include more samples in the training set, this positive effect becomes less signiﬁcant.
In fact, after some point adversarial training actually decreases the standard accuracy.
In Figure 7 in Appendix G we study the behaviour of models trained using adversarial training with different (cid:96) p -bounded adversaries.
We observe a steady decline in standard accuracy as the strength of the adversary increases.
(Note that this still holds if we train on batches that also contain natural examples, as in Kurakin et al. (2016a).
See Appendix B.)
Similar effects were also observed in prior work (Kurakin et al., 2016b; Madry et al., 2017; Dvijotham et al., 2018b; Wong et al., 2018; Xiao et al., 2018b; Su et al., 2018; Babbar & Schölkopf, 2018).
The goal of this work is to illustrate and explain the roots of this phenomenon.
In particular, we would like to understand:
Why does there seem to be a trade-off between standard and adversarially robust accuracy?
As we will show, this effect is not an artifact of our adversarial training methods but in fact is inevitable consequence of different goals of adversarial robustness and standard generalization.
2. A DVERSARIAL ROBUSTNESS
BE WITH STANDARD ACCURACY
As we discussed above, we often observe that employing adversarial training leads to a decrease in a model’s standard accuracy.
In what follows, we show that this phenomenon is a manifestation of an inherent tension between standard accuracy and adversarially robust accuracy.
In particular, we present a theoretical model that demonstrates it.
In fact, this phenomenon can be illustrated in a fairly simple setting which suggests that it is quite prevalent.
Our binary classiﬁcation task Our data model consists of input-label pairs ( x, y ) sampled from a distribution D as follows:
[Equation]
where N ( µ, σ 2 ) is a normal distribution with mean µ and variance σ 2 , and p ≥ 0 .
We chose η to be large enough so that a simple classiﬁer attains high standard accuracy (> 99% ) – e.g. η = Θ(1 / √ d ) will sufﬁce.
The parameter p quantiﬁes how correlated the feature x 1 is with the label.
For the sake of example, we can think of p as being 0 .
This choice is fairly arbitrary; the trade-off between standard and robust accuracy will be qualitatively similar for any p < 1 .
Standard classiﬁcation is easy Note that samples from D consist of a single feature that is moderately correlated with the label and d other features that are only very weakly correlated with it.
Despite the fact that each one of the latter type of features individually is hardly predictive of the correct label, this distribution turns out to be fairly simple to classify from a standard accuracy perspective.
Speciﬁcally, a natural (linear) classiﬁer
[Equation]
where w unif
[Equation]
=
[Equation]
achieves standard accuracy arbitrarily close to 100% , for d large enough.
Indeed, observe that
Pr
[Equation]
η, 1 d (cid:19) >(cid:21) , which is > 99% when η ≥ 3 / √ d .
Adversarially robust classiﬁcation Note that in our discussion so far, we effectively viewed the average of x 2 , . . .
, x d +1 as a single “meta-feature” that is highly correlated with the correct label.
For a standard classiﬁer, any feature that is even slightly correlated with the label is useful.
As a result, a standard classiﬁer will take advantage (and thus rely on) the weakly correlated features x 2 , . . .
, x d + (by implicitly pooling information) to achieve almost perfect standard accuracy.
However, this analogy breaks completely in the adversarial setting.
In particular, an (cid:96) ∞ -bounded adversary that is only allowed to perturb each feature by a moderate ε can effectively override the effect of the aforementioned meta-feature.
For instance, if ε = 2 η , an adversary can shift each weaklycorrelated feature towards − y .
The classiﬁer would now see a perturbed input x (cid:48) such that each of the features x (cid:48) 2 , . . .
, x (cid:48)
d +1 are sampled i.i.d.
from N ( − ηy, 1) (i.e., now becoming anti -correlated with the correct label).
Thus, when ε ≥ 2 η , the adversary can essentially simulate the distribution of the weakly-correlated features as if belonging to the wrong class.
Formally, the probability of the meta-feature correctly predicting y in this setting (4) is
[Equation]
As a result, the simple classiﬁer in (4) that relies solely on these features cannot get adversarial accuracy better than 1% .
Intriguingly, this discussion draws a distinction between robust features ( x 1 ) and non-robust features ( x 2 , . . .
, x d +1 ) that arises in the adversarial setting.
While the meta-feature is far more predictive of the true label, it is extremely unreliable in the presence of an adversary.
