Are adversarial examples inevitable?

Introduction

A number of adversarial attacks on neural networks have been recently proposed. To counter these attacks, a number of authors have proposed a range of defenses. However, these defenses are often quickly broken by new and revised attacks. Given the lack of success at generating robust defenses, we are led to ask a fundamental question: Are adversarial attacks inevitable?

In this paper, we identify a broad class of problems for which adversarial examples cannot be avoided. We also derive fundamental limits on the susceptibility of a classifier to adversarial attacks that depend on properties of the data distribution as well as the dimensionality of the dataset.

Finally, in Section 8, we explore the causes of adversarial susceptibility in real datasets, and the effect of dimensionality. We present an example image class for which there is no fundamental link between dimensionality and robustness, and argue that the data distribution, and not dimensionality, is the primary cause of adversarial susceptibility.

Adversarial examples, first demonstrated in Szegedy et al. (2013) and Biggio et al. (2013), change the label of an image using small and often imperceptible perturbations to its pixels. A number of defenses have been proposed to harden networks against attacks, but historically, these defenses have been quickly broken. Adversarial training, one of the earliest defenses, successfully thwarted the fast gradient sign method (FGSM) (Goodfellow et al., 2014), one of the earliest and simplest attacks. However, adversarial training with FGSM examples was quickly shown to be vulnerable to more sophisticated multi-stage attacks (Kurakin et al., 2016; Tramèr et al., 2017a). More sophisticated defenses that rely on network distillation (Papernot et al., 2016b) and specialized activation functions (Zantedeschi et al., 2017) were also toppled by strong attacks (Papernot et al., 2016a; Tramèr et al., 2017b; Carlini & Wagner, 2016; 2017a).

The ongoing vulnerability of classifiers was highlighted in recent work by Athalye et al. (2018) and Athalye & Sutskever (2017) that broke an entire suite of defenses presented in ICLR 2018 including thermometer encoding (Buckman et al., 2018), detection using local intrinsic dimensionality (Ma et al., 2018), input transformations such as compression and image quilting (Guo et al., 2017), stochastic activation pruning (Dhillon et al., 2018), adding randomization at inference time (Xie et al., 2017), enhancing the confidence of image labels (Song et al., 2017), and using a generative model as a defense (Samangouei et al., 2018).

Rather than hardening classifiers to attacks, some authors have proposed sanitizing datasets to remove adversarial perturbations before classification. Approaches based on auto-encoders (Meng & Chen, 2017) and GANs (Shen et al., 2017) were broken using optimization-based attacks (Carlini & Wagner, 2017b; a).

A number of “certifiable” defense mechanisms have been developed for certain classifiers. Raghunathan et al. (2018) harden a two-layer classifier using semidefinite programming, and Sinha et al. (2018) propose a convex duality-based approach to adversarial training that works on sufficiently small adversarial perturbations with a quadratic adversarial loss. Kolter & Wong (2017) consider training a robust classifier using the convex outer adversarial polytope. All of these methods only consider robustness of the classifier on the training set, and robustness properties often fail to generalize reliably to test examples.

One place where researchers have enjoyed success is at training classifiers on low-dimensional datasets like MNIST (Madry et al., 2017; Sinha et al., 2018). The robustness achieved on more complicated datasets such as CIFAR-10 and ImageNet are nowhere near that of MNIST, which leads some researchers to speculate that adversarial defense is fundamentally harder in higher dimensions – an issue we address in Section 8.

This paper uses well-known results from high-dimensional geometry, specifically isoperimetric inequalities, to provide bounds on the robustness of classifiers. Several other authors have investigated adversarial susceptibility through the lens of geometry. Wang et al. (2018) analyze the robustness of nearest neighbor classifiers, and provide a more robust NN classifier. Fawzi et al. (2018) study adversarial susceptibility of datasets under the assumption that they are produced by a generative model that maps random Gaussian vectors onto images. Gilmer et al. (2018) do a detailed case study, including empirical and theoretical results, of classifiers for a synthetic dataset that lies on two concentric spheres. Simon-Gabriel et al. (2018) show that the Lipschitz constant of untrained networks with random weights gets large in high dimensions. Shortly after the original appearance of our work, Mahloujifar et al. (2018) presented a study of adversarial susceptibility that included both evasion and poisoning attacks. Our work is distinct in that it studies adversarial robustness for arbitrary data distributions, and also that it rigorously looks at the effect of dimensionality on robustness limits.

