Intriguing properties of neural networks

(d) Unit senstive to diagonal straight

stroke.

Figure 1: An MNIST experiment. The figure shows images that maximize the activation of various units

(maximum stimulation in the natural basis direction). Images within each row share semantic properties.

(a) Direction sensitive to upper straight

stroke, or lower round stroke.

(b) Direction sensitive to lower left loop.

(c) Direction senstive to round top stroke.

(d) Direction sensitive to right, upper

round stroke.

Figure 2: An MNIST experiment. The figure shows images that maximize the activations in a random direction

(maximum stimulation in a random basis). Images within each row share semantic properties.

where I is a held-out set of images from the data distribution that the network was not trained on

and ei is the natural basis vector associated with the i-th hidden unit.

Our experiments show that any random direction v ∈ Rn gives rise to similarly interpretable se-

mantic properties. More formally, we find that images x(cid:48) are semantically related to each other, for

many x(cid:48) such that

x(cid:48) = arg max

(cid:104)φ(x), v(cid:105)

x∈I

This suggests that the natural basis is not better than a random basis for inspecting the properties

of φ(x). This puts into question the notion that neural networks disentangle variation factors across

coordinates.

First, we evaluated the above claim using a convolutional neural network trained on MNIST. We

used the MNIST test set for I. Figure 1 shows images that maximize the activations in the natural

basis, and Figure 2 shows images that maximize the activation in random directions. In both cases

the resulting images share many high-level similarities.

Next, we repeated our experiment on an AlexNet, where we used the validation set as I. Figures 3

and 4 compare the natural basis to the random basis on the trained network. The rows appear to be

semantically meaningful for both the single unit and the combination of units.

Although such analysis gives insight on the capacity of φ to generate invariance on a particular

subset of the input distribution, it does not explain the behavior on the rest of its domain. We shall

see in the next section that φ has counterintuitive properties in the neighbourhood of almost every

point form data distribution.

4 Blind Spots in Neural Networks

So far, unit-level inspection methods had relatively little utility beyond confirming certain intuitions

regarding the complexity of the representations learned by a deep neural network [6, 13, 7, 4].

Global, network level inspection methods can be useful in the context of explaining classification

decisions made by a model [1] and can be used to, for instance, identify the parts of the input which

led to a correct classification of a given visual input instance (in other words, one can use a trained

3

(a) Unit sensitive to white flowers.

(b) Unit sensitive to postures.

(c) Unit senstive to round, spiky flowers.

(d) Unit senstive to round green or yellow

objects.

Figure 3: Experiment performed on ImageNet. Images stimulating single unit most (maximum stimulation in

natural basis direction). Images within each row share many semantic properties.

(a) Direction sensitive to white, spread

flowers.

(b) Direction sensitive to white dogs.

(c) Direction sensitive to spread shapes.

(d) Direction sensitive to dogs with brown

heads.

Figure 4: Experiment performed on ImageNet. Images giving rise to maximum activations in a random direc-

tion (maximum stimulation in a random basis). Images within each row share many semantic properties.

model for weakly-supervised localization). Such global analyses are useful in that they can make us

understand better the input-to-output mapping represented by the trained network.

Generally speaking, the output layer unit of a neural network is a highly nonlinear function of its

input. When it is trained with the cross-entropy loss (using the Softmax activation function), it

represents a conditional distribution of the label given the input (and the training set presented so

far). It has been argued [2] that the deep stack of non-linear layers in between the input and the

output unit of a neural network are a way for the model to encode a non-local generalization prior

over the input space. In other words, it is assumed that is possible for the output unit to assign non-

significant (and, presumably, non-epsilon) probabilities to regions of the input space that contain no

training examples in their vicinity. Such regions can represent, for instance, the same objects from

different viewpoints, which are relatively far (in pixel space), but which share nonetheless both the

label and the statistical structure of the original inputs.

