Adversarial machine learning

<h2 id="history">History</h2>
At the MIT Spam Conference in January 2004, <a href="/facts/John_Graham-Cumming/N5xDFBnD">John Graham-Cumming</a> showed that a machine-learning spam filter could be used to defeat another machine-learning spam filter by automatically learning which words to add to a spam email to get the email classified as not spam.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>
In 2004, Nilesh Dalvi and others noted that <a href="/facts/Linear_classifier/vsE823wQ">linear classifiers</a> used in <a href="/facts/Email_filtering/aKxq5uSf">spam filters</a> could be defeated by simple "<a href="/facts/Evasion_(network_security)/eqaKKj89">evasion</a> attacks" as spammers inserted "good words" into their spam emails. (Around 2007, some spammers added random noise to fuzz words within "image spam" in order to defeat <a href="/facts/Optical_character_recognition/kdTyxa1l">OCR</a>-based filters.) In 2006, Marco Barreno and others published "Can Machine Learning Be Secure?", outlining a broad taxonomy of attacks. As late as 2013 many researchers continued to hope that non-linear classifiers (such as <a href="/facts/Support_vector_machine/XobxpdBG">support vector machines</a> and <a href="/facts/Neural_networks/DDhJTfMc">neural networks</a>) might be robust to adversaries, until Battista Biggio and others demonstrated the first gradient-based attacks on such machine-learning models (2012<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>–2013<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a>). In 2012, <a href="/facts/Deep_learning/JLuwD3ea">deep neural networks</a> began to dominate computer vision problems; starting in 2014, Christian Szegedy and others demonstrated that deep neural networks could be fooled by adversaries, again using a gradient-based attack to craft adversarial perturbations.<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a><a class="footnote-ref" id="fnref:11" href="#fn:11">11</a>
Recently, it was observed that adversarial attacks are harder to produce in the practical world due to the different environmental constraints that cancel out the effect of noise.<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a><a class="footnote-ref" id="fnref:13" href="#fn:13">13</a> For example, any small rotation or slight illumination on an adversarial image can destroy the adversariality. In addition, researchers such as Google Brain's Nicholas Frosst point out that it is much easier to make self-driving cars<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a> miss stop signs by physically removing the sign itself, rather than creating adversarial examples.<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a> Frosst also believes that the adversarial machine learning community incorrectly assumes models trained on a certain data distribution will also perform well on a completely different data distribution. He suggests that a new approach to machine learning should be explored, and is currently working on a unique neural network that has characteristics more similar to human perception than state-of-the-art approaches.<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a>
While adversarial machine learning continues to be heavily rooted in academia, large tech companies such as Google, Microsoft, and IBM have begun curating documentation and open source code bases to allow others to concretely assess the <a href="/facts/Robustness_(computer_science)/jHr1dT3m">robustness</a> of machine learning models and minimize the risk of adversarial attacks.<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a><a class="footnote-ref" id="fnref:18" href="#fn:18">18</a><a class="footnote-ref" id="fnref:19" href="#fn:19">19</a>

<h3>Examples</h3>
Examples include attacks in <a href="/facts/Spam_filtering/WqhhDVXj">spam filtering</a>, where spam messages are obfuscated through the misspelling of "bad" words or the insertion of "good" words;<a class="footnote-ref" id="fnref:20" href="#fn:20">20</a><a class="footnote-ref" id="fnref:21" href="#fn:21">21</a> attacks in <a href="/facts/Computer_security/7E5dIy7G">computer security</a>, such as obfuscating malware code within <a href="/facts/Network_packet/xCYf8SJu">network packets</a> or modifying the characteristics of a <a href="/facts/Traffic_flow_(computer_networking)/Yp8UP6ys">network flow</a> to mislead intrusion detection;<a class="footnote-ref" id="fnref:22" href="#fn:22">22</a><a class="footnote-ref" id="fnref:23" href="#fn:23">23</a> attacks in biometric recognition where fake biometric traits may be exploited to impersonate a legitimate user;<a class="footnote-ref" id="fnref:24" href="#fn:24">24</a> or to compromise users' template galleries that adapt to updated traits over time.
Researchers showed that by changing only one-pixel it was possible to fool deep learning algorithms.<a class="footnote-ref" id="fnref:25" href="#fn:25">25</a> Others <a href="/facts/3-D_print/lbvWU3t0">3-D printed</a> a toy turtle with a texture engineered to make Google's object detection <a href="/facts/Artificial_intelligence/lJGauQwX">AI</a> classify it as a rifle regardless of the angle from which the turtle was viewed.<a class="footnote-ref" id="fnref:26" href="#fn:26">26</a> Creating the turtle required only low-cost commercially available 3-D printing technology.<a class="footnote-ref" id="fnref:27" href="#fn:27">27</a>
A machine-tweaked image of a dog was shown to look like a cat to both computers and humans.<a class="footnote-ref" id="fnref:28" href="#fn:28">28</a> A 2019 study reported that humans can guess how machines will classify adversarial images.<a class="footnote-ref" id="fnref:29" href="#fn:29">29</a> Researchers discovered methods for perturbing the appearance of a stop sign such that an autonomous vehicle classified it as a merge or speed limit sign.<a class="footnote-ref" id="fnref:30" href="#fn:30">30</a><a class="footnote-ref" id="fnref:31" href="#fn:31">31</a>
<a href="/facts/McAfee/yIytBc4w">McAfee</a> attacked <a href="/facts/Tesla%2c_Inc./dSyLNM1D">Tesla</a>'s former <a href="/facts/Mobileye/TpLL8KFF">Mobileye</a> system, fooling it into driving 50 mph over the speed limit, simply by adding a two-inch strip of black tape to a speed limit sign.<a class="footnote-ref" id="fnref:32" href="#fn:32">32</a><a class="footnote-ref" id="fnref:33" href="#fn:33">33</a>
Adversarial patterns on glasses or clothing designed to deceive facial-recognition systems or license-plate readers, have led to a niche industry of "stealth streetwear".<a class="footnote-ref" id="fnref:34" href="#fn:34">34</a>
An adversarial attack on a neural network can allow an attacker to inject algorithms into the target system.<a class="footnote-ref" id="fnref:35" href="#fn:35">35</a> Researchers can also create adversarial audio inputs to disguise commands to intelligent assistants in benign-seeming audio;<a class="footnote-ref" id="fnref:36" href="#fn:36">36</a> a parallel literature explores human perception of such stimuli.<a class="footnote-ref" id="fnref:37" href="#fn:37">37</a><a class="footnote-ref" id="fnref:38" href="#fn:38">38</a>
Clustering algorithms are used in security applications. Malware and <a href="/facts/Computer_viruses/IyTC1usq">computer virus</a> analysis aims to identify malware families, and to generate specific detection signatures.<a class="footnote-ref" id="fnref:39" href="#fn:39">39</a><a class="footnote-ref" id="fnref:40" href="#fn:40">40</a>