2 Notation

Problem setup

We also consider a “classifier” function $\mathcal{C}:\Omega\to\{1,2,\ldots,m\}$ that partitions $\Omega$ into disjoint measurable subsets, one for each class label. The classifier we consider is discrete valued – it provides a label for each data point but not a confidence level.

With this setup, we can give a formal definition of an adversarial example.

Consider a point $x\in\Omega$ drawn from class $c,$ a scalar $\epsilon>0,$ and a metric $d.$ We say that $x$ admits an $\epsilon$-adversarial example in the metric $d$ if there exists a point $\hat{x}\in\Omega$ with $\mathcal{C}(\hat{x})\neq c,$ and $d(x,\hat{x})\leq\epsilon.$

In plain words, a point has an $\epsilon$-adversarial example if we can sneak it into a different class by moving it at most $\epsilon$ units in the distance $d$.

We also consider sparse adversarial examples in which only a small subset of pixels are manipulated. This corresponds to the metric $d_{0},$ in which case the constraint $\|x-\hat{x}\|_{0}\leq\epsilon$ means that an adversarial example was crafted by changing at most $\epsilon$ pixels, and leaving the others alone.

The simple case: adversarial examples on the unit sphere

We begin by looking at the case of classifiers for data on the sphere. While this data model may be less relevant than the other models studied below, it provides a straightforward case where results can be proven using simple, geometric lemmas. The more realistic case of images with pixels in $$ will be studied in Section 4.

The idea is to show that, provided a class of data points takes up enough space, nearly every point in the class lies close to the class boundary. To show this, we begin with a simple definition.

The $\epsilon$-expansion of a subset $\mathcal{A}\subset\Omega$ with respect to distance metric $d,$ denoted $\mathcal{A}(\epsilon,d),$ contains all points that are at most $\epsilon$ units away from $\mathcal{A}$. To be precise

We sometimes simply write $\mathcal{A}(\epsilon)$ when the distance metric is clear from context.

Our result provides bounds on the probability of adversarial examples that are independent of the shape of the class boundary. This independence is a simple consequence of an isoperimetric inequality. The classical isoperimetric inequality states that, of all closed surfaces that enclose a unit volume, the sphere has the smallest surface area. This simple fact is intuitive but famously difficult to prove. For a historical review of the isoperimetric inequality and its variants, see Osserman et al. (1978). We will use a special variant of the isoperimetric inequality first proved by Lévy & Pellegrino (1951) and simplified by Talagrand (1995).

The classical isoperimetric inequality is a simple geometric statement, and frequently appears without absolute bounds on the size of the $\epsilon$-expansion of a half-sphere, or with bounds that involve unspecified constants (Vershynin, 2017). A tight bound derived by Milman & Schechtman (1986) is given below. The asymptotic blow-up of the $\epsilon$-expansion of a half sphere predicted by this bound is shown in Figure 3.

The geodesic $\epsilon$-expansion of a half sphere has normalized measure at least

Lemmas 1 and 2 together can be taken to mean that, if a set is not too small, then in high dimensions almost all points on the sphere are reachable within a short $\epsilon$ jump from that set. These lemmas have immediate implications for adversarial examples, which are formed by mapping one class into another using small perturbations. Despite its complex appearance, the result below is a consequence of the (relatively simple) isoperimetric inequality.

Let $V_{c}$ denote the magnitude of the supremum of $\rho_{c}$ relative to the uniform density. This can be written $V_{c}:=s_{n-1}\cdot\sup_{x}\rho_{c}(x).$

Let $f_{c}=\mu_{1}\{x|\mathcal{C}(x)=c\}$ be the fraction of the sphere labeled as $c$ by classifier $\mathcal{C}$.

Choose some class $c$ with $f_{c}\leq\frac{1}{2}$. Sample a random data point $x$ from $\rho_{c}.$ Then with probability at least

$x$ is misclassified by $\mathcal{C},$ or

$x$ admits an $\epsilon$-adversarial example in the geodesic distance.

