=Paper=
{{Paper
|id=Vol-3209/8151
|storemode=property
|title=On the Tradeoff Between Correctness and Completeness in Argumentative Explainable AI
|pdfUrl=https://ceur-ws.org/Vol-3209/8151.pdf
|volume=Vol-3209
|authors=Nico Potyka, Xiang Yin, Francesca Toni
|dblpUrl=https://dblp.org/rec/conf/comma/PotykaYT22
}}
==On the Tradeoff Between Correctness and Completeness in Argumentative Explainable AI==
https://ceur-ws.org/Vol-3209/8151.pdf
On the Tradeoff Between Correctness and
Completeness in Argumentative Explainable AI
Nico Potyka1 , Xiang Yin1 and Francesca Toni1
1
Imperial College London, UK
Abstract
Explainable AI aims at making the decisions of autonomous systems human-understandable. Argumen-
tation frameworks are a natural tool for this purpose. Among them, bipolar abstract argumentation
frameworks seem well suited to explain the effect of features on a classification decision and their
formal properties can potentially be used to derive formal guarantees for explanations. Two particular
interesting properties are correctness (if the explanation says that π affects π , then π affects π ) and
completeness (if π affects π , then the explanation says that π affects π ). The reinforcement property
of bipolar argumentation frameworks has been used as a natural correctness counterpart in previous
work. Applied to the classification context, it basically states that attacking features should decrease
and supporting features should increase the confidence of a classifier. In this short discussion paper, we
revisit this idea, discuss potential limitations when considering reinforcement without a corresponding
completeness property and how these limitations can potentially be overcome.
Keywords
Argumentation, Abstract Bipolar Argumentation, Explainable AI
1. Introduction
Automatic decision making is increasingly driven by black-box machine learning models.
However, their opaqueness raises questions about fairness, reliability and safety. Explanation
methods aim at making the decision process transparent [1]. In recent years, various explanation
methods have been proposed in the argumentation literature, we refer to [2, 3] for an overview.
One easily comprehensible argumentation model are bipolar abstract argumentation frame-
works. They represent arguments in a graph, where nodes correspond to abstract arguments
(entities that can be accepted or rejected) and edges to attack or support relationships between
them. This representation seems well suited to represent the influence of features in a classifica-
tion problem on the class decision. For example, in a credit approval setting, the income may
have a positive effect (support), while existing debts may have a negative effect (attack) on the
decision. This is in line with the idea of the reinforcement property in bipolar argumentation [4],
which roughly states that attacks should decrease and supports should increase the strength of
the addressed argument. More precisely, in a quantitative setting, the effect should be relative
to the strength of the attacker/supporter and our apriori belief in the addressed argument.
1st International Workshop on Argumentation for eXplainable AI (ArgXAI, co-located with COMMA β22), September 12,
2022, Cardiff, UK
$ n.potyka@imperial.ac.uk (N. Potyka); x.yin20@imperial.ac.uk (X. Yin); f.toni@imperial.ac.uk (F. Toni)
Β© 2022 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
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Nico Potyka et al. CEUR Workshop Proceedings 1β8
The faithfulness of an argumentative explanation to the classifier can then naturally be
evaluated by checking to which extent the reinforcement property is satisfied. In recent work,
authors actually guaranteed perfect faithfulness by showing only those connections that respect
reinforcement [5]. While this is a highly desirable correctness property of the explanation, it
leaves the question to which extent the explanation is complete. That is, while it is guaranteed
that all shown relationships have indeed the intended effect, it remains unclear if all effects on
the decision have been captured by the explanation.
We argue that, while reinforcement is a desirable property, it can be too strong when the
arguments in the argumentation framework are not carefully selected. In particular, introducing
a single argument per feature is often not sufficient. One reason is that classifiers are often
non-monotonic and a feature may have a positive effect in one and a negative effect in another
region of the input domain. Another reason is that the effect of features often cannot be captured
independently of the other features. Of course, this does not mean that we should give up
reinforcement. Rather, we should refine the bipolar abstract argumentation frameworks based
on the application domain to capture more complicated effects of features and to improve the
completeness of the explanation.