It is implicit in such arguments that local generalization—in the very proximity of the training

examples—works as expected. And that in particular, for a small enough radius ε > 0 in the vicinity

of a given training input x, an x + r satisfying ||r|| < ε will get assigned a high probability of the

correct class by the model. This kind of smoothness prior is typically valid for computer vision

problems. In general, imperceptibly tiny perturbations of a given image do not normally change the

underlying class.

Our main result is that for deep neural networks, the smoothness assumption that underlies many

kernel methods does not hold. Specifically, we show that by using a simple optimization procedure,

we are able to find adversarial examples, which are obtained by imperceptibly small perturbations

to a correctly classified input image, so that it is no longer classified correctly.

In some sense, what we describe is a way to traverse the manifold represented by the network in an

efficient way (by optimization) and finding adversarial examples in the input space. The adversarial

examples represent low-probability (high-dimensional) “pockets” in the manifold, which are hard to

efficiently find by simply randomly sampling the input around a given example. Already, a variety

of recent state of the art computer vision models employ input deformations during training for

4

increasing the robustness and convergence speed of the models [9, 13]. These deformations are,

however, statistically inefficient, for a given example: they are highly correlated and are drawn from

the same distribution throughout the entire training of the model. We propose a scheme to make this

process adaptive in a way that exploits the model and its deficiencies in modeling the local space

around the training data.

We make the connection with hard-negative mining explicitly, as it is close in spirit: hard-negative

mining, in computer vision, consists of identifying training set examples (or portions thereof) which

are given low probabilities by the model, but which should be high probability instead, cf. [5]. The

training set distribution is then changed to emphasize such hard negatives and a further round of

model training is performed. As shall be described, the optimization problem proposed in this work

can also be used in a constructive way, similar to the hard-negative mining principle.

4.1 Formal description

We denote by f : Rm −→ {1 . . . k} a classifier mapping image pixel value vectors to a discrete

label set. We also assume that f has an associated continuous loss function denoted by lossf :

Rm × {1 . . . k} −→ R+. For a given x ∈ Rm image and target label l ∈ {1 . . . k}, we aim to solve

the following box-constrained optimization problem:

• Minimize (cid:107)r(cid:107)2 subject to:

1. f (x + r) = l

2. x + r ∈ [0, 1]m

The minimizer r might not be unique, but we denote one such x + r for an arbitrarily chosen

minimizer by D(x, l). Informally, x + r is the closest image to x classified as l by f . Obviously,

D(x, f (x)) = f (x), so this task is non-trivial only if f (x) (cid:54)= l. In general, the exact computation

of D(x, l) is a hard problem, so we approximate it by using a box-constrained L-BFGS. Concretely,

we find an approximation of D(x, l) by performing line-search to find the minimum c > 0 for which

the minimizer r of the following problem satisfies f (x + r) = l.

• Minimize c|r| + lossf (x + r, l) subject to x + r ∈ [0, 1]m

This penalty function method would yield the exact solution for D(X, l) in the case of convex

losses, however neural networks are non-convex in general, so we end up with an approximation in

this case.

4.2 Experimental results

Our “minimimum distortion” function D has the following intriguing properties which we will sup-

port by informal evidence and quantitative experiments in this section:

1. For all the networks we studied (MNIST, QuocNet [10], AlexNet [9]), for each sam-

ple, we have always managed to generate very close, visually hard to distinguish, ad-

versarial examples that are misclassified by the original network (see figure 5 and

http://goo.gl/huaGPb for examples).

2. Cross model generalization: a relatively large fraction of examples will be misclassified by

networks trained from scratch with different hyper-parameters (number of layers, regular-

ization or initial weights).

3. Cross training-set generalization a relatively large fraction of examples will be misclassi-

fied by networks trained from scratch on a disjoint training set.