<h2 id="attack-modalities">Attack modalities</h2>
<h3>Taxonomy</h3>
Attacks against (supervised) machine learning algorithms have been categorized along three primary axes:<a class="footnote-ref" id="fnref:41" href="#fn:41">41</a> influence on the classifier, the security violation and their specificity.

<ul><li>Classifier influence: An attack can influence the classifier by disrupting the classification phase. This may be preceded by an exploration phase to identify vulnerabilities. The attacker's capabilities might be restricted by the presence of data manipulation constraints.<a class="footnote-ref" id="fnref:42" href="#fn:42">42</a></li>
<li>Security violation: An attack can supply malicious data that gets classified as legitimate. Malicious data supplied during training can cause legitimate data to be rejected after training.</li>
<li>Specificity: A targeted attack attempts to allow a specific intrusion/disruption. Alternatively, an indiscriminate attack creates general mayhem.</li></ul>
This taxonomy has been extended into a more comprehensive threat model that allows explicit assumptions about the adversary's goal, knowledge of the attacked system, capability of manipulating the input data/system components, and on attack strategy.<a class="footnote-ref" id="fnref:43" href="#fn:43">43</a><a class="footnote-ref" id="fnref:44" href="#fn:44">44</a> This taxonomy has further been extended to include dimensions for defense strategies against adversarial attacks.<a class="footnote-ref" id="fnref:45" href="#fn:45">45</a>

<h3>Strategies</h3>
Below are some of the most commonly encountered attack scenarios.

<h4>Data poisoning</h4>
Poisoning consists of contaminating the training dataset with data designed to increase errors in the output. Given that learning algorithms are shaped by their training datasets, poisoning can effectively reprogram algorithms with potentially malicious intent. Concerns have been raised especially for user-generated training data, e.g. for content recommendation or natural language models. The ubiquity of fake accounts offers many opportunities for poisoning. Facebook reportedly removes around 7 billion fake accounts per year.<a class="footnote-ref" id="fnref:46" href="#fn:46">46</a><a class="footnote-ref" id="fnref:47" href="#fn:47">47</a> Poisoning has been reported as the leading concern for industrial applications.<a class="footnote-ref" id="fnref:48" href="#fn:48">48</a>
On social medias, <a href="/facts/Disinformation/JNY1zmLJ">disinformation campaigns</a> attempt to bias recommendation and moderation algorithms, to push certain content over others.
A particular case of data poisoning is the <a href="/facts/Backdoor_(computing)/4Vry8CKE">backdoor</a> attack,<a class="footnote-ref" id="fnref:49" href="#fn:49">49</a> which aims to teach a specific behavior for inputs with a given trigger, e.g. a small defect on images, sounds, videos or texts.

For instance, <a href="/facts/Intrusion_detection_system/mAw24Fyo">intrusion detection systems</a> are often trained using collected data. An attacker may poison this data by injecting malicious samples during operation that subsequently disrupt retraining.<a class="footnote-ref" id="fnref:50" href="#fn:50">50</a><a class="footnote-ref" id="fnref:51" href="#fn:51">51</a><a class="footnote-ref" id="fnref:52" href="#fn:52">52</a><a class="footnote-ref" id="fnref:53" href="#fn:53">53</a><a class="footnote-ref" id="fnref:54" href="#fn:54">54</a>
Data poisoning techniques can also be applied to <a href="/facts/Text-to-image_model/ez6APLzp">text-to-image models</a> to alter their output, which is used by artists to defend their copyrighted works or their artistic style against imitation.<a class="footnote-ref" id="fnref:55" href="#fn:55">55</a>
Data poisoning can also happen unintentionally through <a href="/facts/Model_collapse/ghoBzIcv">model collapse</a>, where models are trained on synthetic data.<a class="footnote-ref" id="fnref:56" href="#fn:56">56</a>

<h4>Byzantine attacks</h4>
As machine learning is scaled, it often relies on multiple computing machines. In <a href="/facts/Federated_learning/USkKrLa9">federated learning</a>, for instance, edge devices collaborate with a central server, typically by sending gradients or model parameters. However, some of these devices may deviate from their expected behavior, e.g. to harm the central server's model<a class="footnote-ref" id="fnref:57" href="#fn:57">57</a> or to bias algorithms towards certain behaviors (e.g., amplifying the recommendation of disinformation content). On the other hand, if the training is performed on a single machine, then the model is very vulnerable to a failure of the machine, or an attack on the machine; the machine is a <a href="/facts/Single_point_of_failure/royNGgjd">single point of failure</a>.<a class="footnote-ref" id="fnref:58" href="#fn:58">58</a> In fact, the machine owner may themselves insert provably undetectable <a href="/facts/Backdoor_(computing)/4Vry8CKE">backdoors</a>.<a class="footnote-ref" id="fnref:59" href="#fn:59">59</a>
The current leading solutions to make (distributed) learning algorithms provably resilient to a minority of malicious (a.k.a. <a href="/facts/Byzantine_fault/hucMXMBV">Byzantine</a>) participants are based on robust gradient aggregation rules.<a class="footnote-ref" id="fnref:60" href="#fn:60">60</a><a class="footnote-ref" id="fnref:61" href="#fn:61">61</a><a class="footnote-ref" id="fnref:62" href="#fn:62">62</a><a class="footnote-ref" id="fnref:63" href="#fn:63">63</a><a class="footnote-ref" id="fnref:64" href="#fn:64">64</a><a class="footnote-ref" id="fnref:65" href="#fn:65">65</a> The robust aggregation rules do not always work especially when the data across participants has a non-iid distribution. Nevertheless, in the context of heterogeneous honest participants, such as users with different consumption habits for recommendation algorithms or writing styles for language models, there are provable impossibility theorems on what any robust learning algorithm can guarantee.<a class="footnote-ref" id="fnref:66" href="#fn:66">66</a><a class="footnote-ref" id="fnref:67" href="#fn:67">67</a>