2. Probabilistic Classifiers and Correct Argumentative
Explanations
The abstract goal of classification is to map inputs x to outputs π¦. We think of the inputs
as vectors x = (π₯1 , . . . , π₯π ), where the i-th value is taken from some domain π·π . We let
π = Γππ=1 π·π denote the cartesian product of the individual domains. Given an input x =
(π₯1 , . . . , π₯π ), with a slight abuse of notation, we let (xβπ , π₯) = (π₯1 , . . . , π₯πβ1 , π₯, π₯π+1 , . . . , π₯π )
denote the input where the i-th component has been replaced with π₯. The output π¦ is taken
from a finite set πΏ of class labels. A classification problem π = ((π·1 , . . . , π·π ), πΏ, πΈ) consists
of π feature domains π·π , class labels πΏ and a set of training examples πΈ = {(xπ , π¦π ) | 1 β€ π β€
π, xπ β π, π¦π β πΏ}. For example, in a credit approval scenario, the first feature could be Age
with corresponding domain π·1 = N, the second feature Income with domain π·2 = R and the
third feature Debts with domain π·3 = {0, 1}. In this case, the label set could be πΏ = {0, 1}
indicating whether the credit application is accepted or rejected. πΈ then consists of previous
cases, that are used to train a classifier. We will not be concerned with training here and just
assume that a classifier is given and is supposed to be explained.
A probabilistic classifier is a βοΈ function C : π Γ πΏ β [0, 1] that assigns a probability C(x, π¦)
to every pair (x, π¦) such that πβπΏ C(x, π) = 1. C(x, π¦) β [0, 1] can be understood as the
confidence of the classifier that an example with features x belongs to the class π¦. A deterministic
classifier is the special case, where the probabilistic classifier always assigns probability 1 to one
label and 0 to all others. In general, a classification decision can be made by picking the label
with the highest probability or by defining a threshold value for the probability. To simplify
notation, we will often write Cπ¦ (x) instead of C(x, π¦) in the remainder.
A bipolar argumentation graph (BAG) is a tuple (π, Att, Sup) consisting of a set of abstract
arguments and attack and support relationships between them [6]. Here, BAGs are mainly
used as a tool to represent the effects of features on the class label and we will not discuss
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their various semantics and extensions in more detail. For our purposes, the most important
semantical property is reinforcement [4], which basically demands that attackers should weaken
and supporters should strengthen an argument.
In order to explain a classifier, we can associate the underlying classification problem with
a set of abstract arguments that allow us to argue about the classification decision. The most
straightforward way to do this is to introduce one argument per feature and one argument per
class label.
Definition 1 (Naive Classification Arguments). Given a classification problem π =
((π·1 , . . . , π·π ), πΏ, πΈ), the naive classification arguments associated with π are the
β’ π feature arguments π΄πΉπ , 1 β€ π β€ π for the π features and
β’ |πΏ| class arguments π΄πΆ
π , 1 β€ π β€ π for the class labels
We can then build an explanation BAG from a probabilistic classifier by taking the naive
classification arguments and adding support and attack edges from feature arguments to class
arguments.
Roughly speaking, we say that an explanation BAG satisfies reinforcement if the following
two conditions are satisfied:
1. If there is an attack from π΄πΉπ to π΄πΆπ , then increasing (decreasing) the value of the π-th
feature decreases (increases) the probability of the π-th class and
2. If there is an support from π΄πΉπ to π΄πΆπ , then increasing (decreasing) the value of the π-th
feature increases (decreases) the probability of the π-th class.
While there is a natural order for boolean and ordinal features, the definition has to be made
more precise for categorical features. However, it is sufficient for our purposes as we mainly
want to create awareness for potential limitations of this idea and how they can be addressed.