Hence, a tension between standard and adversarial accuracy arises.
Any classiﬁer that aims for high accuracy (say > 99% ) will have to heavily rely on non-robust features (the robust feature provides only, say, 95% accuracy).
However, since the non-robust features can be arbitrarily manipulated, this classiﬁer will inevitably have low adversarial accuracy.
We make this formal in the following theorem proved in Appendix C.
Theorem 2.1 (Robustness-accuracy trade-off) .
Any classiﬁer that attains at least 1 − δ standard accuracy on D has robust accuracy at most p 1 − p δ against an (cid:96) ∞ -bounded adversary with ε ≥ 2 η .
This bound implies that if p < 1 , as standard accuracy approaches 100% ( δ → 0 ), adversarial accuracy falls to 0% .
As a concrete example, consider p = 0 .
95 , then any classiﬁer with standard accuracy more than 1 − δ will have robust accuracy at most 19 δ 1 .
Also it is worth noting that the theorem is tight.
If δ = 1 − p , both the standard and adversarial accuracies are bounded by p which is attained by the classiﬁer that relies solely on the ﬁrst feature.
Additionally, note that compared to the scale of the features ± 1 , the value of ε required to manipulate the standard classiﬁer is very small
( ε = O ( η ) , where η = O (1 / √ d ) ).
On the (non-)existence of an accurate and robust classiﬁer It might be natural to expect that in the regime of inﬁnite data, the standard classiﬁer itself acts as a robust classiﬁer.
Note however, that this is not true for the setting we analyze above.
Here, the trade-off between standard and adversarial accuracy is an inherent trait of the data distribution itself and not due to having insufﬁcient samples.
In this particular classiﬁcation task, we (implicitly) assumed that there does not exist a classiﬁer that is both robust and very accurate (i.e. > 99 % standard and robust accuracy).
Thus, for this task, any classiﬁer that is very accurate (including the Bayes classiﬁer – the classiﬁer minimizing classiﬁcation error having full-information about the distribution) will necessarily be non-robust.
This seemingly goes against the common assumption in adversarial ML that humans are such perfect robust and accurate classiﬁers for standard datasets.
However, note that there is no concrete evidence supporting this assumption.
In fact, humans often have far from perfect performance in vision benchmarks (Karpathy, 2011; 2014; Russakovsky et al., 2015) and are outperformed by ML models in certain tasks (He et al., 2015b; Gastaldi, 2017).
It is plausible that standard ML models are able to outperform humans in these tasks by relying on brittle features that humans are naturally invariant to and the observed decrease in performance might be the manifestation of that.
2. T HE IMPORTANCE OF ADVERSARIAL TRAINING
As we have seen in the distributional model D (3), a classiﬁer that achieves very high standard accuracy (1) will inevitably have near-zero adversarial accuracy.
This is true even when a classiﬁer with reasonable standard and robust accuracy exists.
Hence, in an adversarial setting (2), where the goal is to achieve high adversarial accuracy, the training procedure needs to be modiﬁed.
We now make this phenomenon concrete for linear classiﬁers trained using the soft-margin SVM loss.
Speciﬁcally, in Appendix D we prove the following theorem.
Theorem 2.2 (Adversarial training matters) .
For η ≥ 4 / √ d and p ≤ 0 .
975 (the ﬁrst feature is not perfect), a soft-margin SVM classiﬁer of unit weight norm minimizing the distributional loss achieves a standard accuracy of > 99% and adversarial accuracy of < 1% against an (cid:96) ∞ -bounded adversary of ε ≥ 2 η .
Minimizing the distributional adversarial loss instead leads to a robust classiﬁer that has standard and adversarial accuracy of p against any ε < 1 .
This theorem shows that if our focus is on robust models, adversarial training is crucial to achieve non-trivial adversarial accuracy in this setting.
Simply optimizing the standard accuracy of the model (i.e. standard training) leads to poor robust accuracy.
Soft-margin SVM classiﬁers and the constant 0 . 975 are chosen for mathematical convenience.
Our proofs do not depend on them in a crucial way and can be adapted, in a straightforward manner, to other natural settings, e.g. logistic regression.
Transferability
An interesting implication of our analysis is that standard training produces classiﬁers that rely on features that are weakly correlated with the correct label.
This will be true for any classiﬁer trained on the same distribution.