<h4>Evasion</h4>
Evasion attacks<a class="footnote-ref" id="fnref:68" href="#fn:68">68</a><a class="footnote-ref" id="fnref:69" href="#fn:69">69</a><a class="footnote-ref" id="fnref:70" href="#fn:70">70</a><a class="footnote-ref" id="fnref:71" href="#fn:71">71</a> consist of exploiting the imperfection of a trained model. For instance, spammers and hackers often attempt to evade detection by obfuscating the content of spam emails and <a href="/facts/Malware/5ee7TLFR">malware</a>. Samples are modified to evade detection; that is, to be classified as legitimate. This does not involve influence over the training data. A clear example of evasion is <a href="/facts/Image_spam/VFcKwHdc">image-based spam</a> in which the spam content is embedded within an attached image to evade textual analysis by anti-spam filters. Another example of evasion is given by spoofing attacks against biometric verification systems.<a class="footnote-ref" id="fnref:72" href="#fn:72">72</a>
Evasion attacks can be generally split into two different categories: <a href="/facts/Black_box/hJeAGnIz">black box attacks</a> and <a href="/facts/White_box_(software_engineering)/tFkNbWpA">white box attacks</a>.<a class="footnote-ref" id="fnref:73" href="#fn:73">73</a>

<h4>Model extraction</h4>
Model extraction involves an adversary probing a black box machine learning system in order to extract the data it was trained on.<a class="footnote-ref" id="fnref:74" href="#fn:74">74</a><a class="footnote-ref" id="fnref:75" href="#fn:75">75</a> This can cause issues when either the training data or the model itself is sensitive and confidential. For example, model extraction could be used to extract a proprietary stock trading model which the adversary could then use for their own financial benefit.
In the extreme case, model extraction can lead to model stealing, which corresponds to extracting a sufficient amount of data from the model to enable the complete reconstruction of the model.
On the other hand, membership inference is a targeted model extraction attack, which infers the owner of a data point, often by leveraging the <a href="/facts/Overfitting/5xnFLcMg">overfitting</a> resulting from poor machine learning practices.<a class="footnote-ref" id="fnref:76" href="#fn:76">76</a> Concerningly, this is sometimes achievable even without knowledge or access to a target model's parameters, raising security concerns for models trained on sensitive data, including but not limited to medical records and/or personally identifiable information. With the emergence of <a href="/facts/Transfer_learning/pNz4P2KP">transfer learning</a> and public accessibility of many state of the art machine learning models, tech companies are increasingly drawn to create models based on public ones, giving attackers freely accessible information to the structure and type of model being used.<a class="footnote-ref" id="fnref:77" href="#fn:77">77</a>

<h2 id="categories">Categories</h2>
<h3>Adversarial attacks and training in linear models</h3>
There is a growing literature about adversarial attacks in 
linear models. Indeed, since the seminal work from Goodfellow at al.<a class="footnote-ref" id="fnref:78" href="#fn:78">78</a> studying these models in linear models has been an important tool to understand how adversarial attacks affect machine learning models. 
The analysis of these models is simplified because the computation of adversarial attacks can be simplified in linear regression and classification problems. Moreover, adversarial training is convex in this case.<a class="footnote-ref" id="fnref:79" href="#fn:79">79</a>
Linear models allow for analytical analysis while still reproducing phenomena observed in state-of-the-art models.
One prime example of that is how this model can be used to explain the trade-off between robustness and accuracy.<a class="footnote-ref" id="fnref:80" href="#fn:80">80</a> 
Diverse work indeed provides analysis of adversarial attacks in linear models, including asymptotic analysis for classification <a class="footnote-ref" id="fnref:81" href="#fn:81">81</a> and for linear regression.<a class="footnote-ref" id="fnref:82" href="#fn:82">82</a><a class="footnote-ref" id="fnref:83" href="#fn:83">83</a> And, finite-sample analysis based on Rademacher complexity.<a class="footnote-ref" id="fnref:84" href="#fn:84">84</a>
A result from studying adversarial attacks in linear models is that it closely relates to <a href="/facts/Regularization_(mathematics)/K601A3y2">regularization</a>.<a class="footnote-ref" id="fnref:85" href="#fn:85">85</a> Under certain conditions, it has been shown that

<ul><li>adversarial training of a linear regression model with input perturbations restricted by the infinity-norm closely resembles <a href="/facts/Lasso_(statistics)/R9yE8HDQ">Lasso</a> regression, and that</li>
<li>adversarial training of a linear regression model with input perturbations restricted by the 2-norm closely resembles <a href="/facts/Ridge_regression/Le6KJgJ1">Ridge regression</a>.</li></ul>
<h3>Adversarial deep reinforcement learning</h3>
Adversarial deep reinforcement learning is an active area of research in reinforcement learning focusing on vulnerabilities of learned policies. In this research area, some studies initially showed that reinforcement learning policies are susceptible to imperceptible adversarial manipulations.<a class="footnote-ref" id="fnref:86" href="#fn:86">86</a><a class="footnote-ref" id="fnref:87" href="#fn:87">87</a> While some methods have been proposed to overcome these susceptibilities, in the most recent studies it has been shown that these proposed solutions are far from providing an accurate representation of current vulnerabilities of deep reinforcement learning policies.<a class="footnote-ref" id="fnref:88" href="#fn:88">88</a>

<h3>Adversarial natural language processing</h3>
Adversarial attacks on <a href="/facts/Speech_recognition/z7S7Pgk6">speech recognition</a> have been introduced for speech-to-text applications, in particular for Mozilla's implementation of DeepSpeech.<a class="footnote-ref" id="fnref:89" href="#fn:89">89</a>