The above observations suggest that adversarial examples are somewhat universal and not just the

results of overfitting to a particular model or to the specific selection of the training set. They also

suggest that back-feeding adversarial examples to training might improve generalization of the re-

sulting models. Our preliminary experiments have yielded positive evidence on MNIST to support

this hypothesis as well: We have successfully trained a two layer 100-100-10 non-convolutional neu-

ral network with a test error below 1.2% by keeping a pool of adversarial examples a random subset

of which is continuously replaced by newly generated adversarial examples and which is mixed into

5

(a)

(b)

Figure 5: Adversarial examples generated for AlexNet [9].(Left) is a correctly predicted sample, (center) dif-

ference between correct image, and image predicted incorrectly magnified by 10x (values shifted by 128 and

clamped), (right) adversarial example. All images in the right column are predicted to be an “ostrich, Struthio

camelus”. Average distortion based on 64 examples is 0.006508. Plase refer to http://goo.gl/huaGPb

for full resolution images. The examples are strictly randomly chosen. There is not any postselection involved.

(a)

(b)

Figure 6: Adversarial examples for QuocNet [10]. A binary car classifier was trained on top of the last layer

features without fine-tuning. The randomly chosen examples on the left are recognized correctly as cars, while

the images in the middle are not recognized. The rightmost column is the magnified absolute value of the

difference between the two images.

the original training set all the time. We used weight decay, but no dropout for this network. For

comparison, a network of this size gets to 1.6% errors when regularized by weight decay alone and

can be improved to around 1.3% by using carefully applied dropout. A subtle, but essential detail

is that we only got improvements by generating adversarial examples for each layer outputs which

were used to train all the layers above. The network was trained in an alternating fashion, maintain-

ing and updating a pool of adversarial examples for each layer separately in addition to the original

training set. According to our initial observations, adversarial examples for the higher layers seemed

to be significantly more useful than those on the input or lower layers. In our future work, we plan

to compare these effects in a systematic manner.

For space considerations, we just present results for a representative subset (see Table 1) of the

MNIST experiments we performed. The results presented here are consistent with those on a larger

variety of non-convolutional models. For MNIST, we do not have results for convolutional mod-

els yet, but our first qualitative experiments with AlexNet gives us reason to believe that convolu-

tional networks may behave similarly as well. Each of our models were trained with L-BFGS until

convergence. The first three models are linear classifiers that work on the pixel level with various

weight decay parameters λ. All our examples use quadratic weight decay on the connection weights:

lossdecay = λ (cid:80) w2

i /k added to the total loss, where k is the number of units in the layer. Three

of our models are simple linear (softmax) classifier without hidden units (FC10(λ)). One of them,

FC10(1), is trained with extremely high λ = 1 in order to test whether it is still possible to generate

adversarial examples in this extreme setting as well.Two other models are a simple sigmoidal neural

network with two hidden layers and a classifier. The last model, AE400-10, consists of a single layer

sparse autoencoder with sigmoid activations and 400 nodes with a Softmax classifier. This network

has been trained until it got very high quality first layer filters and this layer was not fine-tuned. The

last column measures the minimum average pixel level distortion necessary to reach 0% accuracy

on the training set. The distortion is measure by

between the original x and distorted

(cid:113) (cid:80)(x(cid:48)

i−xi)2

n

6

(a) Even columns: adver-

sarial examples for a lin-

ear

(std-

(FC) classifier

dev=0.06)

(b) Even columns: adver-

sarial examples for a 200-

200-10 sigmoid network

(stddev=0.063)

(c) Randomly

distorted

samples by Gaussian noise

with stddev=1. Accuracy:

51%.

Figure 7: Adversarial examples for a randomly chosen subset of MNIST compared with randomly distorted

examples. Odd columns correspond to original images, and even columns correspond to distorted counterparts.

The adversarial examples generated for the specific model have accuracy 0% for the respective model. Note

that while the randomly distorted examples are hardly readable, still they are classified correctly in half of the

cases, while the adversarial examples are never classified correctly.

Model Name

Description

Training error

Test error

Av. min. distortion

FC10(10−4)

FC10(10−2)

FC10(1)

FC100-100-10

FC200-200-10

AE400-10

Softmax with λ = 10−4

Softmax with λ = 10−2

Softmax with λ = 1

Sigmoid network λ = 10−5, 10−5, 10−6

Sigmoid network λ = 10−5, 10−5, 10−6

Autoencoder with Softmax λ = 10−6

6.7%

10%

21.2%

0%

0%

0.57%

7.4%

9.4%

20%

1.64%

1.54%

1.9%

0.062

0.1

0.14

0.058

0.065

0.086

Table 1: Tests of the generalization of adversarial instances on MNIST.

FC10(10−4)

FC10(10−2)

FC10(1)

FC100-100-10

FC200-200-10

AE400-10

Av. distortion

FC10(10−4)

FC10(10−2)

FC10(1)

FC100-100-10

FC200-200-10

AE400-10

Gaussian noise, stddev=0.1

Gaussian noise, stddev=0.3

100%

87.1%

71.9%

28.9%

38.2%

23.4%

5.0%

15.6%

11.7%

100%

76.2%

13.7%

14%

16%

10.1%

11.3%

22.7%

35.2%

100%

21.1%

23.8%

24.8%

18.3%

22.7%

2%

35.9%

48.1%

100%

20.3%

9.4%

0%

5%

3.9%

27.3%

47%

6.6%

100%

6.6%

0%

4.3%

2.7%

9.8%

34.4%

2%

2.7%

100%

0.8%

3.1%

0.062

0.1

0.14

0.058

0.065

0.086

0.1

0.3

Table 2: Cross-model generalization of adversarial examples. The columns of the Tables show the error induced

by distorted examples fed to the given model. The last column shows average distortion wrt. original training

set.

x(cid:48) images, where n = 784 is the number of image pixels. The pixel intensities are scaled to be in

the range [0, 1].

In our first experiment, we generated a set of adversarial instances for a given network and fed

these examples for each other network to measure the proportion of misclassified instances. The

last column shows the average minimum distortion that was necessary to reach 0% accuracy on the

whole training set. The experimental results are presented in Table 2. The columns of Table 2 show

the error (proportion of misclassified instances) on the so distorted training sets. The last two rows

are given for reference showing the error induced when distorting by the given amounts of Gaussian

noise. Note that even the noise with stddev 0.1 is greater than the stddev of our adversarial noise

for all but one of the models. Figure 7 shows a visualization of the generated adversarial instances

for two of the networks used in this experiment The general conclusion is that adversarial examples

tend to stay hard even for models trained with different hyperparameters. Although the autoencoder

based version seems most resilient to adversarial examples, it is not fully immune either.

Still, this experiment leaves open the question of dependence over the training set. Does the hardness

of the generated examples rely solely on the particular choice of our training set as a sample or does

this effect generalize even to models trained on completely different training sets?

7

Model

Error on P1

Error on P2

Error on Test

Min Av. Distortion

FC100-100-10: 100-100-10 trained on P1

FC123-456-10: 123-456-10 trained on P1

FC100-100-10’ trained on P2

0%

0%

2.3%

2.4%

2.5%

0%

2%

2.1%

2.1%

0.062

0.059

0.058

Table 3: Models trained to study cross-training-set generalization of the generated adversarial examples. Errors

presented in Table correpond to original not-distorted data, to provide a baseline.

FC100-100-10

FC123-456-10

FC100-100-10’

Distorted for FC100-100-10 (av. stddev=0.062)

Distorted for FC123-456-10 (av. stddev=0.059)

Distorted for FC100-100-10’ (av. stddev=0.058)

Gaussian noise with stddev=0.06

Distorted for FC100-100-10 amplified to stddev=0.1

Distorted for FC123-456-10 amplified to stddev=0.1

Distorted for FC100-100-10’ amplified to stddev=0.1

Gaussian noise with stddev=0.1

100%

6.25%

8.2%

2.2%

100%

96%

27%

2.6%

26.2%

100%

8.2%

2.6%

98%

100%

50%

2.8%

5.9%

5.1%

100%

2.4%

43%

22%

100%

2.7%

Table 4: Cross-training-set generalization error rate for the set of adversarial examples generated for different

models. The error induced by a random distortion to the same examples is displayed in the last row.