Choose a class $c$ with $f_{c}\leq\frac{1}{2}.$ Let $\mathcal{R}=\{x|\mathcal{C}(x)=c\}$ denote the region of the sphere labeled as class $c$ by $\mathcal{C}$, and let $\overline{\mathcal{R}}$ be its complement. $\overline{\mathcal{R}}(\epsilon)$ is the $\epsilon$-expansion of $\overline{\mathcal{R}}$ in the geodesic metric. Because $\overline{\mathcal{R}}$ covers at least half the sphere, the isoperimetric inequality (Lemma 1) tells us that the epsilon expansion is at least as great as the epsilon expansion of a half sphere. We thus have

Now, consider the set $\mathcal{S}_{c}$ of “safe” points from class $c$ that are correctly classified and do not admit adversarial perturbations. A point is correctly classified only if it lies inside $\mathcal{R},$ and therefore outside of $\overline{\mathcal{R}}$. To be safe from adversarial perturbations, a point cannot lie within $\epsilon$ distance from the class boundary, and so it cannot lie within $\overline{R}(\epsilon)$. It is clear that the set $\mathcal{S}_{c}$ of safe points is exactly the complement of $\overline{R}(\epsilon).$ This set has normalized measure

The probability of a random point lying in $\mathcal{S}_{c}$ is bounded above by the normalized supremum of $\rho_{c}$ times the normalized measure $\mu_{1}[\mathcal{S}_{c}].$ This product is given by

We then subtract this probability from 1 to obtain the probability of a point lying outside the safe region, and arrive at equation 1. ∎

It is easily observed that, for any two points $x$ and $y$ on a sphere,

where $d_{\infty}(x,y)$, $d_{2}(x,y)$, and $d_{g}(x,y)$ denote the $l_{\infty}$, Euclidean, and geodesic distance, respectively. From this, we see that Theorem 1 is actually fairly conservative; any $\epsilon$-adversarial example in the geodesic metric would also be adversarial in the other two metrics, and the bound in Theorem 1 holds regardless of which of the three metrics we choose (although different values of $\epsilon$ will be appropriate depending on the norm).

What about the unit cube?

The above result about the sphere is simple and easy to prove using classical results. However, real world images do not lie on the sphere. In a more typical situation, images will be scaled so that their pixels lie in $$, and data lies inside a high-dimensional hypercube (but, unlike the sphere, data is not confined to its surface). The proof of Theorem 1 makes extensive use of properties that are exclusive to the sphere, and is not applicable to this more realistic setting. Are there still problem classes on the cube where adversarial examples are inevitable?

This question is complicated by the fact that geometric isoperimetric inequalities do not exist for the cube, as the shapes that achieve minimal $\epsilon$-expansion (if they exist) depend on the volume they enclose and the choice of $\epsilon$ (Ros, 2001). Fortunately, researchers have been able to derive “algebraic” isoperimetric inequalities that provide lower bounds on the size of the $\epsilon$-expansion of sets without identifying the shape that achieves this minimum (Talagrand, 1996; Milman & Schechtman, 1986). The result below about the unit cube is analogous to Proposition 2.8 in Ledoux (2001), except with tighter constants. For completeness, a proof (which utilizes methods from Ledoux) is provided in Appendix A.

Consider a measurable subset of the cube $\mathcal{A}\subset^{n},$ and a p-norm distance metric $d_{p}(x,y)=\|x-y\|_{p}$ for $p>0.$ Let $\Phi(z)=(2\pi)^{-\frac{1}{2}}\int_{-\infty}^{z}e^{-t^{2}/2}dt,$ and let $\alpha$ be the scalar that satisfies $\Phi(\alpha)=\operatorname{vol}[\mathcal{A}].$ Then

where $p^{*}=\min(p,2).$ In particular, if $\operatorname{vol}(\mathcal{A})\geq 1/2,$ then we simply have

Using this result, we can show that most data samples in a cube admit adversarial examples, provided the data distribution is not excessively concentrated.

Consider a classification problem with $m$ classes, each distributed over the unit hypercube $^{n}$ with density functions $\{\rho_{c}\}_{c=1}^{m}$. Choose a classifier function ${\mathcal{C}:^{n}\to\{1,2,\ldots,m\}}$ that partitions the hypercube into disjoint measurable subsets. Define the following scalar constants:

Let $U_{c}$ denote the supremum of $\rho_{c}$.

Let $f_{c}$ be the fraction of hypercube partitioned into class $c$ by $\mathcal{C}$.