<h2 id="specific-attack-types">Specific attack types</h2>
There are a large variety of different adversarial attacks that can be used against machine learning systems. Many of these work on both <a href="/facts/Deep_learning/JLuwD3ea">deep learning</a> systems as well as traditional machine learning models such as <a href="/facts/Support_vector_machine/XobxpdBG">SVMs</a><a class="footnote-ref" id="fnref:90" href="#fn:90">90</a> and <a href="/facts/Linear_regression/5n998IhK">linear regression</a>.<a class="footnote-ref" id="fnref:91" href="#fn:91">91</a> A high level sample of these attack types include:

<ul><li>Adversarial Examples<a class="footnote-ref" id="fnref:92" href="#fn:92">92</a></li>
<li>Trojan Attacks / Backdoor Attacks<a class="footnote-ref" id="fnref:93" href="#fn:93">93</a></li>
<li>Model Inversion<a class="footnote-ref" id="fnref:94" href="#fn:94">94</a></li>
<li>Membership Inference<a class="footnote-ref" id="fnref:95" href="#fn:95">95</a></li></ul>
<h3>Adversarial examples</h3>
An adversarial example refers to specially crafted input that is designed to look "normal" to humans but causes misclassification to a machine learning model. Often, a form of specially designed "noise" is used to elicit the misclassifications. Below are some current techniques for generating adversarial examples in the literature (by no means an exhaustive list).

<ul><li>Gradient-based evasion attack<a class="footnote-ref" id="fnref:96" href="#fn:96">96</a></li>
<li>Fast Gradient Sign Method (FGSM)<a class="footnote-ref" id="fnref:97" href="#fn:97">97</a></li>
<li>Projected Gradient Descent (PGD)<a class="footnote-ref" id="fnref:98" href="#fn:98">98</a></li>
<li>Carlini and Wagner (C&W) attack<a class="footnote-ref" id="fnref:99" href="#fn:99">99</a></li>
<li>Adversarial patch attack<a class="footnote-ref" id="fnref:100" href="#fn:100">100</a></li></ul>
<h4>Black box attacks</h4>
Black box attacks in adversarial machine learning assume that the adversary can only get outputs for provided inputs and has no knowledge of the model structure or parameters.<a class="footnote-ref" id="fnref:101" href="#fn:101">101</a><a class="footnote-ref" id="fnref:102" href="#fn:102">102</a> In this case, the adversarial example is generated either using a model created from scratch, or without any model at all (excluding the ability to query the original model). In either case, the objective of these attacks is to create adversarial examples that are able to transfer to the black box model in question.<a class="footnote-ref" id="fnref:103" href="#fn:103">103</a>

<h5>Simple Black-box Adversarial Attacks</h5>
Simple Black-box Adversarial Attacks is a query-efficient way to attack black-box image classifiers.<a class="footnote-ref" id="fnref:104" href="#fn:104">104</a> <blockquote>Take a random orthonormal basis 
 
 
 
 
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<h5>Square Attack</h5>
The Square Attack was introduced in 2020 as a black box evasion adversarial attack based on querying classification scores without the need of gradient information.<a class="footnote-ref" id="fnref:106" href="#fn:106">106</a> As a score based black box attack, this adversarial approach is able to query probability distributions across model output classes, but has no other access to the model itself. According to the paper's authors, the proposed Square Attack required fewer queries than when compared to state-of-the-art score-based black box attacks at the time.<a class="footnote-ref" id="fnref:107" href="#fn:107">107</a>
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 ϵ
 
 
 {\displaystyle \min _{{\hat {x}}\in [0,1]^{d}}L(f({\hat {x}}),y),{\text{ s.t. }}||{\hat {x}}-x||_{p}\leq \epsilon }

The result in theory is an adversarial example that is highly confident in the incorrect class but is also very similar to the original image. To find such example, Square Attack utilizes the iterative <a href="/facts/Random_search/b4hcrusx">random search</a> technique to randomly perturb the image in hopes of improving the objective function. In each step, the algorithm perturbs only a small square section of pixels, hence the name Square Attack, which terminates as soon as an adversarial example is found in order to improve query efficiency. Finally, since the attack algorithm uses scores and not gradient information, the authors of the paper indicate that this approach is not affected by gradient masking, a common technique formerly used to prevent evasion attacks.<a class="footnote-ref" id="fnref:110" href="#fn:110">110</a>

<h5>HopSkipJump Attack</h5>
This black box attack was also proposed as a query efficient attack, but one that relies solely on access to any input's predicted output class. In other words, the HopSkipJump attack does not require the ability to calculate gradients or access to score values like the Square Attack, and will require just the model's class prediction output (for any given input). The proposed attack is split into two different settings, targeted and untargeted, but both are built from the general idea of adding minimal perturbations that leads to a different model output. In the targeted setting, the goal is to cause the model to misclassify the perturbed image to a specific target label (that is not the original label). In the untargeted setting, the goal is to cause the model to misclassify the perturbed image to any label that is not the original label. The attack objectives for both are as follows where 
 
 
 
 x
 
 
 {\textstyle x}
 
 is the original image, 
 
 
 
 
 x
 
 ′
 
 
 
 
 {\textstyle x^{\prime }}
 
 is the adversarial image, 
 
 
 
 d
 
 
 {\textstyle d}
 
 is a distance function between images, 
 
 
 
 
 c
 
 ∗
 
 
 
 
 {\textstyle c^{*}}
 
 is the target label, and 
 
 
 
 C
 
 
 {\textstyle C}
 
 is the model's classification class label function:<a class="footnote-ref" id="fnref:111" href="#fn:111">111</a>

 
 
 
 
 
 Targeted:
 
 
 
 min
 
 
 x
 
 ′
 
 
 
 
 d
 (
 
 x
 
 ′
 
 
 ,
 x
 )
 
  subject to 
 
 C
 (
 
 x
 
 ′
 
 
 )
 =
 
 c
 
 ∗
 
 
 
