1 Introduction
Adversarial attacks are some of the most puzzling and burning issues in modern machine learning. An adversarial attack refers to a small, imperceptible change of an input maliciously designed to fool the result of a machine learning algorithm. Since the seminal work of (Szegedy et al., 2014)
exhibiting this intriguing phenomenon in the context of deep learning, a wealth of results have been published on designing attacks
(Goodfellow et al., 2015; Papernot et al., 2016a; MoosaviDezfooli et al., 2016; Kurakin et al., 2016; Carlini & Wagner, 2017; MoosaviDezfooli et al., 2017) and defenses (Goodfellow et al., 2015; Papernot et al., 2016b; Guo et al., 2018; Meng & Chen, 2017; Samangouei et al., 2018; Madry et al., 2018)), or on trying to understand the very nature of this phenomenon (Fawzi et al., 2018b; SimonGabriel et al., 2018; Fawzi et al., 2018a, 2016).Among the defense strategies, randomization has proven effective in many contexts. It consists in injecting random noise (both during training and inference phases) inside the network architecture, i.e. at a given layer of the network, including the input. The noise can be drawn either from Gaussian (Liu et al., 2018; Lecuyer et al., 2018; Rakin et al., 2018), Laplace (Lecuyer et al., 2018), Uniform with fixed parameter (Xie et al., 2018), or Multinomial with fixed parameter (Dhillon et al., 2018) distributions. All these works, despite coming with relevant ideas and showing good empirical results, lack theoretical justifications and guarantees. It is also worth noting that almost all the used noise distributions belong to the Exponential family. This raises the following questions: why randomization works well in practice? what is profound in using the Exponential family?, and to what extent a noise drawn from the Exponential family preserves robustness (in a sense to be defined) to adversarial attacks. Our work answers these questions. The following informal theorem (see Theorem 3 for a more formal version) summarizes our main result:
Theorem (Main resultinformal).
Let us consider
a Neural Network with
layers. For any and for any perturbation we denote the Network truncated at the th layer, and the sensitivity of to . If at prediction time, we inject noise drawn from an Exponential family with parameter , then the sensitivity of the network is in with a nondecreasing function. Therefore, for a small enough (in norm) parameter , the effect of the perturbation to the network can be made negligible.This theorem provides a sensitivity bound on the output of the network for two close inputs. In that sense, it provides stability guarantees to the network: two close inputs will have close output distributions. To prove this theorem, several milestones should be achieved. First, we need to introduce a definition of robustness to adversarial attacks that is suitable to the randomization defense mechanism. As this mechanism can be mathematically described as a nondeterministic querying process, called probabilistic mapping in the sequel, we propose a formal definition of robustness relying on a metric/divergence between probability measures. A key question arises then about the appropriate metric/divergence for our context. This requires tools for comparing divergences with respect to the introduced robustness definition. Renyi divergence reveals to be a measure of choice, since it satisfies most of the desired properties.
The outline of the paper is hence as follows: Section 2 presents the related work on the randomized defense mechanisms. Section 3 details our main results. It introduces a definition for measuring robustness to adversarial attacks in Subsection 3.2, discusses the choice of divergence measures in Subsection 3.3, and states our main theorem in Subsection 3.4. Finally, Section 4 presents experiments supporting our theoretical claims. To ensure the conciseness, proofs have been pushed into the supplementary materials.
2 Related work
Even though several explicit or implicit definitions can be found in the literature, e.g. (Szegedy et al., 2014; Fawzi et al., 2018a; Bubeck et al., 2018), there is no broadly accepted definition of robustness to adversarial examples attacks. Recently (Diochnos et al., 2018) proposed general definitions and a taxonomy of these. The authors divide the definitions from the literature into three categories: errorregion, predictionchange and corrupted instance. In this paper we introduce a definition of robustness that generalizes the one of predictionchange, in the sense that it relies on probabilistic mappings in arbitrary metric spaces, and is not restricted to classification tasks, as discussed in the sequel.