$x$ is misclassified by $\mathcal{C},$ or

$x$ has an adversarial example $\hat{x},$ with $\|x-\hat{x}\|_{p}\leq\epsilon$.

for $p\geq 2,$ where $\hat{\Phi}(z)=\frac{1}{\sqrt{2\pi}}\int_{z}^{\infty}e^{-t^{2}/2}dt\geq\frac{1}{\sqrt{2\pi}z}e^{-z^{2}/2}$ (for $z>0$), and $\alpha=\Phi^{-1}(1-f_{c})$. For this bound to be meaningful with $\epsilon<1$, we need $f_{c}$ to be relatively small, and $\epsilon$ to be roughly $f_{c}$ or smaller. This is realistic for some problems; ImageNet has 1000 classes, and so $f_{c}<10^{-3}$ for at least one class.

What about sparse adversarial examples?

If a point $x$ has an $\epsilon$-adversarial example in this norm, then it can be perturbed into a different class by modifying at most $\epsilon$ pixels (in this case $\epsilon$ is taken to be a positive integer).

Theorem 2 is fairly tight for $p=1$ or $2$. However, the bound becomes quite loose for small $p,$ and in particular it fails completely for the important case of $p=0.$ For this reason, we present a different bound that is considerably tighter for small $p$ (although slightly looser for large $p$).

The case $p=0$ was studied by Milman & Schechtman (1986) (Section 6.2) and McDiarmid (1989), and later by Talagrand (1995; 1996). The proof of the following theorem (appendix B) follows the method used in Section 5 of Talagrand (1996), with modifications made to extend the proof to arbitrary $p$.

Consider a measurable subset of the cube $\mathcal{A}\subset^{n},$ and a p-norm distance metric $d(x,y)=\|x-y\|_{p}$ for any $p\geq 0.$ We have

Using this result, we can prove a statement analogous to Theorem 2, but for sparse adversarial examples. We present only the case of $p=0,$ but the generalization to the case of other small $p$ using Lemma 4 is straightforward.

Consider the problem setup of Theorem 2. Choose some class $c$ with $f_{c}\leq\frac{1}{2}$, and sample a random data point $x$ from the class distribution $\rho_{c}.$ Then with probability at least

$x$ is misclassified by $\mathcal{C},$ or

$x$ can be adversarially perturbed by modifying at most $\epsilon$ pixels, while still remaining in the unit hypercube.

What if we just show that adversarial examples exist?

Tighter bounds can be obtained if we only guarantee that adversarial examples exist for some data points in a class, without bounding the probability of this event.

Let $\operatorname{supp}(\rho_{c})$ denote the support of $\rho_{c}.$ Then there is a point $x$ with $\rho_{c}(x)>0$ that admits an $\epsilon$-adversarial example if

The bound for the case $p=0$ is valid only if $\epsilon\geq\sqrt{n\log 2/2}.$

Discussion: Can we escape fundamental bounds?

There are a number of ways to escape the guarantees of adversarial examples made by Theorems 1-4. One potential escape is for the class density functions to take on extremely large values (i.e., exponentially large $U_{c}$); the dependence of $U_{c}$ on $n$ is addressed separately in Section 8.

In practice, image datasets might lie on low-dimensional manifolds within the cube, and the support of these distributions could have measure zero, making the density function infinite (i.e., $U_{c}=\infty$). The arguments above are still relevant (at least in theory) in this case; we can expand the data manifold by adding a uniform random noise to each image pixel of magnitude at most $\epsilon_{1}.$ The expanded dataset has positive volume. Then, adversarial examples of this expanded dataset can be crafted with perturbations of size $\epsilon_{2}$. This method of expanding the manifold before crafting adversarial examples is often used in practice. Tramèr et al. (2017a) proposed adding a small perturbation to step off the image manifold before crafting adversarial examples. This strategy is also used during adversarial training (Madry et al., 2017).

Adding a “don’t know” class

The analysis above assumes the classifier assigns a label to every point in the cube. If a classifier has the ability to say “I don’t know,” rather than assign a label to every input, then the region of the cube that is assigned class labels might be very small, and adversarial examples could be escaped even if the other assumptions of Theorem 4 are satisfied. In this case, it would still be easy for the adversary to degrade classifier performance by perturbing images into the “don’t know” class.

Feature squeezing

If decreasing the dimensionality of data does not lead to substantially increased values for $U_{c}$ (we see in Section 8 that this is a reasonable assumption) or loss in accuracy (a stronger assumption), measuring data in lower dimensions could increase robustness. This can be done via an auto-encoder (Meng & Chen, 2017; Shen et al., 2017), JPEG encoding (Das et al., 2018), or quantization (Xu et al., 2017).