 
 {\displaystyle {\textbf {Targeted:}}\min _{x^{\prime }}d(x^{\prime },x){\text{ subject to }}C(x^{\prime })=c^{*}}

Untargeted:
 
 
 
 min
 
 
 x
 
 ′
 
 
 
 
 d
 (
 
 x
 
 ′
 
 
 ,
 x
 )
 
  subject to 
 
 C
 (
 
 x
 
 ′
 
 
 )
 ≠
 C
 (
 x
 )
 
 
 {\displaystyle {\textbf {Untargeted:}}\min _{x^{\prime }}d(x^{\prime },x){\text{ subject to }}C(x^{\prime })\neq C(x)}

To solve this problem, the attack proposes the following boundary function 
 
 
 
 S
 
 
 {\textstyle S}
 
 for both the untargeted and targeted setting:<a class="footnote-ref" id="fnref:112" href="#fn:112">112</a>

 
 
 
 S
 (
 
 x
 
 ′
 
 
 )
 :=
 
 
 {
 
 
 
 
 max
 
 c
 ≠
 C
 (
 x
 )
 
 
 
 F
 (
 
 x
 
 ′
 
 
 
 )
 
 c
 
 
 
 −
 F
 (
 
 x
 
 ′
 
 
 
 )
 
 C
 (
 x
 )
 
 
 ,
 
 
 
 (Untargeted)
 
 
 
 
 
 F
 (
 
 x
 
 ′
 
 
 
 )
 
 
 c
 
 ∗
 
 
 
 
 −
 
 max
 
 c
 ≠
 
 c
 
 ∗
 
 
 
 
 
 F
 (
 
 x
 
 ′
 
 
 
 )
 
 c
 
 
 
 ,
 
 
 
 (Targeted)
 
 
 
 
 
 
 
 
 
 {\displaystyle S(x^{\prime }):={\begin{cases}\max _{c\neq C(x)}{F(x^{\prime })_{c}}-F(x^{\prime })_{C(x)},&{\text{(Untargeted)}}\\F(x^{\prime })_{c^{*}}-\max _{c\neq c^{*}}{F(x^{\prime })_{c}},&{\text{(Targeted)}}\end{cases}}}

This can be further simplified to better visualize the boundary between different potential adversarial examples:<a class="footnote-ref" id="fnref:113" href="#fn:113">113</a>

 
 
 
 S
 (
 
 x
 
 ′
 
 
 )
 >
 0
 
 ⟺
 
 
 
 {
 
 
 
 a
 r
 g
 m
 a
 
 x
 
 c
 
 
 F
 (
 
 x
 
 ′
 
 
 )
 ≠
 C
 (
 x
 )
 ,
 
 
 
 (Untargeted)
 
 
 
 
 
 a
 r
 g
 m
 a
 
 x
 
 c
 
 
 F
 (
 
 x
 
 ′
 
 
 )
 =
 
 c
 
 ∗
 
 
 ,
 
 
 
 (Targeted)
 
 
 
 
 
 
 
 
 
 {\displaystyle S(x^{\prime })>0\iff {\begin{cases}argmax_{c}F(x^{\prime })\neq C(x),&{\text{(Untargeted)}}\\argmax_{c}F(x^{\prime })=c^{*},&{\text{(Targeted)}}\end{cases}}}

With this boundary function, the attack then follows an iterative algorithm to find adversarial examples 
 
 
 
 
 x
 
 ′
 
 
 
 
 {\textstyle x^{\prime }}
 
 for a given image 
 
 
 
 x
 
 
 {\textstyle x}
 
 that satisfies the attack objectives.

<ol><li>Initialize 
 
 
 
 x
 
 
 {\textstyle x}
 
 to some point where 
 
 
 
 S
 (
 x
 )
 >
 0
 
 
 {\textstyle S(x)>0}
 
</li>
<li>Iterate below
<ol><li>Boundary search</li>
<li>Gradient update
<ul><li>Compute the gradient</li>
<li>Find the step size</li></ul></li></ol></li></ol>
Boundary search uses a modified <a href="/facts/Binary_search_algorithm/FFmQzunc">binary search</a> to find the point in which the boundary (as defined by 
 
 
 
 S
 
 
 {\textstyle S}
 
) intersects with the line between 
 
 
 
 x
 
 
 {\textstyle x}
 
 and 
 
 
 
 
 x
 
 ′
 
 
 
 
 {\textstyle x^{\prime }}
 
. The next step involves calculating the gradient for 
 
 
 
 x
 
 
 {\textstyle x}
 
, and update the original 
 
 
 
 x
 
 
 {\textstyle x}
 
 using this gradient and a pre-chosen step size. HopSkipJump authors prove that this iterative algorithm will converge, leading 
 
 
 
 x
 
 
 {\textstyle x}
 
 to a point right along the boundary that is very close in distance to the original image.<a class="footnote-ref" id="fnref:114" href="#fn:114">114</a>
However, since HopSkipJump is a proposed black box attack and the iterative algorithm above requires the calculation of a gradient in the second iterative step (which black box attacks do not have access to), the authors propose a solution to gradient calculation that requires only the model's output predictions alone.<a class="footnote-ref" id="fnref:115" href="#fn:115">115</a> By generating many random vectors in all directions, denoted as 
 
 
 
 
 u
 
 b
 
 
 
 
 {\textstyle u_{b}}
 
, an approximation of the gradient can be calculated using the average of these random vectors weighted by the sign of the boundary function on the image 
 
 
 
 
 x
 
 ′
 
 
 +
 
 δ
 
 
 u
 
 b
 
 
 
 
 
 
 {\textstyle x^{\prime }+\delta _{u_{b}}}
 
, where 
 
 
 
 
 δ
 
 
 u
 
 b
 
 
 
 
 
 
 {\textstyle \delta _{u_{b}}}
 
 is the size of the random vector perturbation:<a class="footnote-ref" id="fnref:116" href="#fn:116">116</a>

 
 
 
 ∇
 S
 (
 
 x
 
 ′
 
 
 ,
 δ
 )
 ≈
 
 
 1
 B
 
 
 