Noise injection into algorithms to enhance robustness has been used for ages in detection and signal processing tasks, for example with a physical phenomenon called “stochastic resonance” (Zozor & Amblard, 1999; ChapeauBlondeau & Rousseau, 2004; Mitaim & Kosko, 1998). It has also been extensively studied in several machine learning and optimization fields, e.g. robust optimization (BenTal et al., 2009) and data augmentation techniques (Perez & Wang, 2017). Recently, noise injection techniques have been adopted by the adversarial defense community, especially for neural networks, with very promising results. The first technique explicitly using randomization at inference time as a defense appeared in 2017 during the NIPS defense challenge (Xie et al., 2018). This method uniformly samples from over 12000 geometric transformations of the image to select a substitute image to feed the network. Then (Dhillon et al., 2018) proposed to use stochastic activation pruning based on a multinomial distribution for adversarial defense.
Gaussian noise injection has also been well investigated. Recent papers (Liu et al., 2018; Rakin et al., 2018) propose to inject Gaussian noise directly on the activation of selected layers both at training and inference time. In (Lecuyer et al., 2018), the authors proposed a randomization method by exploiting the link between differential privacy (Dwork et al., 2014) and adversarial robustness. Their framework inheriting some theoretical results from the differential privacy work, is based on injecting Laplace or Gaussian noise at training and inference time. In general, noise drawn from continuous distributions is used to alter the activation of one layer or more, whereas noise drawn from discrete distributions is used to alter either the image or the architecture of the network. However efficient in practice, these methods lack theoretical arguments on every part of the procedure (when/where to inject noise, what noise to use, etc.).
Since the initial discovery of adversarial examples, a wealth of non randomized defense approaches have been proposed, inspired by various machine learning domains such as image reconstruction (Meng & Chen, 2017; Samangouei et al., 2018) or robust learning (Goodfellow et al., 2015; Madry et al., 2018). Even if these methods have their own merits, they fall short to defend against universal attacks. We hypothesize that the randomization strategy is the principled one, hence motivating the current study.
3 Using Exponential family for adversarial robustness
In the following of the paper, we will consider a measurable metric input space . We denote a norm on and we suppose the inputs are samples from a distribution .
3.1 Adversarial attacks problem
Let us consider a classification task^{1}^{1}1Note that the definition of robustness we provide generalizes to other tasks. over . A data has a true label . Let
be a trained classifier over
. The problem of generating an adversarial example from an input writes:(1) 
where is the target class (with ). Equation (1) presents a targeted attack model. For the untargeted attack problem, the condition is changed from to . In this paper, we treat indifferently targeted and untargeted attacks. Figure 1 illustrates the principle of an adversarial attack on : a small perturbation applied on an input fools the classifier. The dashed line represents the boundary decision between two classes. The dashed circle around represents the maximal amount of noise keeping the adversarial example perceptually close to (in the case of images). The inputs and looks similar but they are classified with two different labels.
3.2 A general definition of robustness to adversarial attacks
As we will inject noise in our algorithm in order to defend against adversarial attacks, we need to introduce the notion of “probabilistic mapping”. Let us consider the output space, and a  over .
Definition 1 (Probabilistic mapping).
Let be a measurable space. For any space , a probabilistic mapping from to is a mapping where is the set of probability measures over . To obtain a numerical output out of this mechanism, one needs to sample .
This definition does not depend on the nature of as long as is measurable. In that sense, could be either the label space or any intermediate space corresponding to the outputs of one hidden layer of a neural network. Moreover, any mapping can be considered as a probabilistic mapping, whether it explicitly injects noise (as in (Lecuyer et al., 2018; Rakin et al., 2018; Dhillon et al., 2018)) or not. In fact, any deterministic mapping can be considered as a probabilistic mapping, since it can be characterized by a Dirac measure. Accordingly, the definition of a probabilistic mapping is fully general and equally treats networks with or without noise injection. So far, there exists no definition of robustness against adversarial attacks that comply with the notion of probabilistic mappings. We settle that by generalizing the notion of predictionchange risk initially introduced in (Diochnos et al., 2018) for deterministic classifiers. Given a classifier it is defined as follows:
a where for any , .
In our case, as probabilistic mappings are considered, we need to generalize this notion to probability measures. This leads to the following definition.
Definition 2 (Adversarial robustness).
