Erwan Le Merrer
email: erwan.le-merrer@inria.fr

Gilles Trédan
email: gtredan@laas.fr

The Bouncer Problem: Challenges to Remote Explainability

The concept of explainability is envisioned to satisfy society's demands for transparency on machine learning decisions. The concept is simple: like humans, algorithms should explain the rationale behind their decisions so that their fairness can be assessed.

While this approach is promising in a local context (e.g., the model creator explains it during debugging at training time), we argue that this reasoning cannot simply be transposed in a remote context, where a trained model by a service provider is only accessible to a user through a network and its API. This is problematic as it constitutes precisely the target use-case requiring transparency from a societal perspective.

Through an analogy with a club bouncer (which may provide untruthful explanations upon customer reject), we show that providing explanations cannot prevent a remote service from lying about the true reasons leading to its decisions. More precisely, we prove the impossibility of remote explainability for single explanations, by constructing an attack on explanations that hides discriminatory features to the querying user.

We provide an example implementation of this attack. We then show that the probability that an observer spots the attack, using several explanations for attempting to find incoherences, is low in practical settings. This undermines the very concept of remote explainability in general.

Introduction

Modern decision-making driven by black-box systems now impacts a significant share of our lives [START_REF] De Laat | Algorithmic decision-making based on machine learning from big data: Can transparency restore accountability[END_REF][START_REF] Veale | Logics and practices of transparency and opacity in real-world applications of public sector machine learning[END_REF]. These systems build on user data, and range from recommenders [START_REF] Naumov | Deep learning recommendation model for personalization and recommendation systems[END_REF] (e.g., for personalized ranking of information on websites) to predictive algorithms (e.g., credit default) [START_REF] Veale | Logics and practices of transparency and opacity in real-world applications of public sector machine learning[END_REF]. This widespread deployment, along with the opaque decision process provided by these systems raises concerns about transparency for the general public or for policy makers [START_REF] Goodman | European union regulations on algorithmic decision-making and a "right to explanation[END_REF]. This translated in some jurisdictions (e.g., United States of America and Europe) into a so called right to explanation [START_REF] Goodman | European union regulations on algorithmic decision-making and a "right to explanation[END_REF][START_REF] Selbst | Meaningful information and the right to explanation[END_REF], that states that the output decisions of an algorithm must be motivated.

Explainability of in-house models An already large body of work is interested in the explainability of implicit machine learning models (such as neural network models) [START_REF] Adadi | Peeking inside the blackbox: A survey on explainable artificial intelligence (xai)[END_REF][START_REF] Guidotti | A survey of methods for explaining black box models[END_REF][START_REF] Molnar | Interpretable Machine Learning[END_REF]. Indeed, these models show state-of-art performance when it comes to a task accuracy, but they are not designed to provide explanations -or at least intelligible decision processes-when one wants to obtain more than the output decision of the model. In the context of recommendation, the expression "post hoc explanation" has been coined [START_REF] Zhang | Explainable recommendation: A survey and new perspectives[END_REF]. In general, current techniques for explainability of implicit models take trained in-house models and aim at shedding light on some input features causing salient decisions in their output space. LIME [START_REF] Ribeiro | Explaining the predictions of any classifier[END_REF] for instance builds a surrogate model of a given black-box system that approximates its predictions around a region of interest. The surrogate is created from a new crafted dataset, obtained from the permutation of the original dataset values around the interesting zone (and the observation of the decisions made for this dataset). This surrogate is an explainable model by construction (such as a decision tree), so that it can explain some decision facing some input data. The amount of queries to the black-box model is assumed to be unbounded by LIME and others [START_REF] Galhotra | Fairness testing: Testing software for discrimination[END_REF][START_REF] Lundberg | A unified approach to interpreting model predictions[END_REF], permitting virtually exhaustive queries to it. This reduces their applicability to the inspection of in-house models by their designers.

The temptation to explain decisions to users. As suggested by Andreou et al. [START_REF] Andreou | Investigating Ad Transparency Mechanisms in Social Media: A Case Study of Facebook's Explanations[END_REF], some institutions may apply the same reasoning in order to explain some decisions to their users. Indeed, this would support the will for a more transparent and trusted web by the public. Facebook for instance attempted to offer a form of transparency for the ad mechanism targeting its users, by introducing a "Why I am seeing this" button on received ads. For a user, the decision-making system (here, responsible of selecting relevant ads) is then remote, and can be queried only using inputs (its profile data) and the observation of system decisions. Yet, from a security standpoint, we consider a security model where the remote server (executing the service) is untrusted to the users, in the classic remote execution setup [START_REF] Ateniese | Provable data possession at untrusted stores[END_REF]. Andreou et al. [START_REF] Andreou | Investigating Ad Transparency Mechanisms in Social Media: A Case Study of Facebook's Explanations[END_REF] recently empirically observed in the case of Facebook that these explanations are "incomplete and can be misleading", conjec-turing that malicious service providers can use this incompleteness to hide the true reasons behind their decisions.

In this paper, we question the possibility of such an explanation setup, from a corporate and private model in destination to users: we go one step furtherby demonstrating that remote explainability simply cannot be a reliable guarantee of the lack of discrimination in the decision-making process. In a remote black-box setup such as the one of Facebook, we show that a simple attack, we coin the Public Relations (PR) attack, undermines remote explainability.

The bouncer problem as a parallel for hardness For the sake of the demonstration, we introduce the bouncer problem as an illustration of the difficulty for users to spot malicious explanations. The analogy works as follows: let us picture a bouncer at the door of a club, deciding whoever might enter the club. When he issues a negative decision -refusing the entrance to a given person-, he also provides an explanation for this rejection. However, his explanation might be malicious, in the sense that his explanation does not present the true reasons of this person's rejection. Consider for instance a bouncer discriminating people based on the color of their skin. Of course he will not tell people he refuses the entrance based on that characteristic, since this is a legal offence. He will instead invent a biased explanation that the rejected person is likely to accept.

The classic way to assess a discrimination by the bouncer is for associations to run tests (following the principle of statistical causality [START_REF] Pearl | Causal inference in statistics: An overview[END_REF] for instance): several persons attempt to enter, while they only vary in their attitude or appearance on the possibly discriminating feature (e.g., the color of their skin). Conflicting decisions by the bouncer is then the indication of a possible discrimination and is amenable to the building of a case for prosecution.

We make the parallel with bouncer decisions in this paper by demonstrating that a user cannot trust a single (one-shot) explanation provided by a remote model.Moreover, we show that creating such malicious explanations necessarily creates inconsistent answers for some inputs, and that the only solution to spot those inconsistencies is to issue multiple requests to the service. Unfortunately, we also demonstrate the problem to be hard, in the sense that spotting an inconsistency in such a way is intrinsically not more efficient than for a model creator to exhaustively search on her local model to identify a problem, which is often considered as an intractable process.