As discussed in the introduction, reinforcement is an interesting correctness/faithfulness
property of an explanation. However, we should not only consider correctness, but also com-
pleteness of the explanation. As an extreme example, the empty graph defined over the naive
classification arguments satisfies reinforcement. However, it is not a very interesting explana-
tion because it does not explain anything. In general, a corresponding completeness property is
desirable that explains to which extent the explanation graph captures the existing relationships.
As a first step in this direction, we explain reasons for why an explanation BAG can result in
incomplete explanations.
3. Completeness Problems for Boolean Data
To begin with, let us focus on boolean data. That is, we assume that we have π·π = {0, 1} for
all features. In this case, there is a straightforward (even though computationally expensive)
way to create an explanation BAG that satisfies reinforcement.
Definition 2 (Naive Explanation BAG). The naive explanation BAG for a probabilistic classifier
C : π Γ πΏ β [0, 1] for a classification problem π is the BAG (π, Att, Sup), where
β’ π contains all feature and class arguments for π ,
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β’ there is an attack edge from π΄πΉπ to π΄πΆ π iff for every assignment xβπ to the remaining
features, changing the
(οΈ π-th feature
)οΈ from 0 to 1 decreases the probability of ππ , that is,
πΆ (xβπ , 0), ππ > πΆ (xβπ , 1), ππ ,
(οΈ )οΈ
β’ there is a support edge from π΄πΉπ to π΄πΆ π iff for every assignment xβπ to the remaining
features, changing the π-th feature from 0 to 1 increases the probability of ππ , that is,
πΆ (xβπ , 0), ππ > πΆ (xβπ , 1), ππ ,
(οΈ )οΈ (οΈ )οΈ
β’ there are no other edges.
Let us note that one may also want to introduce attack arguments between the class arguments
(only one class argument should be accepted). We refrain from doing so here because, at
this point, we consider the explanation BAG merely as a visualization of the relationships
between feature and class arguments. However, when applying argumentation semantics to the
explanation BAG, these additional edges may be necessary.
While constructing the naive explanation BAG in a straightforward way takes exponential
time, it is easy to check that it satisfies reinforcement. However, in many cases, it will only
explain a fraction of the actual effects of features because it only looks at the individual effects
of features.
To illustrate the problem, consider the XOR function πππ
(π΄, π΅) that returns 1 if exactly
one of π΄ and π΅ is 1. In this case, our explanation BAG would actually not contain any edges.
However, π΄ and π΅ clearly have an effect on the decision, so that the explanation BAG is not
an accurate explanation even though it satisfies reinforcement. The problem in this example
is that the effect of features cannot be determined independently of each other. While the
XOR function is an extreme example, similar dependencies naturally occur in many datasets.
For example, in a credit approval setting, the highest educational degree may be relevant for
an application if the applicant has a low income (e.g., a student after graduation), but not if
the applicant has a high income. In a medical scenario, a drug may be an effective treatment
for most patients, but could be detrimental for patients with particular medical or physical
conditions.
In order to take account of such joint effects, we need to refine our naive explanation BAG.
Two natural candidate refinements are the following.
1. Increase the BAG: Introduce joint feature arguments that capture the state of multiple
arguments simultaneously.
2. Consider more expressive BAGs: Introduce joint attacks/ joint supports that capture
the joint effect of arguments rather than considering them independently. Joint effects
can be represented, for example, by logical formulas similar to ADFs [7] or by considering
abstract joint edges as considered in classical [8] and weighted SETAFs [9].
4. Completeness Problems for Continuous Data
Let us now look at continuous data. That is, we assume that π·π = [ππ , π’π ], ππ , π’π β R, ππ < π’π is
a real interval for all features. Now, the effect of features may not only depend on the effect of
other features, but the effects of features can actually change on the domain.
To illustrate the problem, consider a simple binary classification problem over a single feature
with domain π· = R. Intuitively, an input should be classified as positive if the value of the
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Figure 1: Function graphs of logistic function ππ (π₯) (left) and non-monotonic classifier C1 (π₯) = ππ (π₯2 )
(right). π₯ is classified as an anomaly if C1 (π₯) β₯ 0.7 (orange line). Increasing π₯ can have a negative (blue
point), neutral (red point) or positive (green point) effect on the probability dependent on the region
where it is located.
feature is βsufficiently far away from 0". A typical example is anomaly detection, where the
feature value corresponds to the deviation of an observation from the mean or median. Consider
the classifier C1 (π₯) = ππ (π₯2 ), where ππ (π§) = 1+exp(βπ§)
1
is the logistic function. Figure 1 shows
the graphs of the functions ππ (π₯) and C1 (π₯).
Intuitively, ππ (π₯) squashes its input between 0 and 1. While the logistic function ππ is
monotonically increasing, our classifier C1 (π₯) = ππ (π₯2 ) is monotonically decreasing for π₯ < 0
and monotonically increasing for π₯ > 0. Hence, similar to our previous XOR example, we
are again in a situation where our feature can neither be characterized as an attacker nor as
a supporter even though it clearly has an effect on the classification. Let us note that non-
monotonic behaviour naturally occurs in many domains. For example, when predicting health
risks, the probability often increases when particular health markers deviate substantially from
a default value. For instance, both underweight and overweight and both low and high blood
pressure could be seen as red flags. For spatial features, even multiple changes in the behaviour
of a classifier can occur. One example are latitude and longitude when predicting property
demand in real estate datasets because areas with high and low popularity are often spread
throughout cities.
In order to deal with non-monotonicity, we basically have to consider value intervals for
features rather than just features. For example, C1 (π₯) in Figure 1 is decreasing on the open
interval (ββ, 0) and increasing on (0, β). However determining these intervals can be chal-
lenging in general. For example, when the classifier is a large neural network, it is hard to tell
at which points the model changes its behaviour. Taking many small intervals will increase
the chance that the model behaves monotonically on the interval, but can potentially blow up
the explanation unnecessarily. Taking too large intervals increase the chance that the model is
non-monotonic on the interval, so that the explanation may miss important effects (reducing
completeness) or may misrepresent the effect (violating reinforcement). However, let us note
that finding good intervals is easier for discrete classifiers like tree ensembles. This is because
they internally discretize continuous features, so that the potential critical points can be found
by traversing all trees in the ensemble.
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To summarize, for continuous features, we do not only have to take account of joint effects
of features, but also of their potential non-monotonicity. In addition to the two refinements
proposed at the end of the previous section, we may consider the following variations to improve
completeness when features are continuous.
1. Increase the BAG: Instead of having a single feature argument per feature, we can
consider multiple arguments that represent the influence of the feature in a particular
region.
2. Consider more expressive BAGs: similar to the boolean setting, joint attacks/supports
can capture the joint effects of continuous feature arguments that take a value in a
particular region or conditional effects based on the state of boolean feature arguments.
5. Towards Characterizing Correct and Complete Explanations
Before closing the paper, let us briefly discuss a sufficient condition under which the naive
explanation BAG can accurately represent a probabilistic classifier. The condition that we
consider here is that the classifier behaves monotonically with respect to every individual
feature (independent of the remaining features). We call this property Strong Monotonicity. To
talk about monotonicity, we have to assume that all domains are ordered. Note that this applies
to boolean features (0 < 1) as well as discrete (π < π + 1) and continuous features. Formally, we
call a classifier monotonically increasing (resp. decreasing) wrt. the label π¦ and the π-th feature iff
for all inputs x β π, and π₯β²π β π·π , π₯π < π₯β²π implies that Cπ¦ (x) β€ Cπ¦ ((xβπ , π₯β²π )) (resp. Cπ¦ (x) β₯
Cπ¦ ((xβπ , π₯))) and π₯π > π₯β²π implies that Cπ¦ (x) β₯ Cπ¦ ((xβπ , π₯β²π )) (resp. Cπ¦ (x) β€ Cπ¦ ((xβπ , π₯))). If
these conditions hold with strict inequality, we call the classifier strictly monotonically increasing
(resp. decreasing) wrt. π¦ and the π-th feature. We call a classifier strongly monotonic if for
every label and for every feature, the classifier is strictly monotonically increasing or strictly
monotonically decreasing.
Let us note that if our data is boolean and the classifier is strongly monotonic, then the naive
explanation BAG accurately captures all effects by definition. If we have continuous features,
then building up the explanation BAG can be difficult because an edge can only be added when
the reinforcement condition can be verified for all domain values. As the domain is infinite,
this is not always possible. However, if the classifier is composed of differentiable functions
like in the case of many neural networks, then it is theoretically possible to apply symbolic
differentiation techniques to compute the partial derivatives with respect to all features. Note
that the classifier is strongly monotonic if and only if all partial derivatives depend only on the
feature itself (independence) and are guaranteed to be always negative (decreasing) or positive
(increasing). The conditions for the naive explanation BAG (οΈcan be checked analogously. π΄πΉπ is
an attacker (supporter) of π΄πΆ π iff the partial derivative of πΆππ x) with respect to xπ depends only
on xπ and is always negative (positive). In this setting (continuous data, differentiable classifier),
the gradient does indeed have several advantages over other feature attribution methods that
evaluate features by numerical scores [10].
Formally, it would be interesting to characterize exactly under which conditions certain
explanation BAGs can correctly and completely capture the behaviour of classification models.
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Intuitively, it seems that the naive explanation BAG can satisfy both reinforcement and complete-
ness if and only if the classifier satisfies a notion (probably a refinement) of strong monotonicity.
However, a formal investigation requires a more rigorous definition of completeness that is out
of scope here.
6. Conclusions
We revisited the notion of reinforcement of bipolar abstract argumentation classifiers. While
the property is important, we believe that it should be considered in combination with a
completeness property that guarantees that reinforcement is not satisfied in a trivial way.
As we illustrated with some examples, reinforcement and completeness cannot be satisfied
simultaneously when the explanation BAG is not sufficiently expressive. We identified two
main reasons for this:
1. The effect of features cannot always be determined independently of the state of other
features.
2. The effect of features can be non-monotonic and may be positive in one region and
negative in another.
For these reasons, a naive explanation BAG that only considers individual features and their
effects on class labels, is necessarily incomplete or incorrect. Possible workarounds include
adding joint-features or joint-attacks and -supports, and discretizing continuous features into
bins on which the classifier behaves monotonically. However, the latter task can be challenging
for some models, and one may have to sacrifice correctness to improve completeness. One
notable exception may be decision tree ensembles where the critical points can directly be
extracted from the trees in the ensemble.
An interesting research direction corresponding to the correctness-completeness-tradeoff
is to characterize for which classification models we can find explanation BAGs that satisfy
both reinforcement and completeness. We conjecture that, for the naive explanation BAG, the
characterizing property of the classification model is some notion of strong monotonicity, that
is, the classifier should be
1. monotonic with respect to every feature,
2. independent of the state of the other features.
Finally, let us note that another way to guarantee an acurate representation of classification
models is to establish a one-to-one relationship between classifiers and argumentation frame-
works. For example, many multilayer perceptrons can be perfectly represented by gradual BAGs
[11]. In this case, we trivially satisfy reinforcement and completeness, but the explanation can be
trivial itself in the sense that it is not easier to comprehend than the original classifier. Clustering
techniques that have recently been considered for argumentation frameworks [12] could be
interesting to obtain more comprehensible explanations. However, it remains to be seen to
which extent such compression techniques allow maintaining reinforcement and completeness.
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Acknowledgements
This research was partially funded by the European Research Council (ERC) under the European
Unionβs Horizon 2020 research and innovation programme (grant agreement No. 101020934).
Francesca Toni was partially funded by J.P. Morgan and by the Royal Academy of Engineering
under the Research Chairs and Senior Research Fellowships scheme. Any views or opinions
expressed herein are solely those of the authors listed.
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