Let be a metric/divergence on . The probabilistic mapping is said to be robust if:
Finally, conversely to the previous work, ours does not restrict neither the task (regression, classification, reinforcement learning, etc.) nor the type of distribution the perturbation is drawn from. As
is an arbitrary space, this notion of robustness for probabilistic mappings is fully general, but, in this paper, our final goal remains to ensure robustness for a classification task.One needs to be careful when considering adversarial robustness regarding this definition: a robust mapping does not necessarily ensures accuracy. In fact, if is the space of labels and
respect the same uniform distribution over
, then for every metric/divergence , one has , and the accuracy will be the one of a random classifier. In the following, robust will mean robust in the sense of Definition 2. Our definition depends on parameters (,,) and on the metric/divergence one chooses to consider between probability measures. Lemma 1 gives some natural insights on the monotony of the robustness according to the parameters, and the probability metric at hand.Lemma 1.
Let u be a probabilistic mapping, and let , and be two metrics/divergences on . If there is a non decreasing function such that , , then the following assertion holds:
The metric/divergence one chooses to consider Definition 2 is intrinsically linked to the notion of robustness that will be preserved.
3.3 On the choice of the metric/divergence
At this point, a natural question to be asked is the choice of the metric/divergence we will choose to defend against adversarial attacks. The main notions that govern the selection of an appropriate metric/divergence are what we call coherence, strength, and computational tractability. A metric/divergence is said to be coherent if it corresponds to the task at hand (e.g. classification tasks are intrinsically linked to discrete/trivial metrics, conversely to regression tasks). The strength of a metric/divergence refers to its ability to cover (dominate) a wide class of others in the sense of Lemma 1. In the following, we will focus on both the total variation metric and the Renyi divergence, that we consider as respectively the most coherent with the classification task using probabilistic mappings, and the strongest divergence we studied. We first discuss how total variation metric is coherent with randomized classifiers but suffers from computational issues. Hopefully, the Renyi divergence provides good guarantees about adversarial robustness, enjoys nice computational properties, in particular when considering Exponential family distributions, and is strong enough to dominate a wide range of metrics/divergences including total variation.
Let and be two measures in , both dominated by a third measure . The trivial distance is the simplest distance one can define between and .
The Trivial distance:
In the deterministic case, it is straightforward to compute (since the numerical output of the algorithm characterizes its associated measure), but this is not the case in general. In fact one might not have access to the true distribution of the mapping, but just to the numerical outputs. Therefore, one needs to consider more sophisticated metrics/divergences, such as the Total variation distance.
The Total variation distance:
The total variation distance is one of the most broadly used probability metrics. It admits several very simple interpretations, and is a very useful tool in many mathematical fields such as probability theory, Bayesian statistics, coupling or transportation theory. In transportation theory, it can be rewritten as the solution of the MongeKantorovich problem with the cost function
:where the infimum is taken over all joint probability measures on with marginals and . According to this interpretation, it seems quite natural to consider the total variation distance as a relaxation of the trivial distance on (see (Villani, 2008) for details). In the deterministic case, the total variation and the trivial distance coincides. In general, the total variation allows a finer analysis of the probabilistic mappings than the trivial distance. But it suffers from a high computational complexity. In the following of the paper we will show how to ensure robustness regarding TV distance.
Finally, denoting by and
the respective probability distributions with respect to
, let us recall the Renyi divergence definition (Rényi, 1961):The Renyi divergence of order :
The Renyi divergence is a generalized measure defined on the interval
, where it equals the KullbackLeibler divergence when
(that will be denoted ), and the maximum divergence when . It also has the very special property of being non decreasing with respect to. This divergence is very common in machine learning, especially in its KullbackLeibler form as it is used widely used as the loss function (cross entropy) of classification algorithms.
The choice of Renyi divergence is motivated by its good properties regarding the bounding of TV distance, good computation with Exponential family distributions and also a good behavior when it comes to ensure robustness for a neural network.
In the following we prove that Renyi divergence implies TVrobustness.
Theorem 1 (Renyirobustness implies TVrobustness).
Let be a probabilistic mapping, then :
for .
An important property about Renyirobustness is what is called the Data processing inequality. It is a wellknown inequality from information theory which states that “postprocessing cannot increase information” (Cover & Thomas, 2012; Beaudry & Renner, 2012). In our case, if we consider a Renyirobust probabilistic mapping, composing it with a deterministic mapping maintains Renyirobustness with the same level.
Theorem 2 (Data processing inequality).
Let consider a probabilistic mapping . Let denote a deterministic mapping. If then probability measure defines a probabilistic mapping .