Rationale and organization of the paper We build a general setup for remote explainability in the next section, that has the purpose of representing actions by a service provider and by users, facing models decisions and explanations. The fundamental blocks for the impossibility proof of a reliable remote explainability, or its hardness for multiple queries are presented in Section 2. We present the bouncer problem in Section 3, that users have to solve in order to detect malicious explanations by the remote service provider. We then illustrate the PR attack, that the malicious provider may execute to remove discriminative explanations to users, on decision trees (Section 4.1). We then practically address the bouncer problem by modeling a user trying to find inconsistencies from a provider decisions based on the German Credit Dataset and a neural network classifier, in Section 4.2. We discuss open problems in Section 5, before reviewing related works in Section 6 and concluding in Section 7. Since we show that remote explainability in its current form is undermined, this work thus aims to be a motivation for researchers to explore the direction of provable explainability, by designing new protocols such as for instance one implying cryptographic means (e.g., such as in proof of ownership for remote storage), or to build collaborative observation systems to spot inconsistencies and malicious explanation systems.

Explainability of remote decisions

In this work, we study classifier models, that will issue decisions given user data. We first introduce the setup we operate in: it is intended to be as general as possible, so that the results drawn from it can apply widely.

General Setup

We consider a classifier C : X → Y that assigns inputs x of the feature space X to a class C(x) = y ∈ Y.

Without loss of generality and to simplify the presentation, we will assume the case of a binary classifier: Y = {0, 1}; the decision is thus the output label returned by the classifier.

Discriminative features and classifiers

To produce a decision, classifiers rely on features (variables) as an input. These are for instance the variables associated to a user profile on a given service platform (e.g., basic demographics, political affiliation, purchase behavior, residential profile [START_REF] Andreou | Investigating Ad Transparency Mechanisms in Social Media: A Case Study of Facebook's Explanations[END_REF]). In our model, we consider that the feature space contains two types of features: discriminatory and legitimate features. The use of discriminatory features allows for exhibiting the possibility of a malicious service provider issuing decisions and biased explanations. This problematic is also referred to as rationalization in a paper by Aïvodji et al [START_REF] Aivodji | Fairwashing: the risk of rationalization[END_REF]. Concretely, we consider discriminatory features to be an arbitrary subset of the input features, such that we can define these as "any feature set the malicious service provider does not want to explain". Two main reasons come to mind:

• Legal: the jurisdiction's law forbids decisions based X l X d X Feature space C l C d C Classifier space Y Decision c 1 c 2
Figure 1: Illustration of our model: we consider binary classifiers, that map the input domain X to labels Y = {0, 1}. Some dimensions of the input space are discriminative X d , which induces a partition on the classifier space. Legitimate classifiers C l that do not rely on discriminative features to issue a label (in green), while others (that is, C d ) can rely on any feature (in red). on a list of criteria1 which are easily found in classifiers input spaces. A service provider risks prosecution upon admitting the use of these.

For instance, features such as age, sex, employment, or the status of foreigner are considered as discriminatory in the work by Hajian et al. [START_REF] Hajian | Rule protection for indirect discrimination prevention in data mining[END_REF], that looks into the German Credit Dataset, that links bank customer features to the accordance or not of a credit.

• Strategical: the service provider wants to hide the use of some features on which its decisions are based. This could be to hide some business secret from competitors (because of the accuracyfairness trade-off [START_REF] Menon | The cost of fairness in binary classification[END_REF] for instance), or to avoid "reward hacking" from users biasing this feature, or to avoid bad press.

Conversely, any feature that is not discriminatory is coined legit. Formally, we partition the classifier input space X along these two types of features: legitimate features X l that the model can legitimately exploit to issue a decision, and discriminative features X d (please refer to Figure 1). In other words X = (X l , X d ), and any input x ∈ X can be decomposed as a pair of legitimate and discriminatory features x = (x l , x d ). We stress that the introduction of such a split in the features is required to build our proof and studies, yet it does not constitute a novel proposal in any way. We assume the input contains at least one legitimate feature: X l = ∅.

We also partition the classifier space accordingly: let C l ⊂ C the space of legitimate classifiers (among all classifiers C), which do not rely on any feature of X d to issue a decision. More precisely, we consider that a classifier is legitimate if and only if arbitrarily changing any discriminative input feature does never change its decision:

C ∈ C l ⇔ ∀x l ∈ X l , ∀x d , x d ∈ X 2 d , C((x l , x d )) = C((x l , x d )).
Observe that therefore, any legitimate classifier C l could simply be defined over input subspace X l ⊂ X . As a slight notation abuse to stress that the value of discriminative features does not matter in this legitimate context, we write C((x l , ∅)), or C(x ∈ X l ) as the decision produced regardless of any discriminative feature. It follows that the space of discriminative classifiers complements the space of legitimate classifiers:

C d = C \ C l .
We can now reframe the main research question we address: Given a set of discriminative features X d , and a classifier C, can we decide if C ∈ C d , in the remote black-box interaction model ?

The remote black-box interaction model We question the remote black-box interaction model (see e.g., paper [START_REF] Tramèr | Stealing machine learning models via prediction apis[END_REF]), where the classifier is exposed to users through a remote API. In other words users can only query the classifier model with an input and obtain a label as an answer (e.g., 0 or 1). In this remote setup, users then cannot collect any specific information about the internals of the classifier model, such as its architecture, its weights, or its training data. This corresponds to a security threat model where two parties are interacting with each other (the user and the remote service), and where the remote model is implemented on a server, belonging to the service operator, that is untrusted by the user.

Requirements for Remote Explainability

Explainability is often presented as a solution to increase the acceptance of AI [START_REF] Adadi | Peeking inside the blackbox: A survey on explainable artificial intelligence (xai)[END_REF], and to potentially prevent discriminative AI behaviour. Let us expose the logic behind this connection.

Explanations using conditional reasoning First, we need to define what is an explanation, to go beyond Miller's definition as an "answer to a why-question" [START_REF] Miller | Explanation in artificial intelligence: Insights from the social sciences[END_REF]. Since the topic of explainability is becoming a hot research field with (to the best of our knowledge) no consensus on a more technical definition of an explanation, we will propose for the sake of our demonstration that an explanation is causally coherent, with respect to the modus ponens rule from deductive reasoning (it stands for "if A is implying B, and A being true, B is true as well") [START_REF] Cummins | Conditional reasoning and causation[END_REF]. For instance, if explanation a explains decision b, it means that in context a, the decision produced will necessarily be b. In this light, we first directly observe the beneficial effect of such explanations on our parallel to club bouncing: while refusing someone, the bouncer may provide him with the reasons of that rejection; the person can then correct their behaviour in order to be accepted on next attempt. Second, this modus ponens explanation form is also sufficient to prove non-discrimination. For instance, if a does not involve discriminating arguments (which can be checked by the user as a is a sentence), and a ⇒ b, then decision b is not discriminative in case a. On the contrary, if a does involve discriminating arguments, then decision b is taken on a discriminative basis, and is therefore a discriminative decision. In other words, this property of an explanation is enough to reveal discrimination.

To sum up, any explanation framework that behaves "logically" (i.e., fits the modus ponens [START_REF] Cummins | Conditional reasoning and causation[END_REF]) -which is in our view a rather mild assumption-is enough to establish the discriminative basis of a decision. We believe this is the rationale of the statement "transparency can improve users trust in AI systems". In fact, this logical behaviour is not only sufficient to establish discrimination, it is also necessary: assume a framework providing explanation a for decision b such that we do not have a ⇒ b. Since a and b are not connected anymore, a does not bring any information about b.

While this logical behaviour is desirable for users, unfortunately in a remote context they cannot check whether a ⇒ b is in general true because they are only provided with a particular explanation a leading to a particular decision b. They cannot check that a being true leads in all contexts to b being true.

Requirements on the user side for checking explanations In a nutshell, a user in a remote interaction can verify that in her context a is true, and b is true, which is compatible with the a ⇒ b relation of an explanation fitting the modus ponens. Let us formalise what can a user check regarding the explanation she collects. A user that queries a classifier C with an input x gets two elements: the decision (inferred class) y = C(x) and an explanation a such that a explains y. Formally, upon request x, a user collects y and a = exp C (y, x).

We assume that such a user can check that a is apropos (i.e., appropriate): a corresponds to her input x. Formally, we write a ∈ A(x). This allows us to formally write a non-discriminatory explanation as a ∈ A(x l ). This forbids lying by explaining an input that is different than x.

We also assume that the user can check the explanation is consequent: user can check that a is compatible with y. This forbids crafting explanations that are incoherent w.r.t. the decision (like a bouncer that would explain why you can enter in while leaving the door locked).

To produce such explanations, we assume the existence of an explanation framework exp C producing explanations for classifier C (this could for instance by the LIME framework [START_REF] Ribeiro | Explaining the predictions of any classifier[END_REF]). The explanation a explaining decision y in context x by classifier C is written a = exp C (y, x).

Having defined the considered model for exposing our results, we stress that this model aims at constraining the provider as much as possible (e.g., explanations must be as complete as possible, are always provided, must always be coherent with decision, etc.). The intuition being that if we prove the possibility of malicious explanations in this constrained case, then the implementation in all less constrained cases, such as for incomplete explanation [START_REF] Andreou | Investigating Ad Transparency Mechanisms in Social Media: A Case Study of Facebook's Explanations[END_REF], or example-based explanations will only be easier.

To sum up on the explanation model: explanations fitting the modus ponens allow users to detect discrimination. Unfortunately in a remote context, users cannot check whether explanations do fit the modus ponens. However they can check the veracity of the explanation and the decision in their particular experience. This is the space we exploit for our attack, by generating malicious explanations that appear correct to the user (yet that do not fit the modus ponens).

Limits of Remote Explainability:

The PR (Public Relations) Attack

We articulate our demonstration of the limits of explainability in a remote setup by showing that a malicious service provider can hide the use of discriminating features for issuing its decisions, while conforming to the mild explainability framework we described in the previous subSection. Such a malicious provider thus wants to i) produce decisions based on discriminative features and to ii) produce non-discriminatory explanations to avoid prosecution.

A first approach could be to manipulate the explanation directly. It might however be difficult to do so while keeping the explanation convincing and true in an automated way. In this paper, we follow another strategy that instead consists in inventing a legitimate classifier that will then be explained.

A Generic Attack Against Remote Explainability

We coin this attack the Public Relations attack (noted PR). The idea is rather simple: upon reception of an input x, first compute discriminative decision C(x). Then train a surrogate model C that is non-discriminative, and such that C (x) = y. Explain C (x), and return this explanation along with C(x).

Figure 2 illustrates a decision based solely on legitimate features (A.), a provider giving an explanation that includes discriminatory features (B.), and the attack by a malicious provider (C.). In all three scenarios, a user is querying a remote service with inputs x, and obtaining decisions y each along with an explanation. In case B., the explanation exp C reveals the use of discriminative features X d ; this provider is prone to complaints. To avoid these, the malicious provider (C.) leverages the PR attack, by first computing C(x) using its discriminative classifier C. Then, based on the legitimate features x l of the input, and its final (discriminative) decision y, it derives a classifier C for the explanation.

Core to the attack is the ability to derive such classifier C : Definition 1 (PR attack). Given an arbitrary classifier C ∈ C d , a PR attack is a function that finds for an arbitrary input x a classifier C :

PR(C, x, C(x)) → C , (1)
x = (x l , ∅) such that C satisfies two properties:

C → y Remote x y, exp C (y, (x l , ∅)) User A. x = (x l , x d ) C → y Remote x y, exp C (y, (x l , x d )) Discriminated User B. (x l , x d ) C → y PR(C, (x l , x d ), y) → C s.t. C (x l ) = y x y, exp C (y, (x l , ∅)) Discriminated & Fooled User C.
• coherence: C (x l ) = y. • legitimacy: C ∈ C l .
Informally, coherence ensures that the explanation (derived from C ) appears consequent to the user observing decision y, while legitimacy ensures the explanation will appear to the user as originating from the "modus ponens" explanation of a non-discriminating classifier.

Effectiveness of the attack: Let us consider the perspective of a user whom upon request x collects a y answer along with an explanation a. Observe that a = exp C (y, x) is apropos since it directly involves x : a ∈ A(x). Since we have C (x) = y it is also consequent. Finally, observe that since C ∈ C l , then a ∈ A(x l ): a is non-discriminatory. So from the user perspective, she collects an apropos and consequent explanation that could originate from the logical explanation of a legitimate classifier.

Existence of the attack: We note that crafting a classifier C satisfying the first property is trivial since it only involves a single data point x. An example solution is the Dirac delta function of the form:

C (x ) = C ((x l , x d )) = δ x l ,x l if y = 1 1 -δ x l ,x l if y = 0 ,
where δ is the Dirac delta function. Informally, this solution corresponds to defining the classifier that would only answer "bounce" to this specific input x, and answer "enter" to any other input. While a corresponding intuitive explanation could be "because it is specifically you", explaining this very specific function might not fit any explainability framework. To alleviate this concern, we provide an example implementation of a PR attack that produces legitimate decision trees from discriminating ones in section 4.1.