 ∑
 
 b
 =
 1
 
 
 B
 
 
 ϕ
 (
 
 x
 
 ′
 
 
 +
 
 δ
 
 
 u
 
 b
 
 
 
 
 )
 
 u
 
 b
 
 
 
 
 {\displaystyle \nabla S(x^{\prime },\delta )\approx {\frac {1}{B}}\sum _{b=1}^{B}\phi (x^{\prime }+\delta _{u_{b}})u_{b}}

The result of the equation above gives a close approximation of the gradient required in step 2 of the iterative algorithm, completing HopSkipJump as a black box attack.<a class="footnote-ref" id="fnref:117" href="#fn:117">117</a><a class="footnote-ref" id="fnref:118" href="#fn:118">118</a><a class="footnote-ref" id="fnref:119" href="#fn:119">119</a>

<h4>White box attacks</h4>
White box attacks assumes that the adversary has access to model parameters on top of being able to get labels for provided inputs.<a class="footnote-ref" id="fnref:120" href="#fn:120">120</a>

<h5>Fast gradient sign method</h5>
One of the first proposed attacks for generating adversarial examples was proposed by Google researchers <a href="/facts/Ian_Goodfellow/ycEznz5D">Ian J. Goodfellow</a>, Jonathon Shlens, and Christian Szegedy.<a class="footnote-ref" id="fnref:121" href="#fn:121">121</a> The attack was called fast gradient sign method (FGSM), and it consists of adding a linear amount of in-perceivable noise to the image and causing a model to incorrectly classify it. This noise is calculated by multiplying the sign of the gradient with respect to the image we want to perturb by a small constant epsilon. As epsilon increases, the model is more likely to be fooled, but the perturbations become easier to identify as well. Shown below is the equation to generate an adversarial example where 
 
 
 
 x
 
 
 {\textstyle x}
 
 is the original image, 
 
 
 
 ϵ
 
 
 {\textstyle \epsilon }
 
 is a very small number, 
 
 
 
 
 Δ
 
 x
 
 
 
 
 {\textstyle \Delta _{x}}
 
 is the gradient function, 
 
 
 
 J
 
 
 {\textstyle J}
 
 is the loss function, 
 
 
 
 θ
 
 
 {\textstyle \theta }
 
 is the model weights, and 
 
 
 
 y
 
 
 {\textstyle y}
 
 is the true label.<a class="footnote-ref" id="fnref:122" href="#fn:122">122</a>

 
 
 
 a
 d
 
 v
 
 x
 
 
 =
 x
 +
 ϵ
 ⋅
 s
 i
 g
 n
 (
 
 Δ
 
 x
 
 
 J
 (
 θ
 ,
 x
 ,
 y
 )
 )
 
 
 {\displaystyle adv_{x}=x+\epsilon \cdot sign(\Delta _{x}J(\theta ,x,y))}

One important property of this equation is that the gradient is calculated with respect to the input image since the goal is to generate an image that maximizes the loss for the original image of true label 
 
 
 
 y
 
 
 {\textstyle y}
 
. In traditional <a href="/facts/Gradient_descent/pFFrek0F">gradient descent</a> (for model training), the gradient is used to update the weights of the model since the goal is to minimize the loss for the model on a ground truth dataset. The Fast Gradient Sign Method was proposed as a fast way to generate adversarial examples to evade the model, based on the hypothesis that neural networks cannot resist even linear amounts of perturbation to the input.<a class="footnote-ref" id="fnref:123" href="#fn:123">123</a><a class="footnote-ref" id="fnref:124" href="#fn:124">124</a><a class="footnote-ref" id="fnref:125" href="#fn:125">125</a> FGSM has shown to be effective in adversarial attacks for image classification and skeletal action recognition.<a class="footnote-ref" id="fnref:126" href="#fn:126">126</a>

<h5>Carlini & Wagner (C&W)</h5>
In an effort to analyze existing adversarial attacks and defenses, researchers at the University of California, Berkeley, <a href="/facts/Nicholas_Carlini/jxSXUukQ">Nicholas Carlini</a> and <a href="/facts/David_A._Wagner/ewC269Uo">David Wagner</a> in 2016 propose a faster and more robust method to generate adversarial examples.<a class="footnote-ref" id="fnref:127" href="#fn:127">127</a>
The attack proposed by Carlini and Wagner begins with trying to solve a difficult non-linear optimization equation:<a class="footnote-ref" id="fnref:128" href="#fn:128">128</a>

 
 
 
 min
 (
 
 |
 
 
 |
 
 δ
 
 |
 
 
 
 |
 
 
 p
 
 
 )
 
  subject to 
 
 C
 (
 x
 +
 δ
 )
 =
 t
 ,
 x
 +
 δ
 ∈
 [
 0
 ,
 1
 
 ]
 
 n
 
 
 
 
 {\displaystyle \min(||\delta ||_{p}){\text{ subject to }}C(x+\delta )=t,x+\delta \in [0,1]^{n}}

Here the objective is to minimize the noise (
 
 
 
 δ
 
 
 {\textstyle \delta }
 
), added to the original input 
 
 
 
 x
 
 
 {\textstyle x}
 
, such that the machine learning algorithm (
 
 
 
 C
 
 
 {\textstyle C}
 
) predicts the original input with delta (or 
 
 
 
 x
 +
 δ
 
 
 {\textstyle x+\delta }
 
) as some other class 
 
 
 
 t
 
 
 {\textstyle t}
 
. However instead of directly the above equation, Carlini and Wagner propose using a new function 
 
 
 
 f
 
 
 {\textstyle f}
 
 such that:<a class="footnote-ref" id="fnref:129" href="#fn:129">129</a>

 
 
 
 C
 (
 x
 +
 δ
 )
 =
 t
 
 ⟺
 
 f
 (
 x
 +
 δ
 )
 ≤
 0
 
 
 {\displaystyle C(x+\delta )=t\iff f(x+\delta )\leq 0}

This condenses the first equation to the problem below:<a class="footnote-ref" id="fnref:130" href="#fn:130">130</a>