For any if is  robust then is also is  robust.
Data processing inequality will allow us later to inject additive noise after in a neural network and to ensure Renyirobustness.
3.4 Our main result: Exponential family ensures Renyirobustness
For now, the question of what class of noise to add is treated ad hoc, we choose here to investigate one particular class of noise, namely Exponential family distributions, and demonstrate their interest. Let us first recall what the Exponential family is. Without loss of generality, we can restrict our study to an output space ).
Definition 3 (Exponential family).
Let be an open convex set of , and . Let be a measure dominated by (either by the Lesbegue or counting measure), it is said to be part of the Exponential family of parameter (denoted
) if it has the following probability density function:

is a sufficient statistic

a carrier measure (either for Lebesgue or counting measure)

Our main result is the following: by injecting noise from an exponential family distribution, we ensure Renyirobustness up to a certain value.
Theorem 3 (Exponential family ensures robustness).
Let be an open convex subset of . Let be a mapping such that . Let
be a random variable. We denote
the probability measure of the random variable .
If where and have nondecreasing modulus of continuity and .
Then for any , defines a probabilistic mapping that is  robust with .

If is a centered Gaussian random variable with a non degenerated matrix parameter . Then for any , defines a probabilistic mapping that is  robust with .
In simpler words, the previous theorem ensures stability in the neural network regarding the distribution of the output. Intuitively, if two inputs are close with respect to , the output distributions of the network will be close regarding Renyi divergence. So the predicted labels of two close inputs of the network are “more likely” to be the same.
Let us consider a deterministic feed forward neural network
with layers, where corresponds to the neural network truncated at layer . To obtain a robust probabilistic classifier, for any input , we add a noise to layer : where is a random variable from the Exponential family. Then according to Theorems 2 and 3, after adding the noise to the th layer, the whole network defines a probabilistic mapping satisfying Renyirobustness. Figure 2 illustrates our noise injection defense mechanism. refers to the perturbed th layer of networks each respecting robustness, for small enough values of , and . represents the maximal amount of noise, the probabilistic mapping is robust against. It defines a ball (with radius ) on which the outputs of the network are stable. Any example falling within the ball defined by will be mapped close to the mapped version of as shown for in Figure 2. Otherwise, any example out of this ball may be mapped farther (potentially crossing the decision boundary). Depending on the magnitude of , the mapping could be made more robust as shown for in Figure 2. To summarize, if the outputs of the mapping are stable on a ball that includes the set of adversarial examples visually imperceptible, the probabilistic mapping will be robust to adversarial examples.3.5 On the need for injecting noise in the training phase
So far, we have designed an algorithm for neural networks to ensure robustness at inference time. But simply injecting noise at inference time destroys the accuracy of the algorithm. Thus one needs to also inject noise during the training phase as well. The justification comes from the distribution shift (Sugiyama & Kawanabe, 2012). Distribution shift occurs when the training distribution differs from the test distribution. This implies that the hypothesis minimizing the empirical risk is not consistent, i.e. it does not converge to the true model as the training size increases. A way to circumvent that is to ensure that training and test distributions matches using importance weighting (in the case of covariateshift) or with noise injection in training and test phases (in our case).
4 Experiments
Our main theoretical finding can be summarized as follows: if one adds random noise drawn from an Exponential family distribution at inference time, the network’s predictions are made locally stable to small changes in the input. Hence, small perturbations on the image result in small changes on the predictions. Accordingly, the accuracy of such a network should be close when evaluated either on natural images or their adversarial substitutes. Note that, robustness is not a synonym of accuracy and the local stability does not fix a poorly performing network. In order to build a robust and accurate network, we will use a stateofthe art architecture (developed for natural images) and make it robust through randomized procedures. We hereafter present experiments that confirm the theoretical evidences on the effectiveness of noise injection as a defense method. From these experiments, we discuss the impact of noise injection on the classifiers’ accuracy, and what noise standard deviation would be a reasonable tradeoff in terms of robustness and accuracy.
4.1 Adversarial attacks
As reference adversarial attacks, we consider the followings:
Fast Gradient Method attack. The idea of the fast gradient method with norm () is to find a linear approximation of the following optimization problem:
where is the loss function (usually ) and the label of . Assuming to be small, the previous problem can be relaxed as follows:
The special case of is called Fast Gradient Sign Method (FGSM, (Goodfellow et al., 2015)).