Dirac here only constitutes an example proving the existence of PR attack functions. It is important to realise that many such C PR attack functions exists: any function X l → Y that satisfies one easy condition: Proof. Pick x l ∈ X l and y = C(x) the decision with which our PR attack function must be coherent. Since C l is a set of functions defined over X l , any particular function C in C l is defined at x l . Let us partition the function space C l according to the value these functions take at x l : let A : {C ∈ C l s.t. C(x l ) = y} and B : {C ∈ C l s.t. C(x l ) = ȳ}. We have C l = A ∪ B.

C (x) = y. Proposition 1. Let C l : X l → Y = {0,
Let not : A → B be a "negation function" that associate for each function C ∈ A its negation not(C) ∈ B s.t. not(C)(x) = 1 -C(x). Observe that not • not = Id: not defines a bijection between A and B (any function in A has exactly one unique corresponding function in B and vice versa). Since not is a bijection, we deduce

|A| = |B| = |C l |/2.
Since A contains all possible legitimate functions (A ⊂ C l ) that are coherent with C(x l ) = y, A = PR.

Thus |PR| = |C l |/2
In other words, PR attack functions are easy to find: if one could sample C l uniformly at random, since C (x) = y is equally likely than C (x) = ȳ, each sample would yield a PR attack function with probability 1/2.

We have presented the framework and an attack necessary to question the possibility of remote explainability. We next discuss the possibility for a user to spot that an explanation is malicious and obtained by a PR attack. We stress that if a user cannot, then the very concept of remote explainability is at stake.

The bouncer problem: spotting PR attacks

We presented in the previous section a general setup for remote explainability. We now formalise our research question regarding the possibility of a user to spot an attack in that setup.

Definition 2 (The bouncer problem (BP )). Using requests that each returns a decision y i = C(x i ) and an explanation exp C (y i , x), we denote by BP ( ), decide if C ∈ C d .

An Impossibility Result for One-Shot Explanations

We already know that using a single input point is insufficient:

Observation 1. BP (1) has no solution.

Proof. The Dirac construction above always exists.

Indeed, constructions like the introduced Dirac function, or the tree pruning construct a PR attack that produces explainable decisions. Given a single explanation on model C (i.e., = 1) the user cannot distinguish between the use of a model (C in case A.), or the one of a crafted model by a PR attack (C in case C.), since it is consequent. This means that such a user cannot spot the use of hidden discriminatory features due to the PR attack by the malicious provider.

We observed that a user cannot spot a PR attack, with BP (1). This is already problematic, as it gives a formal proof on why Facebook ad explanation system cannot be trusted [START_REF] Andreou | Investigating Ad Transparency Mechanisms in Social Media: A Case Study of Facebook's Explanations[END_REF].

The Hardness of Multiple Queries for Explanation

To address the case BP ( > 1), we observe that a PR attack generates a new model C for each request; in consequence, an approach to detect that attack is to detect the impossibility (using multiples queries) of a single model C to produce coherent explanations for a set of observed decisions. We here study this approach. Interestingly, classifiers and bouncers share this property that their outputs are all mutually exclusive (each input is mapped to exactly one class). Thus we have Enter ⇒ Bounce (with Enter and Bounce the positive or negative decision to for instance enter a place). In which case it is impossible to have a ⇒ Enter and a ⇒ Bounce. Note that this relation assumes a "logical" explainer. On a non logical explainer, since we cannot say a ⇒ Enter given a and Enter, we cannot detect such attack. Note also that non-mutually exclusive outputs (e.g. in the case of recommenders where recommending item a does not imply not recommending item b) are not bound by this rule.

A potential problem for the PR attack is a decision conflict, in which a could explain both b and b its opposite. For instance, imagine a bouncer refusing you the entrance of a club because, say, you have white shoes. Then, if the bouncer is coherent, he should refuse the entrance to anyone wearing white shoes, and if you witness someone entering with white shoes, you could argue against the lack of coherence of the bouncer decisions. We build on those incoherences to spot PR attacks.

In order to examine the case BP ( ), where > 1, we first define the notion of an incoherent pair :

Definition 3 (Incoherent Pair -IP). Let x 1 = (x 1 l , x 1 d ), x 2 = (x 2 l , x 2 d ) ∈ X = X l × X d
be a two input points in the feature space. x 1 and x 2 form an incoherent pair for classifier C iff they both have the same legit feature values in X l and yet end up being classified differently:

x

1 l = x 2 l ∧ C(x 1 ) = C(x 2 )
. For convenience we write

(x 1 , x 2 ) ∈ IP C .
Finding such an IP is a powerful proof of PR attack on the model by the provider. Intuitively, this is a formalization of an intuitive reasoning: "if you let others enter with white shoes then this was not the true reason for my rejection": Proposition 2. Only decisions resulting from a model crafted by a PR attack (1) can exhibit incoherent pairs:

IP C = ∅ ⇒ C ∈ C d . Proof. We prove the contra-positive form C ∈ C d ⇒ IP C = ∅. Let C ∈ C d . Therefore C ∈ C l , and by defini- tion: ∀x l ∈ X l , ∀x d , x d ∈ X 2 d , C((x l , x d )) = C((x l , x d )). By contradiction assume IP C = ∅. Let (x 1 , x 2 ) ∈ IP C : x 1 l = x 2 l ∧ C(x 1 ) = C(x 2 ). This directly contradicts C ∈ C l . Thus IP C = ∅.
We can show that there is always a pair of inputs allowing to detect a discriminative classifier C ∈ C d . Proposition 3. A classifier C , resulting from a PR attack, always has at least one incoherent pair:

C ∈ C d ⇒ IP C = ∅.
Proof. We prove the contrapositive form IP C = ∅ ⇒ C ∈ C d . Informally, the strategy here is to prove that if no such pair exists, this means that decisions are not based on discriminative features in X d , and thus the provider had no interest in conducting a PR attack on the model; the considered classifier is not discriminating.

Assume that IP C = ∅. Let x ∅ ∈ X d , and let C l : X l → Y be a legitimate classifier such that C l (x l ) = C((x l , x ∅ )).

Since Proof. Straightforward: C ∈ C d ⇒ IP C = ∅, and since IP ⊆ X × X testing the whole input space will necessarily exhibit such an incoherent pair. This last result is rather weakly positive: even though any PR attack is eventually detectable, in practice it is impossible to exhaustively explore the input space of modern classifiers due to their dimension. This remark also further questions the opportunity of remote explainability.

IP C = ∅, this means that ∀x 1 , x 2 ∈ X , x 1 l = x 2 l ⇒ C(x 1 ) = C(x 2 ). In particular ∀x ∈ X , C(x = (x l , x d )) = C l (x l , x ∅ ). Thus C = C l ;
Moreover, it is important to observe that while finding an IP proves the presence of a PR attack, it is not an efficient technique to prove the absence of a PR attack, which is probably the use case interesting users the most. Section 4.2 details this approach on a concrete dataset.

This concludes the theoretical perspective of this paper. To sum up, an explainer that could allow to spot classifier discrimination should behave logically, this is what is expected by the users. However, they can only check the properties of the provided explanation with regards to input x, which leaves room for malicious providers. One such provider can just "invent" a legit explainer whose decision matches the discriminative one for input x. Fortunately, this technique can be detected. This detection is however difficult in practice, as we will illustrate next.

Illustration and Experiments

In this section, we instantiate concretely some of the points raised by our theoretical perspective. We first illustrate the ease of finding PR attack functions on binary decision trees by presenting an algorithm that implements a PR attack. We then focus on detection, and evaluate in practice the hardness of finding incoherent pairs on the German Credit dataset.

Illustration using Decision Trees

In this section, we embody the previous observations and approaches on the concrete case of decision tree classifiers. The choice of decision trees is motivated first because of its recognised importance (e.g., C4.5 ranked number one of the top ten data mining algorithms [START_REF] Wu | Top 10 algorithms in data mining[END_REF]). Second, there is a wide consensus on their explainability, that is straightforward [START_REF] Molnar | Interpretable Machine Learning[END_REF]: a path in the tree "naturally" lists the attributes considered by the algorithm to establish a classification. Finally, the simplicity of crafting PR attacks on those make them good candidates for an illustration and argues for the practical implementability of such an attack.

We denote T as the set of tree-based classifiers. We do not need any assumption on how the tree is built (e.g., C4.5 [START_REF] Quinlan | C4. 5: programs for machine learning[END_REF]). Regarding explainability, we here only need to assume that decision trees are explainable: ∀C ∈ T , exp C exists.

Let C ∈ T ∩ C d be a discriminatory binary tree classifier. Each internal node n ∈ V (C) tests incoming examples based on a feature n.label. Each internal node is connected to exactly two sons in the tree, named n.r and n.l for right and left. Depending on the (binary) result of this test, the example will continue on either of these paths. We denote the father of an internal node by n.f ather (the root node r is the only node such that r.f ather = ∅).

Algorithm 1 presents a PR attack on binary decision trees. To ease its presentation, we assume that given an input x, n.r (right) will by convention always be the branch taken after evaluating x on n. The algorithm starts by initializing the target decision tree C as a copy of C. Then, it selectively removes all nodes involving discriminative features, and replaces them with the subtree the target example x would take. To do so, Algorithm 1 removes each discriminative node n i by connecting n i-1 and n i+1 . While this approach would be problematic in the general case (we would loose the n i .l subtree), in the context of x we know the explored branch is n i .r, so we simply reconnect this branch, and replace the left subtree by a dummy output.

An example is presented in Figure 3: the discriminative classifier C is queried for the explanation exp C (C(x), x) of input x. To produce an answer for a discriminative feature such as the age, it first applies Algorithm 1 on C, given the query x. If x < 60 (upper right in Figure 3), the explanation exp C has simply became a node with the age limit, leading to an "Enter" decision. In case x ≥ 60, the explanation node is a legit one ("Disguised"), leading to the "Bounce" decision. Both explanation then do not exhibit the fact that the provider relied on a discriminative feature in C. This exhibits that BP (1) does not have a solution.

Comparing both versions of C easily yields solutions for BP (2), for instance (Disguised, withsocks, Age = 49) and (Disguised, withsocks, Age = 62). Proof. To prove the statement, we need to prove that:

• C is legitimate • C is coherent: C (x l ) = y • C is explainable
First, observe that any nodes of C containing discriminative features is removed line 7. Thus, C only takes decisions based on features in X l : C is legitimate.

Second, observe that by construction since x = (x l , x d ), and since any discriminative node n is replaced by this right (n.r) outcome which is the one that corresponds to x d . In other words, ∀x l ∈ X l , C (x l ) = C((x l , x d )): C behaves like C where discriminative features are evaluated at x d . This is true in particular for

x l : C (x l ) = C((x l , x d )) = C(x) = y.
Finally, observe that C is a valid decision tree. Therefore, according to our explainability framework, C is explainable.

Interestingly, the presented attack can be efficient as it only involves pruning part of the target tree. In the worst case, this one has Ω(2 d ) elements, but in practice decision trees are rarely that big.

Finding IPs on a Neural Model: the German Credit Dataset

We now take a closer look at the detectability of the attack, namely: how difficult is it to spot an IP ? We illustrate this by experimenting on the German Credit Dataset.

Experimental setup We leverage Keras over Ten-sorFlow to learn a neural network-based model for the German Credit Dataset [START_REF]Statlog[END_REF]. While we could have used any relevant type of classifier for our experiments, general current focus is on neural networks regarding explainability. The bank dataset classifies client profiles (1, 000 of them), described by a set of attributes, as good or bad credit risks. Multiple techniques have been employed to model the credit risks on that dataset, which range from 76.59% accuracy for a SVM to 78.90% for a hybrid between genetic algorithm and a neural network [START_REF] Oreski | Genetic algorithm-based heuristic for feature selection in credit risk assessment[END_REF].

The dataset is composed of 24 features (some categorical ones, such as sex, of status, were set to numerical). This thus constitutes a low dimensional dataset as compared to current applications (observations in [START_REF] Andreou | Investigating Ad Transparency Mechanisms in Social Media: A Case Study of Facebook's Explanations[END_REF] reported up to 893 features for the sole application of ad placement on user feeds on Facebook). Furthermore, modern classifiers are currently dealing with up to 512 × 512 × 3 dimensions [START_REF] Brock | Large scale GAN training for high fidelity natural image synthesis[END_REF], which permit a significant increase in data processing and thus the capability to expand the amount of features taken into account for decision-making.

The neural network we built2 is inspired by the one proposed [START_REF] Khashman | Neural networks for credit risk evaluation: Investigation of different neural models and learning schemes[END_REF] in 2010, and that reached 73.17% accuracy. It is a simple multi-layer perceptron, with a single hidden layer of 23 neurons (with sigmoid activations), and a single output neuron for the binary classification of the input profile to "risky" or not. In this experiment we use the Adam optimizer and a learning rate of 0.1 (leading to much faster convergence than in [START_REF] Khashman | Neural networks for credit risk evaluation: Investigation of different neural models and learning schemes[END_REF]), with a validation split of 25%. We create 30 models, with an average accuracy of 76.97%@100 epochs on the validation set (with a standard deviation of 0.92%).

In order to generate input profiles, we consider two scenarios. In A) we consider a scenario where a user sets a random value in a discriminative feature to try to find an IP. This yields rather artificial user profiles (that may be detected as such by the remote service provider). To have an aggregated view of this scenario, we proceed as follows. For each of the 30 models, we randomly select 50 users as a test set (not used for training the previous models). We then repeat 500 times the following: we select a random user among the 50 and select a random discriminative feature among the four, to set a random (uniform) value in it (belonging to the domain of each selected feature, e.g., from 18 to 100 in the age feature). This creates a set of 15, 000 fake profiles as inputs.

In B), in order to have a more realistic scenario where profiles are created from real data from the dataset, we now proceed as follows. We also select 50 profiles from the dataset as a test set, so we can perform our core experiment: the four discriminatory features of each of those profiles are sequentially replaced by the ones of the 49 remaining profiles; each resulting test profile is fed to the model for prediction. (This permits to test the model with realistic values in those features. This process creates 2450 profiles to search of an IP). We count the number of times the output risk label has switched, as compared to the original untouched profile fed to the model. We repeat this operation on the 30 models to observe deviations.

The low probability of findings IPs at random

In the case of scenario A), we compare the original label with the one obtained from each crafted input. We obtain 8.09% of IPs (standard deviation of 4.08).

Figure 4 depicts for scenario B) the proportion of label changes over the total number of test queries; recall that a label change while considering two inputs constitutes an IP. We observe that if we just change one of the four features, we obtain on average 1.86%, 0.27%, 1.40%, 2.27% labels changes (for the employment, sex/status, age, foreigner features, respectively), while 4.25% if the four features are simultaneously changed. (Standard deviations are of 1.48%, 0, 51%, 1, 65%, 2, 17% and 3, 13%, respectively).

This probability of 4.25% is higher than our deterministic lower bound BP (|X |) (Proposition 4), hinting that this discriminating classifier is easier to spot that the worst-case one. Moreover, since not finding an IP after some requests does not guarantee the absence of a discriminating behaviour, we now look at the user-side perspective: testing the absence of discrimination of a remote service. It turns out that we can compute an expectation of the number of queries for such a user to find an IP.

Users can query the service with inputs, until they are confident enough that such pair does not exist. Assuming one seeks a 99% confidence level -that is, less than one percent of chances to falsely detect a discriminating classifier as non-discriminating-, and using the detection probabilities of Figure 4, we can compute the associated p-values. A user testing a remote service based on those hypotheses would need to craft respectively 490, 2555, 368, 301 and 160 (for the employment, sex/status, age, foreigner, and all four respectively) pairs in the hope to decide on the existence or not of an IP, as presented in Figure 5 (please note the log-scale on the y-axis).

Those experiments highlight the hardness to experimentally check for PR attacks.

Discussion

We now list in this section several consequences of the findings in this paper, and some open questions.

Findings and Applicability

We have shown that a malicious provider can always craft a fake explanation to hide its use of discriminatory features, by creating a surrogate model for providing an explanation to a given user. An impossibility result follows, for a user to detect such an attack while using a single explanation. The detection by a user, or a group of users, is possible only in the case of multiple and deliberate queries (BP ( > 1)); this process may require an exhaustive search of the input space.

However, we see that our practical experiment on the German Credit Dataset is far from this bound. Intuitively the probability of finding an IP is proportional to the "discrimination level" of a classifier. While quantifying such level is a difficult task, we explore a possible connection in the next paragraph.

We note that the malicious providers have another advantage for covering PR attacks. Since multiple queries must be issued to spot inconsistencies via IP pairs, basic rate limiting mechanisms for queries may block and ban the incriminated users. Defenses of this kind, for preventing attacks on online machine services exposing APIs, are being proposed [START_REF] Hou | Ml defense: Against prediction api threats in cloud-based machine learning service[END_REF]. This adds another layer of complexity for the observation of misbehaviour.

Connection with Disparate Impact

We now briefly relate our problem to disparate impact: a recent article [START_REF] Feldman | Certifying and removing disparate impact[END_REF] proposes to adopt "a generalization of the 80 percent rule advocated by the US Equal Employment Opportunity Commission (EEOC)" as a criteria for disparate impact. This notion of disparate impact proposes to capture discrimination through the variation of outcomes of an algorithm under scrutiny when applied to different population groups.

More precisely, let α be the disparity ratio. The authors propose the following formula, here adapted to our notations [START_REF] Feldman | Certifying and removing disparate impact[END_REF]:

α = P(y|x d = 0) P(y|x d = 1)
, where X d = {0, 1} is the discriminative space reduced to a binary discriminatory variable. Their approach is to consider that if α < 0.8 then the tested algorithm could be qualified as discriminative.

To connect disparate impact to our framework, we can conduct the following strategy. Consider a classifier C having a disparate impact α, and producing a binary decision C(x) ∈ {0 = "bounce", 1 = "enter"}. We search for Incoherent Pairs as follows: first, pick x ∈ X l a set of legit features. Then take a = (x, x d = 0), representing the discriminated group, and b = (x, x d = 1) representing the undiscriminated group. Then test C on both a and b: if C(a) = C(b) then (a, b) is an IP. The probability P of finding an IP in this approach can be written as P(IP ). Let A (resp. B) be the event "a enters" (resp. "b enters").

We Using conditional probabilities, we have P(A ∩ B) = P(B|A).P(A). Thus P(IP ) = P(B)(1 + α -2α.P(B|A)).

Since the conditional probability P(B|A) is difficult to assess without further hypotheses on C, let us investigate two extreme scenarios:

• Independence: A and B are completely independent events, even though a and b share their legit features in X l . This scenario, which is not very realistic, could model purely random decisions with respect to attributes from X d . In this scenario P(B|A) = P(B).

• Dependence: A ⇒ B: if A is selected despite its membership to the discriminated group (a = (x, 0)), then necessarily b must be selected, as it can only be "better" from C's perspective. In this scenario P(B|A) = 1. Figure 6 represents the numerical evaluation of our two scenarios. First, it shows that the probability of finding an IP strongly depends on the probability of a success for the non-discriminated group P(B). Indeed, since the discriminated group has an even lower probability of success, a low success probability for the non-discriminated group implies frequent cases where both a and b are failures, which does not constitutes an IP.

In the absence of disparate impact (α = 1), both scenarios provide very different results: the independence scenario easily identifies IPs -which is coherent with the "random" nature of the independence assumption-. This underlines the unrealistic nature of the independence scenario in this context. With a high disparate impact however (e.g., α = 0.1), the discriminated group has a high probability of failure. Therefore the probability of finding an IP is very close to the simple probability of the non-discriminated group having a success P(B), regardless of the considered scenario.

The dependence scenario nicely illustrates a natural connection: the higher the disparate impact, the higher the probability to find an IP. While this only constitutes a thought experiment, we believe this highlights possible connections with standard discrimination measures and conveys the intuition that in practice, the probability of finding IPs exposing a PR attack strongly depends on the intensity of the discrimination hidden by that PR attack.

Open Problems for Remote Explainability

On the test efficiency It is common for fairness assessment tools to leverage testing. As the features that are considered discriminating are often precise [START_REF] Galhotra | Fairness testing: Testing software for discrimination[END_REF][START_REF] Hajian | Rule protection for indirect discrimination prevention in data mining[END_REF], the test queries for fairness assessment can be targeted and some notions of efficiency in terms of the amount of requests can be derived. This may be done by sampling the feature space under question for instance (as in work by Galhotra et al. [START_REF] Galhotra | Fairness testing: Testing software for discrimination[END_REF]).

Yet, it appears that with current applications such as social networks [START_REF] Andreou | Investigating Ad Transparency Mechanisms in Social Media: A Case Study of Facebook's Explanations[END_REF], users spend a considerable amount of time online, producing more and more data that turn into features, and also are the basis to the generation of other meta-features. In that context, the full scope of features, discriminating or not, may not be clear to a user. This makes exhaustive testing even theoretically unreachable, due to the very likely non-complete picture of what providers are using to issue decisions. This is is another challenge on the way to remote explainability, if providers are not willing to release a complete and precise list of all attributes leveraged in their system.

Towards a provable explainability? Some other computing applications, such as data storage or intensive processing also have questioned the possibility of malicious service providers in the past. Motivated by the plethora of offers in the cloud computing domain and the question of quality of service, protocols such as proof of data possession [START_REF] Ateniese | Provable data possession at untrusted stores[END_REF], or proof-based verifiable computation [START_REF] Braun | Verifying computations with state[END_REF], assume that the service provider might be malicious. A solution to still have services executed remotely in this context is then to rely on cryptographic protocols to formally verify the work performed remotely. To the best of our knowledge, no such provable process logic has been adapted to explainability. That is certainly an interesting development to come.

Related Work

Explaining in-house models As a consequence of the major impact of machine learning models in many areas of our daily life, the notion of explainability has been pushed by policy makers and regulators. Many works address explainability of inspected model decisions on a local setup (please refer to surveys [START_REF] Datta | Algorithmic transparency via quantitative input influence: Theory and experiments with learning systems[END_REF][START_REF] Guidotti | A survey of methods for explaining black box models[END_REF][START_REF] Molnar | Interpretable Machine Learning[END_REF]) -some specifically for neural network models [START_REF] Yeh | Representer point selection for explaining deep neural networks[END_REF]-, where the number of requests to the model is unbounded. Regarding the question of fairness, a recent work specifically targets the fairness and discrimination of in-house softwares, by developing a testing-based method [START_REF] Galhotra | Fairness testing: Testing software for discrimination[END_REF].

Dealing with remote models The case of models available through a remote black-box interaction setup is particular, as external observers are bound to scarce data (labels corresponding to inputs, while being limited in the number of queries to the black-box [START_REF] Tramèr | Stealing machine learning models via prediction apis[END_REF]). Adapting the explainability reasoning to models available in a black box setup is of a major societal interest: Andreou et al. [START_REF] Andreou | Investigating Ad Transparency Mechanisms in Social Media: A Case Study of Facebook's Explanations[END_REF] shown that Facebook's explanations for their ad platform are incomplete and sometimes misleading. They also conjecture that malicious service providers can "hide" sensitive features used, by explaining decisions with very common ones. In that sense, our paper is exposing the hardness of explainability in that setup, confirming that malicious attacks are possible. Milli et al. [START_REF] Milli | Model reconstruction from model explanations[END_REF] provide a theoretical ground for reconstructing a remote model (a two-layer ReLu neural network) from its explanations and input gradients; if further research proves the approach practical for current applications, this technique may help to infer the use of discriminatory features in use by the service provider.

Operating without trust: the domain of security

In the domain of security and cryptography, some similar setups have found a large body of work to solve the trust problem in remote interacting systems. In proof of data possession protocols [START_REF] Ateniese | Provable data possession at untrusted stores[END_REF], a client executes a cryptographic protocol to verify the presence of her data on a remote server; the challenge that the storage provider responds to assesses the possession or not of some particular piece of data. Protocols can give certain or probabilistic guarantees. In proof-based verifiable computation [START_REF] Braun | Verifying computations with state[END_REF], the provider returns the results of a queried computation, along with a proof for that computation. The client can then check that the computation indeed took place. These schemes, along with this paper exhibiting attacks on remote explainability, motivate the need for the design of secure protocols.

Discrimination and bias detection approaches

Our work is complementary to classic discrimination detection in automated systems. In contrast to works on fairness [START_REF] Binns | Fairness in machine learning: Lessons from political philosophy[END_REF] that attempt to identify and measure discrimination from systems, our work does not aim at spotting discrimination, as we have shown it can be hidden by the remote malicious provider. We instead are targeting the occurrence of incoherent explanations produced by such a provider in the will to cover its behavior, which is a a completely different nature than fairness based test suites. Galhotra et al. [START_REF] Galhotra | Fairness testing: Testing software for discrimination[END_REF], inspired by statistical causality [START_REF] Pearl | Causal inference in statistics: An overview[END_REF], for instance propose to create input datasets for observing discrimination on some specific features by the system under test.

While there are numerous comments and proposals for good practices when releasing models that may include forms of bias [START_REF] Mitchell | Model cards for model reporting[END_REF], the automatic detection of bias on the user side is also of interest for the community. For instance, researchers seek to detect the Simpson's Paradox [START_REF] Blyth | On simpson's paradox and the sure-thing principle[END_REF] in the data [START_REF] Alipourfard | Using simpson's paradox to discover interesting patterns in behavioral data[END_REF]. Another work makes use of causal graphs to detect [START_REF] Zhang | Achieving nondiscrimination in data release[END_REF] a potential discrimina-tion in the data, while authors propose in [START_REF] Hajian | A methodology for direct and indirect discrimination prevention in data mining[END_REF] to purge the data so that direct and/or indirect discriminatory decision rules are converted to legitimate classification rules. Some works are specific to some applications, such as financial ones [START_REF] Zhang | Fairness assessment for artificial intelligence in financial industry[END_REF]. Note that those approaches by definition require an access to the training data, which is a too restrictive assumption in the context of our target contribution.

The work in [START_REF] Tan | Distilland-compare: Auditing black-box models using transparent model distillation[END_REF] proposes to leverage transfer learning (or distillation) to mimic the behaviour of a black box model, here a credit scoring model. A collection campaign is assumed to provide a labeled dataset with the risk scores, as produced by the model and the ground-truth outcome. From this dataset is trained a model aiming at mimicking the black box as close as possible. Both models are then compared on their outcome, and a method to estimate the confidence interval for the variance of results is presented. The trained model can then be queried to assess potential bias. This approach proves solid guarantees when one assumes that the dataset is extracted from a black box that does not aim to bias its outputs to prevent audits of that form.

The rationalization of explanations More closely related to our work is the recent paper by Aivodji et al. [START_REF] Aivodji | Fairwashing: the risk of rationalization[END_REF], that introduces the concept of rationalization, in which a black-box algorithm is approximated by a surrogate model that is "fairer" that the original blackbox. In our terminology, they craft C models that optimise arbitrary fairness objectives. To achieve this, they explore decision tree models trained using the black-box decisions on a predefined set of inputs. This produces another argument against black-box explainability in a remote context. The main technical difference with our tree algorithm section 4.1 is that their surrogates C optimises an exterior metric (fairness) at the cost of some coherence (fidelity in the authors' terminology). In contrast, our illustration section 4.1 produces surrogates with perfect coherence that do not optimise any exterior metric such as fairness. In our model, spotting an incoherence (i.e., the explained model produces a y while the black-box produces a ȳ) would directly provide a proof of manipulation and reveal the trickery. Interestingly, the incoherent pair approach fully applies in the context of their model surrogates, as it arises as soon as more than one surrogate is used (regardless of the explanation). Our paper focuses on the userside observation of explanations, and users ability to discover such attacks. We rigorously prove that single queries are not sufficient to determine a manipulation, and that the problem is hard even in the presence of multiple queries and observations.

Conclusion

In this paper, we studied explainability in a remote context, which is sometimes presented as a way to satisfy society's demand for transparency facing automated de-cisions. We prove it is unwise to blindly trust those explanations: like humans, algorithms can easily hide the true motivations of a decision when asked for explanation. To that end, we presented an attack that generates explanations to hide the use of an arbitrary set of features by a classifier. While this construction applies to any classifier queried in a remote context, we also presented a concrete implementation of that attack on decision trees. On the defensive side, we have shown that such a manipulation cannot be spotted by one-shot requests, which is unfortunately the nominal use-case. However, the proof of such trickery (pairs of classifications that are not coherent) necessarily exists. We further evaluated in a practical scenario the probability of finding such pairs, which is low. The attack is thus arguably impractical to detect for an isolated user.

We conclude that this must consequently question the whole concept of the explainability of a remote model operated by a third party provider, at the very least. A research direction is to develop secure schemes in which the involved parties can trust the exchanged information about decisions and their explainability, as enforced by new protocols. A second line of research may be the collaboration of users observations for spotting the attack in an automated way. Indeed, as done by Chen et al. [START_REF] Chen | Peeking beneath the hood of uber[END_REF] for understanding the impact of Uber surge pricing on passengers and drivers (by deploying 43 Uber accounts that act as clients), data can be put in common to retrieve information more reliably. The anonymization of users data if a pool of measurements is made public is for sure a crucial point to ensure scalable observation of black box decision-making systems. We believe this is an interesting development to come, in relation with the promises of AI and automated decisions processes.

Figure 2 :

 2 Figure 2: (A.) A provider using a model that does not leverage discriminatory features. (B.) A discriminative model divulges its use of a discriminating feature. (C.) The PR attack principle, undermining remote explainability: the black-box builds a surrogate model C for each new request x, that decides y based on x l features only. It explains y using C .

  1} the set of all possible legit classifiers. Let PR ⊂ C l be the set of possible PR attack functions. We have |PR| = |C l |/2: half of all possible legit classifiers are PR attack functions.

  by the definition of a PR attack being only applied to a model that uses discriminatory features, this leads to C ∈ C \ C d , i.e., C ∈ C d . Which directly applies to our problem: Proposition 4 (Detectability lower bound). BP (|X |) is solvable.

Algorithm 1 : 7 C 8 C 9 else 10 C

 178910 PR attack on a discriminative decision binary tree C Input: C, x = (x l , x d ) 1 y = C(x) ; // Find discriminative decision 2 Let {n0, . . . n l } be breadth first ordering of the nodes of C; 3 Let C = C ; // Initialise surrogate 4 for node i = 0 to l do 5 if ni.label ∈ X d then 6 C .ni.f ather.r = ni.r ; // Reconnect ni father to right son = C \ {ni} ; // Remove discriminating node = C \ {ni.l subtree} ; // Remove left subtree .ni.l = ȳ ; // Keep legit node, add dummy terminal node 11 end 12 end 13 return y, exp C (y, (x l , ∅))

Proposition 5 .Figure 3 :

 53 Figure 3: Illustration of a possible implementation (Algorithm 1) of the PR attack: instead of having to explain the use of a discriminative feature (age in this case) in the classifier C, two non-discriminative classifiers (C , on the right) are derived. Depending on the age feature in the request, a C is then selected to produce legit explanations. The two dashed circles represent an Incoherent Pair, that users might seek to detect a malicious explanation.

Figure 4 :

 4 Figure 4: Percentage of label changes when swapping the discriminative features in the test set data for scenario B). Bars indicate standard deviations. Those indicate the low probability to spot a PR attack on the provider model).

Figure 5 :

 5 Figure 5: Confidence level as a function of the number of tested input pairs, based on the German Credit detection probability in Figure 4. The dashed line represents a 99% confidence level.

  can develop: P(IP ) = P(C(a) = C(b)) = P(A ∩ B) + P(A ∩ B) = P(A) -P(A ∩ B) + P(B) -P(A ∩ B) = P(B)(1 + α) -2P(A ∩ B), since α = P(A)/P(B).

Figure 6 :

 6 Figure 6: Probability to find an Incoherent Pair (IP), as a function of P(B) the probability of success for a nondiscriminated group. α represents the disparity ratio.

For instance in the U.K.: https://www.gov.uk/ discrimination-your-rights.

Code is made available at: https://github.com/ erwanlemerrer/bouncer_problem.