Hence, the adversarial examples that are created by perturbing each feature in the direction of − y will transfer across classiﬁers trained on independent samples from the distribution.
This constitutes an interesting manifestation of the generally observed phenomenon of transferability (Szegedy et al., 2013) and might hint at its origin.
Empirical examination In Section 2.1, we showed that the trade-off between standard accuracy and robustness might be inevitable.
To examine how representative our theoretical model is of real-world datasets, we also experimentally investigate this issue on MNIST (LeCun et al., 1998) as it is amenable to linear classiﬁers.
Interestingly, we observe a qualitatively similar behavior.
For instance, in Figure 5(b) in Appendix E, we see that the standard classiﬁer assigns weight to even weakly-correlated features.
(Note that in settings with ﬁnite training data, such brittle features could
O r i g i n a l
S t a n d a r d - t r a i n e d 2 - t r a i n e d (a) MNIST
O r i g
i n a l bird airplane frog
S t a n d a r d - t r a i n e d 2 - t r a i n e d (b) CIFARS a m p l e insect dog primate
N a t u r a l - t r a i n e d 2 - t r a i n e d (c) Restricted ImageNet arise even from noise – see Appendix E.) The robust classiﬁer on the other hand does not assign any weight beyond a certain threshold.
Further, we ﬁnd that it is possible to obtain a robust classiﬁer by directly training a standard model using only features that are relatively well-correlated with the label (without adversarial training).
As expected, as more features are incorporated into the training, the standard accuracy is improved at the cost of robustness (see Appendix E Figure 5(c)).
U NEXPECTED BENEFITS OF ADVERSARIAL ROBUSTNESS
In Section 2, we established that robust and standard models might depend on very different sets of features.
We demonstrated how this can lead to a decrease in standard accuracy for robust models.
In this section, we will argue that the representations learned by robust models can also be beneﬁcial.
At a high level, robustness to adversarial perturbations can be viewed as an invariance property of a model.
A model that achieves small loss for all perturbations in the set ∆ , will necessarily have learned representations that are invariant to such perturbations.
Thus, robust training can be viewed as a method to embed certain invariances in a model.
Since we also expect humans to be invariant to these perturbations (e.g. small (cid:96) p -bounded changes of the pixels), robust models will be more aligned with human vision than standard models.
In this section, we present evidence supporting the view.
Loss gradients in the input space align well with human perception As a starting point, we want to investigate which features of the input most strongly affect the prediction of the classiﬁer both for standard and robust models.
To this end, we visualize the gradients of the loss with respect to individual features (pixels) of the input in Figure 2.
We observe that gradients for adversarially trained networks align well with perceptually relevant features (such as edges) of the input image.
In contrast, for standard networks, these gradients have no coherent patterns and appear very noisy to humans.
We want to emphasize that no preprocessing was applied to the gradients (other than scaling and clipping for visualization).
On the other hand, extraction of interpretable information from the gradients of standard networks has so far only been possible using additional sophisticated techniques (Simonyan et al., 2013;
Yosinski et al., 2015; Olah et al., 2017).
This observation effectively outlines an approach to train models that align better with human perception by design .
By encoding the correct prior into the set of perturbations ∆ , adversarial
Original
Standard -trained 2 -trained    (a) MNIST airplane
Original bird
Standard bird -trained bird 2 -trained dog deer bird deer (b) CIFARprimate
Original dog
Standard dog -trained dog 2 -trained bird turtle dog cat
[Figure]
Additional examples are visualized in Figure 8 and 9 of Appendix G training alone might be sufﬁcient to yield interpretable gradients.
We believe that this phenomenon warrants an in-depth investigation and we view our experiments as only exploratory.
Adversarial examples exhibit salient data characteristics Given how the gradients of standard and robust models are concentrated on qualitatively different input features, we want to investigate how the adversarial examples of these models appear visually.
To ﬁnd adversarial examples, we start from a given test image and apply Projected Gradient Descent (PGD; a standard ﬁrst-order optimization method) to ﬁnd the image of highest loss within an (cid:96) p -ball of radius ε around the original image 2 .
This procedure will change the pixels that are most inﬂuential for a particular model’s predictions and thus hint towards how the model is making its predictions.
The resulting visualizations are presented in Figure 3 (details in Appendix A).
Surprisingly, we can observe that adversarial perturbations for robust models tend to produce salient characteristics of another class.
In fact, the corresponding adversarial examples for robust models can often be perceived as samples from that class.
This behavior is in stark contrast to standard models, for which adversarial examples appear as noisy variants of the input image.
These ﬁndings provide additional evidence that adversarial training does not necessarily lead to gradient obfuscation (Athalye et al., 2018).
Following the gradient changes the image in a meaningful way and (eventually) leads to images of different classes.
Hence, the robustness of these models does not stem from having gradients that are ill-suited for ﬁrst-order methods.
Smooth cross-class interpolations via gradient descent By linearly interpolating between the original image and the image produced by PGD we can produce a smooth, “perceptually plausible” interpolation between classes (Figure 4).
Such interpolation have thus far been restricted to generative models such as GANs (Goodfellow et al., 2014a) and VAEs (Kingma & Welling, 2013), involved manipulation of learned representations (Upchurch et al., 2016), and hand-designed methods (Suwajanakorn et al., 2015; Kemelmacher-Shlizerman, 2016).
In fact, we conjecture that the similarity of these inter-class trajectories to GAN interpolations is not a coincidence.
We postulate that the saddle point problem that is key in both these approaches may be at the root of this effect.
We hope that future research will investigate this connection further and explore how to utilize the loss landscape of robust models as an alternative method to smoothly interpolate between classes.
R ELATED WORK
Due to the large body of related work, we will only focus on the most relevant studies here and defer the full discussion to Appendix F. Fawzi et al.
(2018b) prove upper bounds on the robust of classiﬁers and exhibit a standard vs. robust accuracy trade-off for a speciﬁc classiﬁer families on a synthetic task.
Their setting also (implicitly) utilizes the notion of robust and non-robust features, however these features have small magnitude rather than weak correlation.
Ross & Doshi-Velez (2017) propose regularizing the gradient of the classiﬁer with respect to its input.
They ﬁnd that the resulting classiﬁers have more interpretable gradients and targeted adversarial examples resemble the target class for digit and character recognition tasks.
There has been recent of work proving upper bounds on classiﬁer robustness (Gilmer et al., 2018; Schmidt et al., 2018; Fawzi et al., 2018a).
However, this work is orthogonal to ours as it does not differentiate standard and robust classiﬁers.
C ONCLUSIONS AND FUTURE DIRECTIONS
In this work, we show that the goal of adversarially robust generalization might fundamentally be at odds with that of standard generalization.
Speciﬁcally, we identify an inherent trade-off between the standard accuracy and adversarial robustness of a model, that provably manifests in a concrete, simple setting.
This trade-off stems from intrinsic differences between the feature representations learned by standard and robust models.
Our analysis also explains the drop in standard accuracy observed when employing adversarial training in practice.
Moreover, it emphasizes the need to develop robust training methods, since robustness is unlikely to arise as a consequence of standard training.
We discover that even though adversarial robustness comes at a price, it has some unexpected beneﬁts.
Robust models learn meaningful feature representations that align well with salient data characteristics.
The root of this phenomenon is that the set of adversarial perturbations encodes some prior for human perception.
Thus, classiﬁers that are robust to these perturbations are also necessarily invariant to input modiﬁcations that we expect humans to be invariant to.
We demonstrate a striking consequence of this phenomenon: robust models yield clean feature interpolations similar to those obtained from generative models such as GANs (Goodfellow et al., 2014b).
This emphasizes the possibility of a stronger connection between GANs and adversarial robustness.
Finally, our ﬁndings show that the interplay between adversarial robustness and standard classiﬁcation might be more nuanced that one might expect.
This motivates further work to fully undertand the relative costs and beneﬁts of each of these notions.
A E XPERIMENTAL SETUP A. D ATASETS
We perform our experimental analysis on the MNIST (LeCun et al., 2010), CIFAR-10 (Krizhevsky & Hinton, 2009) and (restricted) ImageNet (Deng et al., 2009) datasets.
For binary classiﬁcation, we ﬁlter out all the images from the MNIST dataset other than the “5” and “7” labelled examples.
For the ImageNet dataset [Table] adversarial training is signiﬁcantly harder since the classiﬁcation problem is challenging by itself and standard classiﬁers are already computationally expensive to train.
We thus restrict our focus to a smaller subset of the dataset.
We group together a subset of existing, semantically similar ImageNet classes into 8 different super-classes, as shown in Table 1.
We train and evaluate only on examples corresponding to these classes.
[Table]
A. M ODELS
• Binary MNIST (Section 2.2): We train a linear classiﬁer with parameters w ∈ R 784 , b ∈ R on the dataset described in Section A.1 (labels − 1 and +1 correspond to images labelled as “5” and “7” respectively).
We use the cross-entropy loss and perform 100 epochs of gradient descent in training.
• MNIST: We use the simple convolution architecture from the TensorFlow tutorial (TFM, 2017) 3 .
• CIFAR-10:
We consider a standard ResNet model (He et al., 2015a).
It has 4 groups of residual layers with ﬁlter sizes (16, 16, 32, 64) and 5 residual units each 4 .
• Restricted ImageNet: We use a ResNet-50 (He et al., 2015a) architecture using the code from the tensorpack repository (Wu et al., 2016).
We do not modify the model architecture, and change the training procedure only by changing the number of examples per “epoch” from 1,280,000 images to 76,800 images.
A. A DVERSARIAL TRAINING
We perform adversarial training to train robust classiﬁers following Madry et al.
Speciﬁcally, we train against a projected gradient descent (PGD) adversary, starting from a random initial perturbation of the training data.
We consider adversarial perturbations in (cid:96) p norm where p = { 2 , ∞} .
Unless otherwise speciﬁed, we use the values of ε provided in Table 2 to train/evaluate our models. [Table] [Table]
A DVERSARIAL EXAMPLES FOR LARGE ε
The images we generated for Figure 3 were allowed a much larger perturbation from the original sample in order to produce visible changes to the images.
These values are listed in Table 3.
Since
[Table]
these levels of perturbations would allow to truly change the class of the image, training against such strong adversaries would be impossible.
Still, we observe that smaller values of ε sufﬁces to ensure that the models rely on the most robust (and hence interpretable) features.
B M IXING NATURAL AND ADVERSARIAL EXAMPLES IN EACH BATCH
In order to make sure that the standard accuracy drop in Figure 7 is not an artifact of only training on adversarial examples, we experimented with including unperturbed examples in each training batch, following the recommendation of (Kurakin et al., 2016a).
We found that while this slightly improves the standard accuracy of the classiﬁer, it decreases it’s robust accuracy by a roughly proportional amount, see Table 4.
C P ROOF OF T HEOREM 2.
The main idea of the proof is that an adversary with ε = 2 η is able to change the distribution of features x 2 , . . .
, x d +1 to reﬂect a label of − y instead of y by subtracting εy from each variable.
Hence any information that is used from these features to achieve better standard accuracy can be used by the adversary to reduce adversarial accuracy.
We deﬁne G + to be the distribution of x 2 , . . .
, x d + when y =
+
and G − to be that distribution when y = − 1 .
We will consider the setting where ε = 2 η and ﬁx the adversary that replaces x i by x i − yε for each i ≥ 2 .
This adversary is able to change G + to G − in the adversarial setting and vice-versa.
Consider any classiﬁer f ( x ) that maps an input x to a class in {− 1 , +1 } .
Let us ﬁx the probability that this classiﬁer predicts class +1 for some ﬁxed value of x 1 and distribution of x 2 , . . .
, x d +1 .
Concretely, we deﬁne p ij to be the probability of predicting +1 given that the ﬁrst feature has sign i and the rest of the features are distributed according to G j .
Formally,
[Equation]
[Table]
Using these deﬁnitions, we can express the standard accuracy of the classiﬁer as
[Equation]
Similarly, we can express the accuracy of this classiﬁer against the adversary that replaces G + with
G − (and vice-versa) as
[Equation]
For convenience we will deﬁne a = 1 − p ++ + p −− and b = 1 − p − + + p + − .
Then we can rewrite standard accuracy :
[Equation]
adversarial accuracy :
[Equation]
We are assuming that the standard accuracy of the classiﬁer is at least 1 − δ for some small δ .
This implies that
[Equation]
Since p ij are probabilities, we can guarantee that a ≥ 0 .
Moreover, since p ≥ 0 . 5 , we have p/ (1 − p ) ≥ 1 .
We use these to upper bound the adversarial accuracy by
[Equation]
D P ROOF OF T HEOREM 2.
We consider the problem of ﬁtting the distribution D of (3) by using a standard soft-margin SVM classiﬁer.
Speciﬁcally, this can be formulated as:
[Equation]
for some value of λ .
We will assume that we tune λ such that the optimal solution w ∗ has (cid:96) 2 -norm of 1 .
This is without much loss of generality since our proofs can be adapted to the general case.
We will refer to the ﬁrst term of (5) as the margin term and the second term as the regularization term.
First we will argue that, due to symmetry, the optimal solution will assign equal weight to all the features x i for i = 2 , . . .
, d + 1 .
Lemma D.1.
Consider an optimal solution w ∗ to the optimization problem (5) .
Then,
[Equation]
+ 1 } .
Proof.
Assume that ∃ i, j ∈ { 2 , ..., d + 1 } such that w ∗ i (cid:54) =
w ∗ j .
Since the distribution of x i and x j are identical, we can swap the value of w i and w j , to get an alternative set of parameters ˆ w that has the same loss function value ( ˆ w j = w i , ˆ w i = w j , ˆ w
Moreover, since the margin term of the loss is convex in w , using Jensen’s inequality, we get that averaging w ∗ and ˆ w will not increase the value of that margin term.
Note, however, that (cid:107) w ∗ + ˆ w(cid:107) 2 < (cid:107) w ∗ (cid:107) 2 , hence the regularization loss is strictly smaller for the average point.
This contradicts the optimality of w ∗ .
Since every optimal solution will assign equal weight to all x i for k ≥ 2 , we can replace these features by their sum (and divide by √ d for convenience).
We will deﬁne
[Equation]
which, by the properties of the normal distribution, is distributed as
[Equation]
By assigning a weight of v to that combined feature the optimal solutions can be parametrized as
[Equation]
where the regularization term of the loss is λ ( w 21 + v 2 ) / 2 .
Recall that our chosen value of η is 4 / √ d , which implies that the contribution of vz is distributed normally with mean 4 yv and variance v 2 .
By the concentration of the normal distribution, the probability of vz being larger than v is large.
We will use this fact to show that the optimal classiﬁer will assign on v at least as much weight as it assigns on w 1 .
Lemma D.2.
Consider the optimal solution ( w ∗ 1 , v ∗ ) of the problem (5) .
Then
[Equation]
Proof.
Assume for the sake of contradiction that v ∗ < 1 / √ 2 .
Then, with probability at least 1 − p , the ﬁrst feature predicts the wrong label and without enough weight, the remaining features cannot compensate for it.
Concretely,
[Equation]
We will now show that a solution that assigns zero weight on the ﬁrst feature ( v = 1 and w 1 = 0 ), achieves a better margin loss.
[Equation]
Hence, as long as p ≤ 0 . 975 , this solution has a smaller margin loss than the original solution.
Since both solutions have the same norm, the solution that assigns weight only on v is better than the original solution ( w ∗ 1 , v ∗ ) , contradicting its optimality.
We have established that the learned classiﬁer will assign more weight to v than w 1 .
Since z will be at least y with large probability, we will show that the behavior of the classiﬁer depends entirely on z .
Lemma D.3.
The standard accuracy of the soft-margin SVM learned for problem (5) is at least 99% .
Proof.
By Lemma D.2, the classiﬁer predicts the sign of w 1 x 1 + vz where vz ∼ N (4 yv, v 2 ) and v ≥ 1 / √ 2 .
Hence with probability at least 99% , vzy > 1 / √ 2 ≥ w 1 and thus the predicted class is y (the correct class) independent of x 1 .
We can utilize the same argument to show that an adversary that changes the distribution of z has essentially full control over the classiﬁer prediction.
Lemma D.4.
The adversarial accuracy of the soft-margin SVM learned for (5) is at most 1% against an (cid:96) ∞ -bounded adversary of ε = 2 η .
Proof.
Observe that the adversary can shift each feature x i towards y by 2 η .
This will cause z to be distributed as
[Equation]
Therefore with probability at least 99% , vyz < − y ≤ − w
and the predicted class will be − y (wrong class) independent of x 1 .
It remains to show that adversarial training for this classiﬁcation task with ε > 2 η will results in a classiﬁer that has relies solely on the ﬁrst feature.
Lemma D.5.
Minimizing the adversarial variant of the loss (5) results in a classiﬁer that assignsweight to features x i for i ≥ 2 .
Proof.
The optimization problem that adversarial training solves is
[Equation]
which is equivalent to
[Equation]
Consider any optimal solution w for which w i > 0 for some i > 2 .
The contribution of terms depending on w i to 1 − yw (cid:62) x + ε (cid:107) w (cid:107) 1 is a normally-distributed random variable with mean 2 η − ε ≤ 0 .
Since the mean is non-positive, setting w i to zero can only decrease the margin term of the loss.
At the same time, setting w i to zero strictly decreases the regularization term, contradicting the optimality of w .
Clearly, such a classiﬁer will have standard and adversarial accuracy of p against any ε < 1 since such a value of ε is not sufﬁcient to change the sign of the ﬁrst feature.
This concludes the proof of the theorem.
E R OBUSTNESS - ACCURACY TRADE - OFF : A N EMPIRICAL EXAMINATION
Our theoretical analysis shows that there is an inherent tension between standard accuracy and adversarial robustness.
At the core of this trade-off is the concept of robust and non-robust features.
The robustness of a feature is characterized by the strength of its correlation with the correct label.
It is natural to wonder whether this concept of robust features is an artifact of our theoretical analysis or if it manifests more broadly.
We thus investigate this issue experimentally on a dataset that is amenable to linear classiﬁers, MNIST (LeCun et al., 1998) (details in Appendix A).
Recall the goal of standard classiﬁcation for linear classiﬁers is to predict accurately, i.e. y = sign ( w (cid:62) x ) .
Hence the correlation of a feature i with the true label, computed as | E [ yx i ] | , quantiﬁes how useful this feature is for classiﬁcation.
In the adversarial setting, against an ε (cid:96) ∞ -bounded adversary we need to ensure that y = sign ( w (cid:62) x − εy (cid:107) w (cid:107) 1 ) .
In that case we expect a feature i to be helpful if | E [ yx i ] | ≥ ε .
[Figure]
This calculation suggests that in the adversarial setting, there is an implicit threshold on feature correlations imposed by the threat model (the perturbation allowed to the adversary).
While standard models may utilize all features with non-zero correlations, a robust model cannot rely on features with correlation below this threshold.
In Figure 5(b) we visualize the correlation of each pixel (feature) in the MNIST dataset along with the learned weights of the standard and robust classiﬁers.
As expected, we see that the standard classiﬁer assigns weights even to weakly-correlated pixels so as to maximize prediction conﬁdence.
On the other hand, the robust classiﬁer does not assign any weight below a certain correlation threshold which is dictated by the adversary’s strength ( ε ) (Figures 5(a, b))
Interestingly, the standard model assigns non-zero weight even to very weakly correlated pixels (Figure 5(a)).
In settings with ﬁnite training data, such non-robust features could arise from noise.
(For instance, in N tosses of an unbiased coin, the expected imbalance between heads and tails is O ( √ N ) with high probability.)
A standard classiﬁer would try to take advantage of even this “hallucinated” information by assigning non-zero weights to these features.
E. A N ALTERNATIVE PATH TO ROBUSTNESS ?
The analysis above highlights an interesting trade-off between the predictive power of a feature and its vulnerability to adversarial perturbations.
This brings forth the question – Could we use these insights to train robust classiﬁers with standard methods (i.e. without performing adversarial training)?
As a ﬁrst step, we train a (standard) linear classiﬁer on MNIST utilizing input features (pixels) that lie above a given correlation threshold (see Figure 5(c)).
As expected, as more non robust features are incorporated in training, the standard accuracy increases at the cost of robustness.
Further, we observe that a standard classiﬁer trained in this manner using few robust features attains better robustness than even adversarial training.
This results suggest a more direct (and potentially better) method of training robust networks in certain settings.
F A DDITIONAL RELATED WORK
Fawzi et al.
(2016) derive parameter-dependent bounds on the robustness of any ﬁxed classiﬁer.
Our results focus on the statistical setting itself and provide lower bounds for all classiﬁers learned in this setting.
Wang et al.
(2017) analyze the adversarial robustness of nearest neighbor classiﬁers.
Instead we focus on lower bounds that are inherent to the statistical setting itself and apply to all classiﬁers.
Schmidt et al.
(2018) study the generalization aspect of adversarially robustness.
They show that the number of samples needed to achieve adversarially robust generalization is polynomially larger in the dimension than the number of samples needed to ensure standard generalization.
In the limit of inﬁnite samples, a simple linear classiﬁer can achieve perfect standard and adversarial accuracy.
In contrast, our results are orthogonal and capture the optimization aspect of the problem, persisting in the limit of inﬁnite data.
We show that fundamentally different classiﬁer are needed to achieve high accuracy in the adversarial setting.
Gilmer et al.
(2018) demonstrate a setting where even a small amount of standard error implies that most points provably have a misclassiﬁed point close to them.
In this setting, achieving perfect standard accuracy (easily achieved by a simple classiﬁer) is sufﬁcient to achieve perfect adversarial robustness.
In contrast, our work focuses on a setting where adversarial training (provably) matters and there exists a trade-off between standard and adversarial accuracy.
Xu & Mannor (2012) explore the connection between robustness and generalization, showing that, in a certain sense, robustness can imply generalization.
This direction is orthogonal to our, since we work in the limit of inﬁnite data, optimizing the distributional loss directly.
Fawzi et al.
(2018a) prove lower bounds on the robustness of any classiﬁer based on certain generative assumptions.
Since these bounds apply to all classiﬁers, independent of architecture and training procedure, they fail to capture the situation we face in practice where robust optimization can signiﬁcantly improve the adversarial robustness of standard classiﬁers (Madry et al., 2017; Kolter &
Wong, 2017; Raghunathan et al., 2018; Sinha et al., 2017).
A recent work (Bubeck et al., 2018) turns out to (implicitly) rely on the distinction between robust and non-robust features in constructing a distribution for which adversarial robustness is hard from a different, computational point of view.
Goodfellow et al.
(2014b) observed that adversarial training results in feature weights that depend on fewer input features (similar to Figure 5(a)).
Additionally, it has been observed that for naturally trained RBF classiﬁers on MNIST, targeted adversarial attacks resemble images of the target class (Goodfellow, 2015).
Su et al.
(2018) empirically observe a similar trade-off between the accuracy and robustness of standard models across different deep architectures on ImageNet.
Babbar & Schölkopf (2018) study an extreme multi-label problem and observe that for classes with relatively few examples, (cid:96) 1 -regularization (which corresponds to adversarial training for linear models) is helpful, while for classes with more samples, it is harmful to the model accuracy
[Figure]
G O MITTED FIGURES
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[Table] [Figure] [Table] [Figure]
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[Figure]
against an (cid:96) ∞ -bounded adversary as a function of the size of the training dataset.
We observe that in the low-data regime, adversarial training has an effect similar to data augmentation and helps with generalization in certain cases (particularly on MNIST).
However, in the limit of sufﬁcient training data, we see that the standard accuracy of robust models is less than that of the standard model ( ε train = 0 ), which supports the theoretical analysis in Section 2.1.
[Figure]
O r i g i n a l
S t a n d a r d         - t r a i n e d         2 - t r a i n e d        (a) MNIST cat
O r i g i n a l ship ship airplane frog
frog automobile frog cat bird S t a n d a r d automobile bird bird bird bird frog bird frog frog - t r a i n e d automobile automobile bird deer horse truck deer bird frog 2 - t r a i n e d automobile automobile ship deer truck cat cat deer (b) CIFARdog
O r i g i n a l cat primate primate frog primate insect fish primate primate
S t a n d a r d dog frog dog turtle cat primate primate dog cat - t r a i n e d dog frog dog turtle bird frog insect dog cat 2 - t r a i n e d dog frog dog turtle bird cat cat dog
[Figure]
O r i g i n a l
S t a n d a r d         - t r a i n e d         2 - t r a i n e d        (a) MNIST cat
O r i g i n a l ship ship airplane frog
frog automobile frog cat frog
[Figure]
bird bird deer bird truck bird bird frog - t r a i n e d airplane truck deer deer deer dog deer frog bird 2 - t r a i n e d automobile automobile ship deer truck cat deer frog (b) CIFAR-
[Figure]
l frog dog bird primate crab primate turtle insect
[Figure]
insect
[Figure]
insect turtle
[Figure]
O r i g i n a l
S t a n d a r d - t r a i n e d 2 - t r a i n e d
[Figure]
O r i g i n a l dog horse truck ship dog horse ship frog horse airplane
S t a n d a r d - t r a i n e d 2 - t r a i n e d
[Figure]
S a m p l e primate primate primate cat dog dog primate bird primate insect
N a t u r a l - t r a i n e d 2 - t r a i n e d
[Figure]