Carlini & Wagner attack. The Carlini & Wagner attack () introduced in (Carlini & Wagner, 2017) writes as:
where is a function such that iff where is the target class. The authors listed some functions:
where is the softmax function and a positive constant.
Instead of using boxconstrained LBFGS (Szegedy et al., 2014) like in the original attack, they use a new variable for :
Then they use binary search to optimize the constant and Adam or SGD for computing the optimum solution.
4.2 Experimental setting
Dataset. We use CIFAR10 dataset, which is composed of training samples and test sample images with resolution and 3 channels.
Architecture. We used a ResNet architecture (He et al., 2016), as it is considered to be stateofart on image classification. More precisely, we use a wide residual network (Zagoruyko & Komodakis, 2016) with layers. It is more memory efficient than very deep residual network and has comparably good performance. We trained every networks with a cross entropy loss.
Training procedure. Our inference procedure consists in adding noise to the network, which can be seen as modifying the test distribution, resulting in a distribution shift. It imposes the training procedure to also inject noise. During training, at each iteration we inject additive noise drawn from an Exponential family distribution on the activation of the first layer of the network. This method has the advantage of being independent of the training method. It only depends on the architecture of the network.
Inference procedure. For a given , the prediction procedure in a classical neural network consists in passing through the successive layers of the network and selects the output label with the maximum a posteriori rule on the activation of the last layer. Our method only differs in one way: when feeding the network with , we add random noise drawn from the same distribution to the same layer as the training phase.
Type of noise. Our experimental framework makes use of noises drawn from continuous distributions (to alter the activations). Being more difficult to compare, the case where a noise is drawn from discrete distribution is left for future work. Indeed, their use heavily depends on the considered transformation (either on the architecture or the image). Therefore, we investigate the noise injections drawn from the following distributions:

Gaussian distribution:
For , . 
Laplace distribution:
For , . 
Exponential distribution:
For , . 
Transformed Weibull ditribution:
For , where .
All these distributions belong to the Exponential family, and respect the hypotheses of Theorem 3.
Evaluation protocol. For every type of noise, we evaluate the accuracy of the obtained classifier on natural images. Then we use an attack to construct an adversarial example for every image in the initial set, and reevaluate the classifier on them. The obtained accuracy is called the accuracy under attack. To be able to make a global analysis, for every type of noise, we reproduce the above procedure for several attacks and noise’s standard deviations. To benchmark the defense mechanisms, we trained a network without injecting noise, neither on training phase nor on inference phase. Note that, the point of our experiment is not to present a stateofthe art classifier but to investigate the use of Exponential family noises as a defense method. This is why we voluntarily use classical and simple methods.
training  inference  No attack  0.3  0.3  0.03  

No  No  0.904  0.893  0.601  0.347  0.029 
Yes  No  0.855  0.854  0.778  0.517  0.030 
Yes  Yes  0.853  0.855  0.807  0.596  0.396 
4.3 Implementation details
Architecture
We learn every model by using a stochastic gradient descent with a momentum of
, and a LeakyReLU with slope
during epochs. The gradient descent is performed using a staircase learning rate starting at . We also apply a Parseval normalization to every convolutions layers (Cisse et al., 2017) during training. Finally, we initialize every layer with a normal random distribution with , where is the layer’s width. We did not use any data augmentation technique nor any pretraining.Adversarial attacks. To evaluate the robustness of defense method, we apply attacks to both defended and undefended networks. First, we deploy Fast Gradient Methods with norm for (). This attacks depends on a parameter bounding the perturbation norm. We choose to investigate noises that are small enough (respectively and ) for the perception of the attacked image to hold. Second, we use the Carlini and Wagner attack (). represent the state of the art of adversarial attacks. It is a powerful and iterative attack that does not have any parameter since it automatically finds the appropriated perturbation bound.
4.4 Results
For every type of noise, we compare the accuracy of the methods for several level of noise intensity, and attacks. Tables 1 and 2 and Figure 7 present our result for every investigated noises on CIFAR10 dataset. The yaxes represents the accuracy (natural image line), and the accuracy under attack (other lines), i.e. the percentage of good response of the network fed with classical/adversarial examples.