 
 
 
 min
 (
 
 |
 
 
 |
 
 δ
 
 |
 
 
 
 |
 
 
 p
 
 
 )
 
  subject to 
 
 f
 (
 x
 +
 δ
 )
 ≤
 0
 ,
 x
 +
 δ
 ∈
 [
 0
 ,
 1
 
 ]
 
 n
 
 
 
 
 {\displaystyle \min(||\delta ||_{p}){\text{ subject to }}f(x+\delta )\leq 0,x+\delta \in [0,1]^{n}}

and even more to the equation below:<a class="footnote-ref" id="fnref:131" href="#fn:131">131</a>

 
 
 
 min
 (
 
 |
 
 
 |
 
 δ
 
 |
 
 
 
 |
 
 
 p
 
 
 +
 c
 ⋅
 f
 (
 x
 +
 δ
 )
 )
 ,
 x
 +
 δ
 ∈
 [
 0
 ,
 1
 
 ]
 
 n
 
 
 
 
 {\displaystyle \min(||\delta ||_{p}+c\cdot f(x+\delta )),x+\delta \in [0,1]^{n}}

Carlini and Wagner then propose the use of the below function in place of 
 
 
 
 f
 
 
 {\textstyle f}
 
 using 
 
 
 
 Z
 
 
 {\textstyle Z}
 
, a function that determines class probabilities for given input 
 
 
 
 x
 
 
 {\textstyle x}
 
. When substituted in, this equation can be thought of as finding a target class that is more confident than the next likeliest class by some constant amount:<a class="footnote-ref" id="fnref:132" href="#fn:132">132</a>

 
 
 
 f
 (
 x
 )
 =
 (
 [
 
 max
 
 i
 ≠
 t
 
 
 Z
 (
 x
 
 )
 
 i
 
 
 ]
 −
 Z
 (
 x
 
 )
 
 t
 
 
 
 )
 
 +
 
 
 
 
 {\displaystyle f(x)=([\max _{i\neq t}Z(x)_{i}]-Z(x)_{t})^{+}}

When solved using gradient descent, this equation is able to produce stronger adversarial examples when compared to fast gradient sign method that is also able to bypass defensive distillation, a defense that was once proposed to be effective against adversarial examples.<a class="footnote-ref" id="fnref:133" href="#fn:133">133</a><a class="footnote-ref" id="fnref:134" href="#fn:134">134</a><a class="footnote-ref" id="fnref:135" href="#fn:135">135</a><a class="footnote-ref" id="fnref:136" href="#fn:136">136</a>

<h2 id="defenses">Defenses</h2>

Researchers have proposed a multi-step approach to protecting machine learning.<a class="footnote-ref" id="fnref:137" href="#fn:137">137</a>

<ul><li>Threat modeling – Formalize the attackers goals and capabilities with respect to the target system.</li>
<li>Attack simulation – Formalize the optimization problem the attacker tries to solve according to possible attack strategies.</li>
<li>Attack impact evaluation</li>
<li>Countermeasure design</li>
<li>Noise detection (For evasion based attack)<a class="footnote-ref" id="fnref:138" href="#fn:138">138</a></li>
<li>Information laundering – Alter the information received by adversaries (for model stealing attacks)<a class="footnote-ref" id="fnref:139" href="#fn:139">139</a></li></ul>
<h3>Mechanisms</h3>
A number of defense mechanisms against evasion, poisoning, and privacy attacks have been proposed, including:

<ul><li>Secure learning algorithms<a class="footnote-ref" id="fnref:140" href="#fn:140">140</a><a class="footnote-ref" id="fnref:141" href="#fn:141">141</a><a class="footnote-ref" id="fnref:142" href="#fn:142">142</a></li>
<li>Byzantine-resilient algorithms<a class="footnote-ref" id="fnref:143" href="#fn:143">143</a><a class="footnote-ref" id="fnref:144" href="#fn:144">144</a></li>
<li>Multiple classifier systems<a class="footnote-ref" id="fnref:145" href="#fn:145">145</a><a class="footnote-ref" id="fnref:146" href="#fn:146">146</a></li>
<li>AI-written algorithms.<a class="footnote-ref" id="fnref:147" href="#fn:147">147</a></li>
<li>AIs that explore the training environment; for example, in image recognition, actively navigating a 3D environment rather than passively scanning a fixed set of 2D images.<a class="footnote-ref" id="fnref:148" href="#fn:148">148</a></li>
<li>Privacy-preserving learning<a class="footnote-ref" id="fnref:149" href="#fn:149">149</a><a class="footnote-ref" id="fnref:150" href="#fn:150">150</a></li>
<li>Ladder algorithm for <a href="/facts/Kaggle/XuEmk22y">Kaggle</a>-style competitions</li>
<li>Game theoretic models<a class="footnote-ref" id="fnref:151" href="#fn:151">151</a><a class="footnote-ref" id="fnref:152" href="#fn:152">152</a><a class="footnote-ref" id="fnref:153" href="#fn:153">153</a></li>
<li>Sanitizing training data</li>
<li>Adversarial training<a class="footnote-ref" id="fnref:154" href="#fn:154">154</a><a class="footnote-ref" id="fnref:155" href="#fn:155">155</a></li>
<li>Backdoor detection algorithms<a class="footnote-ref" id="fnref:156" href="#fn:156">156</a></li>
<li>Gradient masking/obfuscation techniques: to prevent the adversary exploiting the gradient in white-box attacks. This family of defenses is deemed unreliable as these models are still vulnerable to black-box attacks or can be circumvented in other ways.<a class="footnote-ref" id="fnref:157" href="#fn:157">157</a></li>
<li><a href="/facts/Ensemble_learning/GvFsd3Qc">Ensembles</a> of models have been proposed in literature, which have shown to be ineffective against evasion attacks<a class="footnote-ref" id="fnref:158" href="#fn:158">158</a> but effective against data poisoning attacks.<a class="footnote-ref" id="fnref:159" href="#fn:159">159</a></li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Pattern_recognition/o1s38opj">Pattern recognition</a></li>
<li><a href="/facts/Fawkes_(image_cloaking_software)/tC88VIdA">Fawkes (image cloaking software)</a></li>
<li><a href="/facts/Generative_adversarial_network/SGNuGrlo">Generative adversarial network</a></li></ul>

<h2 id="external-links">External links</h2>
<ul><li><a href="https://atlas.mitre.org/">MITRE ATLAS: Adversarial Threat Landscape for Artificial-Intelligence Systems</a></li>
<li><a href="https://csrc.nist.gov/publications/detail/nistir/8269/draft">NIST 8269 Draft: A Taxonomy and Terminology of Adversarial Machine Learning</a></li>
<li>NIPS 2007 Workshop on <a href="https://web.archive.org/web/20120108072159/http://nips.cc/Conferences/2007/Program/event.php?ID=615">Machine Learning in Adversarial Environments for Computer Security</a></li>
<li><a href="http://pralab.diee.unica.it/en/ALFASVMLib">AlfaSVMLib</a> <a href="https://web.archive.org/web/20200924221614/http://pralab.diee.unica.it/en/ALFASVMLib">Archived</a> 2020-09-24 at the <a href="/facts/Wayback_Machine/nmQ3a6JC">Wayback Machine</a> – Adversarial Label Flip Attacks against Support Vector Machines</li>
<li>Laskov, Pavel; Lippmann, Richard (2010). "Machine learning in adversarial environments". Machine Learning. 81 (2): 115–119. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fs10994-010-5207-6">10.1007/s10994-010-5207-6</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:12567278">12567278</a>.</li>
<li>Dagstuhl Perspectives Workshop on "<a href="https://www.dagstuhl.de/en/seminars/seminar-calendar/seminar-details/12371">Machine Learning Methods for Computer Security</a>"</li>
<li>Workshop on <a href="http://aisec.cc/">Artificial Intelligence and Security</a>, (AISec) Series</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Kianpour, Mazaher; Wen, Shao-Fang (2020). "Timing Attacks on Machine Learning: State of the Art". Intelligent Systems and Applications. Advances in Intelligent Systems and Computing. Vol. 1037. pp. 111–125. doi:10.1007/978-3-030-29516-5_10. ISBN 978-3-030-29515-8. S2CID 201705926. <a href="978-3-030-29515-8" target="_blank">978-3-030-29515-8</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Siva Kumar, Ram Shankar; Nyström, Magnus; Lambert, John; Marshall, Andrew; Goertzel, Mario; Comissoneru, Andi; Swann, Matt; Xia, Sharon (May 2020). "Adversarial Machine Learning-Industry Perspectives". 2020 IEEE Security and Privacy Workshops (SPW). pp. 69–75. doi:10.1109/SPW50608.2020.00028. ISBN 978-1-7281-9346-5. S2CID 229357721. <a href="978-1-7281-9346-5" target="_blank">978-1-7281-9346-5</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Goodfellow, Ian; McDaniel, Patrick; Papernot, Nicolas (25 June 2018). "Making machine learning robust against adversarial inputs". Communications of the ACM. 61 (7): 56–66. doi:10.1145/3134599. ISSN 0001-0782.[permanent dead link] <a href="https://doi.org/10.1145%2F3134599" target="_blank">https://doi.org/10.1145%2F3134599</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Geiping, Jonas; Fowl, Liam H.; Huang, W. Ronny; Czaja, Wojciech; Taylor, Gavin; Moeller, Michael; Goldstein, Tom (2020-09-28). Witches' Brew: Industrial Scale Data Poisoning via Gradient Matching. International Conference on Learning Representations 2021 (Poster). <a href="https://openreview.net/forum?id=01olnfLIbD" target="_blank">https://openreview.net/forum?id=01olnfLIbD</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">El-Mhamdi, El Mahdi; Farhadkhani, Sadegh; Guerraoui, Rachid; Guirguis, Arsany; Hoang, Lê-Nguyên; Rouault, Sébastien (2021-12-06). "Collaborative Learning in the Jungle (Decentralized, Byzantine, Heterogeneous, Asynchronous and Nonconvex Learning)". Advances in Neural Information Processing Systems. 34. arXiv:2008.00742. <a href="https://proceedings.neurips.cc/paper/2021/hash/d2cd33e9c0236a8c2d8bd3fa91ad3acf-Abstract.html" target="_blank">https://proceedings.neurips.cc/paper/2021/hash/d2cd33e9c0236a8c2d8bd3fa91ad3acf-Abstract.html</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Tramèr, Florian; Zhang, Fan; Juels, Ari; Reiter, Michael K.; Ristenpart, Thomas (2016). Stealing Machine Learning Models via Prediction {APIs}. 25th USENIX Security Symposium. pp. 601–618. ISBN 978-1-931971-32-4. <a href="978-1-931971-32-4" target="_blank">978-1-931971-32-4</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">"How to beat an adaptive/Bayesian spam filter (2004)". Retrieved 2023-07-05. <a href="http://blog.jgc.org/2023/07/how-to-beat-adaptivebayesian-spam.html" target="_blank">http://blog.jgc.org/2023/07/how-to-beat-adaptivebayesian-spam.html</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Biggio, Battista; Nelson, Blaine; Laskov, Pavel (2013-03-25). "Poisoning Attacks against Support Vector Machines". arXiv:1206.6389 [cs.LG]. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Biggio, Battista; Corona, Igino; Maiorca, Davide; Nelson, Blaine; Srndic, Nedim; Laskov, Pavel; Giacinto, Giorgio; Roli, Fabio (2013). "Evasion Attacks against Machine Learning at Test Time". Advanced Information Systems Engineering. Lecture Notes in Computer Science. Vol. 7908. Springer. pp. 387–402. arXiv:1708.06131. doi:10.1007/978-3-642-40994-3_25. ISBN 978-3-642-38708-1. S2CID 18716873. <a href="978-3-642-38708-1" target="_blank">978-3-642-38708-1</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Szegedy, Christian; Zaremba, Wojciech; Sutskever, Ilya; Bruna, Joan; Erhan, Dumitru; Goodfellow, Ian; Fergus, Rob (2014-02-19). "Intriguing properties of neural networks". arXiv:1312.6199 [cs.CV]. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
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</ol>

Adversarial machine learning open-in-new

Adversarial machine learning