To study cross-training-set generalization, we have partitioned the 60000 MNIST training images

into two parts P1 and P2 of size 30000 each and trained three non-convolutional networks with

sigmoid activations on them: Two, FC100-100-10 and FC123-456-10, on P1 and FC100-100-10 on

P2. The reason we trained two networks for P1 is to study the cumulative effect of changing the

hypermarameters and the training sets at the same time. Models FC100-100-10 and FC100-100-

10 share the same hyperparameters: both of them are 100-100-10 networks, while FC123-456-10

has different number of hidden units. In this experiment, we were distorting the elements of the

test set rather than the training set. Table 3 summarizes the basic facts about these models. After

we generate adversarial examples with 100% error rates with minimum distortion for the test set,

we feed these examples to the each of the models. The error for each model is displayed in the

corresponding column of the upper part of Table 4. In the last experiment, we magnify the effect of

our distortion by using the examples x + 0.1 x(cid:48)−x

rather than x(cid:48). This magnifies the distortion

(cid:107)x(cid:48)−x(cid:107)2

on average by 40%, from stddev 0.06 to 0.1. The so distorted examples are fed back to each of the

models and the error rates are displayed in the lower part of Table 4. The intriguing conclusion is

that the adversarial examples remain hard for models trained even on a disjoint training set, although

their effectiveness decreases considerably.

4.3 Spectral Analysis of Unstability

The previous section showed examples of deep networks resulting from purely supervised training

which are unstable with respect to a peculiar form of small perturbations. Independently of their

generalisation properties across networks and training sets, the adversarial examples show that there

exist small additive perturbations of the input (in Euclidean sense) that produce large perturbations

at the output of the last layer. This section describes a simple procedure to measure and control the

additive stability of the network by measuring the spectrum of each rectified layer.

Mathematically, if φ(x) denotes the output of a network of K layers corresponding to input x and

trained parameters W , we write

φ(x) = φK(φK−1(. . . φ1(x; W1); W2) . . . ; WK) ,

where φk denotes the operator mapping layer k − 1 to layer k. The unstability of φ(x) can be

explained by inspecting the upper Lipschitz constant of each layer k = 1 . . . K, defined as the

constant Lk > 0 such that

∀ x, r , (cid:107)φk(x; Wk) − φk(x + r; Wk)(cid:107) ≤ Lk(cid:107)r(cid:107) .

The resulting network thus satsifies (cid:107)φ(x) − φ(x + r)(cid:107) ≤ L(cid:107)r(cid:107), with L = (cid:81)K

k=1 Lk.

A half-rectified layer (both convolutional or fully connected) is defined by the mapping

φk(x; Wk, bk) = max(0, Wkx+bk). Let (cid:107)W (cid:107) denote the operator norm of W (i.e., its largest singu-

8

Layer

Conv. 1

Conv. 2

Conv. 3

Conv. 4

Conv. 5

FC. 1

FC. 2

FC. 3

Size

Stride

Upper bound

3 × 11 × 11 × 96

96 × 5 × 5 × 256

256 × 3 × 3 × 384

384 × 3 × 3 × 384

384 × 3 × 3 × 256

9216 × 4096

4096 × 4096

4096 × 1000

4

1

1

1

1

N/A

N/A

N/A

2.75

10

7

7.5

11

3.12

4

4

Table 5: Frame Bounds of each rectified layer of the network from [9].

lar value). Since the non-linearity ρ(x) = max(0, x) is contractive, i.e. satisfies (cid:107)ρ(x)−ρ(x+r)(cid:107) ≤

(cid:107)r(cid:107) for all x, r; it follows that

(cid:107)φk(x; Wk)−φk(x+r; Wk)(cid:107) = (cid:107) max(0, Wkx+bk)−max(0, Wk(x+r)+bk)(cid:107) ≤ (cid:107)Wkr(cid:107) ≤ (cid:107)Wk(cid:107)(cid:107)r(cid:107) ,

and hence Lk ≤ (cid:107)Wk(cid:107). On the other hand, a max-pooling layer φk is contractive:

∀ x , r , (cid:107)φk(x) − φk(x + r)(cid:107) ≤ (cid:107)r(cid:107) ,

since its Jacobian is a projection onto a subset of the input coordinates and hence does not expand

the gradients. Finally, if φk is a contrast-normalization layer

φk(x) =

(cid:16)

x

(cid:15) + (cid:107)x(cid:107)2

(cid:17)γ ,

one can verify that

∀ x , r , (cid:107)φk(x) − φk(x + r)(cid:107) ≤ (cid:15)−γ(cid:107)r(cid:107)

for γ ∈ [0.5, 1], which corresponds to most common operating regimes.

It results that a conservative measure of the unstability of the network can be obtained by simply

computing the operator norm of each fully connected and convolutional layer. The fully connected

case is trivial since the norm is directly given by the largest singular value of the fully connected

matrix. Let us describe the convolutional case. If W denotes a generic 4-tensor, implementing a

convolutional layer with C input features, D output features, support N × N and spatial stride ∆,

W x =

(cid:40) C

(cid:88)

c=1

xc (cid:63) wc,d(n1∆, n2∆) ; d = 1 . . . , D

,

(cid:41)

where xc denotes the c-th input feature image, and wc,d is the spatial kernel corresponding to input

feature c and output feature d, by applying Parseval’s formula we obtain that its operator norm is

given by

sup

ξ∈[0,N ∆−1)2

where A(ξ) is a D × (C · ∆2) matrix whose rows are

(cid:107)W (cid:107) =

(cid:107)A(ξ)(cid:107) ,

(1)

∀ d = 1 . . . D , A(ξ)d =

(cid:16)

∆−2

(cid:100)wc,d(ξ + l · N · ∆−1) ; c = 1 . . . C , l = (0 . . . ∆ − 1)2(cid:17)

,

and (cid:100)wc,d is the 2-D Fourier transform of wc,d:

(cid:88)

(cid:100)wc,d(ξ) =

u∈[0,N )2

wc,d(u)e−2πi(u·ξ)/N 2

.

Table 5 shows the upper Lipschitz bounds computed from the ImageNet deep convolutional network

of [9], using (1). It shows that instabilities can appear as soon as in the first convolutional layer.

These results are consistent with the exsitence of blind spots constructed in the previous section,

but they don’t attempt to explain why these examples generalize across different hyperparameters

or training sets. We emphasize that we compute upper bounds: large bounds do not automatically

translate into existence of adversarial examples; however, small bounds guarantee that no such ex-

amples can appear. This suggests a simple regularization of the parameters, consisting in penalizing

each upper Lipschitz bound, which might help improve the generalisation error of the networks.

9

5 Discussion

We demonstrated that deep neural networks have counter-intuitive properties both with respect to

the semantic meaning of individual units and with respect to their discontinuities. The existence of

the adversarial negatives appears to be in contradiction with the network’s ability to achieve high

generalization performance. Indeed, if the network can generalize well, how can it be confused

by these adversarial negatives, which are indistinguishable from the regular examples? Possible

explanation is that the set of adversarial negatives is of extremely low probability, and thus is never

(or rarely) observed in the test set, yet it is dense (much like the rational numbers), and so it is found

near every virtually every test case. However, we don’t have a deep understanding of how often

adversarial negatives appears, and thus this issue should be addressed in a future research.

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