Baseline accuracies. As presented in Table 1, we note that the network performs very well on natural images with an accuracy of . attack is not powerful enough to be significantly discussed. But adding adversarial noise from other attacks on images decreases the accuracy of the trained network. Our theoretical work considers that robustness is obtained when noise is injected at inference time. Accordingly, adding noise only during training can give a good accuracy, but the accuracy under attack remains low, especially for powerful attacks such as that makes the accuracy drops from to . Moreover, regarding Section 3, only adding noise at inference phase will lead to poor accuracy. Indeed, this technique makes one fall in a distribution shift since noisy images and natural images are not drawn from the same distributions. Therefore, it both theoretically and empirically justifies to inject noise both at training and testing phases, in order to preserve both accuracy and robustness. This technique presents a significant increase over the others regarding the accuracy under attack, especially for attack, where mixing noise at training and inference is times more effective.
St. Dev.  No attack  0.3  0.3  0.03  

0.01  0.890  0.883  0.722  0.414  0.515 
0.06  0.878  0.874  0.798  0.473  0.473 
0.21  0.841  0.835  0.800  0.555  0.437 
0.32  0.832  0.826  0.808  0.647  0.439 
0.52  0.806  0.800  0.795  0.706  0.444 
0.72  0.780  0.774  0.766  0.701  0.430 
0.93  0.770  0.763  0.758  0.713  0.407 
1.13  0.747  0.739  0.736  0.692  0.388 
1.34  0.726  0.724  0.713  0.687  0.389 
1.54  0.704  0.697  0.690  0.664  0.362 
Robustness by noise injection At a first glance (see Figure 7), the effectiveness of noise injection as a defense method seems natural. In fact, regardless of the type of noise, the noise injection defenses largely outperform the undefended network. We observe the gap between the accuracy and the accuracy under attack, and use this gap to measure the robustness of the different noise injections under the proposed attack. With respect to the amount of injected noise, the classifiers are increasingly robust. Note also that when the amount of noise is big enough, the gap becomes constant.
Robustness/Accuracy tradeoff. When injecting noise as a defense mechanism, regardless of the distribution it is drawn from, we observe that accuracy (and accuracy under attack) decreases when noise intensity grows. Note that, for a strong enough noise intensity both accuracy and accuracy under attack (regardless of the attack) decrease with the same slope. This represents an evidence that at some point, making the quantity of noise grow only impacts the accuracy of the method, and not the robustness anymore (this corresponds to both constant robustness and decreasing accuracy). In that sense, the noise needs to be calibrated to preserve both accuracy and robustness against adversarial attacks, i.e. it needs to be large enough to preserve robustness and small enough to preserve accuracy. Analyzing Figure 7, we find that, in practice, for the evaluated noise, small standard deviations seem to already represent good tradeoffs between robustness and accuracy.
Comparison of noise distributions. Globally, any noise drawn from any distribution of the Exponential family constitutes an efficient defense against adversarial attacks. According to Figure (a)a, Laplace noise performs poorly in comparison to other distributions: the degradation of accuracy with respect to the noise intensity is too fast to be satisfactory. Transformed Weibull (see Figure (d)d) and Gaussian (see Figure (b)b) distributions have similar performances. This is not surprising regarding how similar the shapes of the two distributions are, but we find this quite interesting. In fact, Gaussian noise is the most used in the literature, and transformed Weibull could constitute a nice alternative to it. Finally, the exponential noise (see Figure (c)c and Table 2) is the one which defends the best the accuracy under attack. This result is quite surprising, since we expected symmetric distributions to better protect the accuracy during the defense mechanism.
Note that we did not investigate noises from families other than the Exponential one, as it would not have brought any valuable insight. Indeed, our main theorem derives robustness from injection of noise from the Exponential family but it does not draw any conclusion for other families.
5 Conclusion and future work
In this work, we bring a theoretically wellgrounded framework in order to understand why previous methods based on noise injection were in practice effective against adversarial attacks. While the very article is a theoretical analysis, it also paves the way to novel defense mechanisms using noises from yet unexplored distributions (e.g. Weibull or exponential).
Our theoretical analysis mainly focused on the robustness of the methods but our numerical experiments validated that the accuracy was slightly altered by noise injection. Hence, we demonstrated the practical applicability of the approach. Note also that as we only used a vanilla ResNet21, using tricks of the trade for neural networks and the noise injection, the accuracy could be further improved.