Computational hardness

It may be computationally hard to craft adversarial examples because of local flatness of the classification function, obscurity of the classifier function, or other computational difficulties. Computational hardness could prevent adversarial attacks in practice, even if adversarial examples still exist.

Experiments & Effect of Dimensionality

In this section, we discuss the relationship between dimensionality and adversarial robustness, and explore how the predictions made by the theorems above are reflected in experiments.

It is commonly thought that high-dimensional classifiers are more susceptible to adversarial examples than low-dimensional classifiers. This perception is partially motivated by the observation that classifiers on high-resolution image distributions like ImageNet are more easily fooled than low resolution classifiers on MNIST (Tramèr et al., 2017a). Indeed, Theorem 2 predicts that high-dimensional classifiers should be much easier to fool than low-dimensional classifiers, assuming the datasets they classify have comparable probability density limits $U_{c}.$ However, this is not a reasonable assumption; we will see below that high dimensional distributions may be more concentrated than their low-dimensional counterparts.

We study the effects of dimensionality with a thought experiment involving a “big MNIST” image distribution. Given an integer expansion factor $b,$ we can make a big MNIST distribution, denoted $b$-MNIST, by replacing each pixel in an MNIST image with a $b\times b$ array of identical pixels. This expands an original $28\times 28$ image into a $28b\times 28b$ image. Figure 4a shows that, without adversarial training, a classifier on big MNIST is far more susceptible to attacks than a classifier trained on the original MNIST Only the fully-connected layer is modified to handle the difference in dimensionality between datasets..

However, each curve in Figure 4a only shows the attack susceptibility of one particular classifier. In contrast, Theorems 1-4 describe the fundamental limits of susceptibility for all classifiers. These limits are an inherent property of the data distribution. The theorem below shows that these fundamental limits do not depend in a non-trivial way on the dimensionality of the images in big MNIST, and so the relationship between dimensionality and susceptibility in Figure 4a results from the weakness of the training process.

Likewise, if all $b$-MNIST classifiers have $b\epsilon$-adversarial examples with probability $p$ for some $b\geq 1$, then all classifiers on the original MNIST distribution have $\epsilon$-adversarial examples with probability $p$.

We get a better picture of the fundamental limits of MNIST by considering classifiers that are hardened by adversarial trainingAdversarial examples for MNIST/CIFAR-10 were produced as in Madry et al. (2017) using 100-step/20-step PGD. (Figure 4b). These curves display several properties of fundamental limits predicted by our theorems. As predicted by Theorem 5, the $112\times 112$ classifer curve is twice as wide as the $56\times 56$ curve, which in turn is twice as wide as the $28\times 28$ curve. In addition, we see the kind of “phase transition” behavior predicted by Theorem 2, in which the classifier suddenly changes from being highly robust to being highly susceptible as $\epsilon$ passes a critical threshold. For these reasons, it is reasonable to suspect that the adversarially trained classifiers in Figure 4b are operating near the fundamental limit predicted by Theorem 2.

Theorem 5 shows that increased dimensionality does not increase adversarial susceptibility in a fundamental way. But then why are high-dimensional classifiers so easy to fool? To answer this question, we look at the concentration bound $U_{c}$ for object classes. The smallest possible value of $U_{c}$ is 1, which only occurs when images are “spread out” with uniform, uncorrelated pixels. In contrast, adjacent pixels in MNIST (and especially big MNIST) are very highly correlated, and images are concentrated near simple, low-dimensional manifolds, resulting in highly concentrated image classes with large $U_{c}$. Theory predicts that such highly concentrated datasets can be relatively safe from adversarial examples.

We can reduce $U_{c}$ and dramatically increase susceptibility by choosing a more “spread out” dataset, like CIFAR-10, in which adjacent pixels are less strongly correlated and images appear to concentrate near complex, higher-dimensional manifolds. We observe the effect of decreasing $U_{c}$ by plotting the susceptibility of a $56\times 56$ MNIST classifier against a classifier for CIFAR-10 (Figure 4, right). The former problem lives in $3136$ dimensions, while the latter lives in $3072,$ and both have 10 classes. Despite the structural similarities between these problems, the decreased concentration of CIFAR-10 results in vastly more susceptibility to attacks, regardless of whether adversarial training is used. The theory above suggests that this increased susceptibility is caused at least in part by a shift in the fundamental limits for CIFAR-10, rather than the weakness of the particular classifiers we chose.

Informally, the concentration limit $U_{c}$ can be interpreted as a measure of image complexity. Image classes with smaller $U_{c}$ are likely concentrated near high-dimensional complex manifolds, have more intra-class variation, and thus more apparent complexity. An informal interpretation of Theorem 2 is that “high complexity” image classes are fundamentally more susceptible to adversarial examples, and Figure 4 suggests that complexity (rather than dimensionality) is largely responsible for differences we observe in the effectiveness of adversarial training for different datasets.

So…are adversarial examples inevitable?

The question of whether adversarial examples are inevitable is an ill-posed one. Clearly, any classification problem has a fundamental limit on robustness to adversarial attacks that cannot be escaped by any classifier. However, we have seen that these limits depend not only on fundamental properties of the dataset, but also on the strength of the adversary and the metric used to measure perturbations. This paper provides a characterization of these limits and how they depend on properties of the data distribution. Unfortunately, it is impossible to know the exact properties of real-world image distributions or the resulting fundamental limits of adversarial training for specific datasets. However, the analysis and experiments in this paper suggest that, especially for complex image classes in high-dimensional spaces, these limits may be far worse than our intuition tells us.

Acknowledgements

T. Goldstein and his students were generously supported by DARPA Lifelong Learning Machines (FA8650-18-2-7833), the Office of Naval Research (N00014-17-1-2078), the DARPA Young Faculty Award program (D18AP00055), and the Sloan Foundation. The work of C. Studer was supported in part by Xilinx Inc., and by the US NSF under grants ECCS-1408006, CCF-1535897, CCF-1652065, and CNS-1717559.

References

Appendix A Proof of lemma 3

We now prove Lemma 3. To do this, we begin with a classical isoperimetric inequality for random Gaussian variables. Unlike the case of a cube, tight geometric isoperimetric inequalities exist in this case. We then prove results about the cube by creating a mapping between uniform random variables on the cube and random Gaussian vectors.

which is the cumulative density of a Gaussian curve.

The following Lemma was first proved in Sudakov & Tsirelson (1974), and an elementary proof was given in Bobkov et al. (1997).

Using this result we can now give a proof of Lemma 3.

Since $\frac{\partial}{\partial z_{i}}\Phi(z)\leq\frac{1}{\sqrt{2\pi}},$ we also have

where we have used the identity $\|u\|_{p}\leq n^{1/\min(p,2)-1/2}\|u\|_{2}.$

Now, consider any set $\mathcal{A}$ in the cube, and let $\mathcal{B}=\Phi^{-1}(\mathcal{A}).$ From equation 10, we see that

where $\alpha=\Phi^{-1}(\sigma[\mathcal{A}]).$

To obtain the simplified formula in the theorem, we use the identity

which is valid for $x>0,$ and can be found in Abramowitz & Stegun (1965).

Appendix B Proof of Lemma 4

Consider a probability space $\Omega$ with measure $\mu.$ For $g:\Omega\to,$ we have

Our proof of Lemma 3 follows the three-step process of Talagrand illustrated in Talagrand (1995). We begin by proving the bound

where $f(x,\mathcal{A})=\min_{y\in\mathcal{A}}\sum_{i}|x_{i}-y_{i}|^{p}=d_{p}^{p}(x,\mathcal{A})$ is a measure of distance from $\mathcal{A}$ to $x$, and $\alpha,t$ are arbitrary positive constants. Once this bound is established, a Markov bound can be used to obtain the final result. Finally, constants are tuned in order to optimize the tightness of the bound.

We start by proving the bound in equation 12 using induction on the dimension. The base case for the induction is $n=1,$ and we have

We now prove the result for $n$ dimensions using the inductive hypothesis. We can upper bound the integral by integrating over “slices” along one dimension. Let $\mathcal{A}\subset^{n}.$ Define

Clearly, the distance from $(\omega,z)$ to $\mathcal{A}$ is at most the distance from $z$ to $\mathcal{A}_{\omega},$ and so

We also have that the distance from $x$ to $\mathcal{A}$ is at most one unit greater than the distance from $x$ to $\mathcal{B}$. This gives us

Now, we apply lemma 6 to equation 13 with $g(\omega)=\alpha[\mathcal{A}_{\omega}]/\alpha[\mathcal{B}]$ to arrive at equation 12.

The second step of the proof is to produce a Markov inequality from equation 12. For the bound in equation 12 to hold, we need

The third step is to optimize the bound by choosing constants. We minimize the right hand side by choosing $t=\frac{4\alpha\epsilon^{p}}{n(\alpha+1)}$ to get

Now, we can simply choose $\alpha=1$ to get the simple bound

or we can choose the optimal value of $\alpha=\sqrt{\frac{2\epsilon^{2p}}{n\log(1/\sigma)}}-1,$ which optimizes the bound in the case $\epsilon^{2p}\geq\frac{n}{2}\log(1/\sigma(\mathcal{A})).$ We arrive at

This latter bound is stronger than we need to prove Lemma 3, but it will come in handy later to prove Theorem 4.

Appendix C Proof of Theorems 2 and 3

We combine the proofs of these results since their proofs are nearly identical. The proofs closely follow the argument of Theorem 1.

The set $\overline{\overline{\mathcal{R}}(\epsilon;h)}$ contains all points that are correctly classified and safe from adversarial perturbations. This region has volume at most $\delta$, and the probability of a sample from the class distribution $\rho_{c}$ lying in this region is at most $U_{c}\delta.$ We then subtract this from 1 to obtain the mass of the class distribution lying in the “unsafe” region $\overline{\mathcal{R}}_{c}.$

Appendix D Proof of Theorem 4

Let $\mathcal{A}$ denote the support of $p_{c},$ and suppose that this support has measure $\operatorname{vol}[\mathcal{A}]=\eta.$ We want to show that, for large enough $\epsilon,$ the expansion $\mathcal{A}(\epsilon,d_{p})$ is larger than half the cube. Since class $c$ occupies less than half the cube, this would imply that $\mathcal{A}(\epsilon,d_{p})$ overlaps with other classes, and so there must be data points in $\mathcal{A}$ with $\epsilon$-adversarial examples.

We start with the case $p>0,$ where we bound $\mathcal{A}(\epsilon,d_{p})$ using equation 2 of Lemma 3. To do this, we need to approximate $\Phi^{-1}(\eta).$ This can be done using the inequality

which holds for $\alpha<0.$ Rearranging, we obtain

Now, if $\alpha=\Phi^{-1}(\eta),$ then $\Phi(\alpha)=\eta,$ and equation 19 gives us $\alpha\geq-\sqrt{\log\frac{1}{4\eta^{2}}}.$ Plugging this into equation 2 of Lemma 3, we get

The quantity on the left will be greater than $\frac{1}{2},$ thus guaranteeing adversarial examples, if

This can be re-arranged to obtain the desired result.

In the case $p=0,$ we need to use equation 17 from the proof of Lemma 3 in Appendix B, which we restate here

This bound is valid, and produces a non-vacuous guarantee of adversarial examples, if

Appendix E Proof of Theorem 5

Assume that any MNIST classifier can be fooled by perturbations of size at most $\epsilon$ with probability at least $p$. To begin, we put a bound on the susceptibility of any $b$-MNIST classifier (for $b\geq 1$) under this assumption. We can classify MNIST images by upsampling them to resolution $28b\times 28b$ and feeding them into a high-resolution “back-end” classifier. After upsampling, an MNIST image with perturbation of norm $\epsilon$ becomes a $28b\times 28b$ image with perturbation of norm $b\epsilon.$ The classifier we have constructed takes low-resolution images as inputs, and so by assumption it is fooled with probability at least $p$. However, the low-resolution classifier is fooled only when the high-resolution “back-end” classifier is fooled, and so the high-resolution classifier is fooled with probability at least $p$ at well. Note that we can build this two-scale classifier using any high-resolution classifier as a back-end, and so this bound holds uniformly over all high-resolution classifiers.

Likewise, suppose we classify $b$-MNIST images (for integer $b\geq 1$) by downsampling them to the original $28\times 28$ resolution (by averaging pixel blocks) and feeding them into a “back-end” low-resolution classifier. After downsampling, a $28b\times 28b$ image with perturbation of norm $b\epsilon$ becomes a $28\times 28$ image with perturbation of norm at most $\epsilon.$ Whenever the high-resolution classifier is fooled, it is only because the back-end classifier is fooled by a perturbation of size at most $\epsilon,$ and this happens with probability at least $p$.