In future work, we plan to investigate more complex architectures, other noise injection schemes and even the combination with other defenses (Madry et al., 2018; Goodfellow et al., 2015). But more importantly, we aim at developing new family of noise (respecting conditions of Theorem 3) further impeding the loss of accuracy.
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1 Notations
Let us consider an output space , a  over . We denote the set of probability measures over . Let be a second measurable space, and a measurable function from to . Finally Let us consider two measures on .
Dominated measure: is said to be dominated by (denoted ) if and only if for all , . If is dominated by , there is a measurable function such that for all , . is called the RadonNikodym derivative and is denoted .
Pushforward measure: the pushforward measure of by (denoted ) is the measure on such that .
Convolution product: the convolution of with , denoted is the pushforward measure of by the addition on . since the convolution between function is defined accordingly, we use indifferently for measures and simple functions.
2 Main proofs
Lemma 1.
Let u be the probabilistic mapping, and let , and be two metrics on . If there is a non decreasing function such that , , then the following assertion holds:
Proof.
Let consider a probabilistic mapping , , and , one has Hence . By inverting the inequality, one gets the expected result. ∎
Proposition 1 ((Kraft, 1969)).
Given two probability measures and on , on has
Proposition 2 ((Vajda, 1970)).
Given two probability measures and on , on has
Theorem 1 (Renyirobustness implies TVrobustness).
Let be a probabilistic mapping, then :
Proof.
Given two probability measures and on , and one wants to find a bound on as a functional of .
Using Proposition 1, on has
It suffices to solve a 2nd degree equation to get that
One thus finally gets:
Moreover, using Proposition 2, one gets:
For simplicity, and since the second part of the right hand equation is non increasing given , and since one gets:
Hence, one gets:
By combining the two results, one gets:
To conclude for it suffices to use Lemma 1, and the monotony of Renyi divergence regarding . ∎
Theorem 2 (Data processing inequality).
Let consider a probabilistic mapping . Let denote a deterministic algorithm. If then probability measure defines a probabilistic mapping .
For any if is  robust then is also is  robust.
Proof.
Let consider a  robust algorithm. Let us also take , and . Without loss of generality, we consider that , and are dominated by the same measure . Finally let us take a measurable mapping from to . For the sake of readability we denote and .
Since , one has . Hence the transfer theorem, the generalized Jensen’s inequality for conditional expectation, and the property of the conditional expectation with regard to the regular expectation, one has
Simply using the transfer theorem, one gets  
Since one easily gets the following:  
Finally, by using the Jensen inequality, and the property of the conditional expectation, one has  
∎
Theorem 3 (Exponential family ensures robustness).
Let be an open convex subset of . Let a mapping such that . Let be a random variable. We denote the probability measure of the random variable .

If where and have nondecreasing modulus of continuity and .
Then for any , defines a probabilistic mapping that is  robust with .

If is a centered Gaussian random variable with a non degenerated matrix parameter . Then for any , defines a probabilistic mapping that is  robust with .
Proof.
Let consider the probabilistic mapping constructed from noise injection respectively drawn from 1) exponential family with nondecreasing modulus of continuity, and 2) a non degenerate Gaussian. Let us also take , and . Without loss of generality, we consider that , and are dominated by the same measure . Let us also denote, the RadonNikodym derivative of the noise drawn in 1) with respect to , the RadonNikodym derivative of the noise drawn in 2) with respect to and the Dirac function in a mapping any element if it equals and 0 otherwise.
1)
2) Since the is non degenerated the Gaussian measure accept a pdf with respect to the Lebesgue measure, hence for all s.t
∎
3 Additional results on the strength of the Renyi divergence
Let us consider an output space , a  over , and three measures on , with in the set of probability measures over denoted . One has and one denotes and the RadonNikodym derivatives with respect to .
The Separation distance:
The Hellinger distance:
The Prokhorov metric:
The Discrepancy metric:
Lemma 2.
Given two probability measures and on the Separation metric and the Renyi divergence satisfy the following relation:
Proof.
The function is negative on , therefore for any one has , hence ∎
Proposition 3 ((Gibbs & Su, 2002)).
Given two probability measures and on , the Wasserstein metric and the Total Variation distance satisfy the following relation: