ERIC ED615536: Fair-Capacitated Clustering

2021·0.47 MB·English

by  ERIC

Fair-Capacitated Clustering
Tai Le Quy Arjun Roy Gunnar Friege
LeibnizUniversityHannover LeibnizUniversityHannover LeibnizUniversityHannover
Hannover,Germany Hannover,Germany Hannover,Germany
[email protected] [email protected] [email protected]
hannover.de
Eirini Ntoutsi
FreieUniversitätBerlin,Germany
Berlin,Germany
[email protected]
ABSTRACT tion of students [15] to education admission decisions [21].
Traditionally,clusteringalgorithmsfocusonpartitioningthe Recently, the issue of bias and discrimination in ML-based
data into groups of similar instances. The similarity objec- decision-making systems is receiving a lot of attention [28]
tive, however, is not sufficient in applications where a fair- astherearemanyrecordedincidentsofdiscrimination(e.g.,
representation ofthegroupsintermsofprotectedattributes recidivism prediction [20], grades prediction [4, 14]) caused
like gender or race, is required for each cluster. Moreover, by such systems against individuals or groups or people on
in many applications, to make the clusters useful for the the basis of protected attributes like gender, race etc. Bias
end-user, a balanced cardinality among the clusters is re- in education is not a new problem. There is already a long
quired. Our motivation comes from the education domain literatureondifferentsourcesofbiasineducation[24]orstu-
where studies indicate that students might learn better in dents’ data analysis [3] as well as studies on racial bias [31]
diverse student groups and of course groups of similar car- and gender bias [22]. However, ML-based decision-making
dinality are more practical e.g., for group assignments. To systemshavethepotentialtoamplifyprevalentbiasesorcre-
this end, we introduce the fair-capacitated clustering prob- ate new ones and therefore, fairness-aware ML approaches
lem thatpartitionsthedataintoclustersofsimilarinstances are required also for the educational domain.
while ensuring cluster fairness and balancing cluster cardi-
nalities. We propose a two-step solution to the problem: i) In this work, we focus on fairness in clustering, as in edu-
we rely on fairlets to generate minimal sets that satisfy the cational activities, group assignments [8] and student team
fair constraint and ii) we propose two approaches, namely achievementdivisions[30]areimportanttoolsthathelpstu-
hierarchical clustering and partitioning-based clustering, to dentsworkingtogethertowardssharedlearninggoals. Clus-
obtain the fair-capacitated clustering. Our experiments on tering is an effective solution for partitioning students into
threeeducationaldatasetsshowthatourapproachesdeliver groups of similar instances [3, 26]. Traditional algorithms,
well-balanced clusters in terms of both fairness and cardi- however, focus solely on the similarity objective and do not
nality while maintaining a good clustering quality. consider the fairness of the resulting clusters w.r.t. pro-
tectedattributeslikegender. However,studiesindicatethat
students might learn better in diverse groups, e.g., mixed-
Keywords
gendergroups[11,32]. Lately,fair-clusteringsolutionshave
fair-capacitated clustering, fair clustering, capacitated clus- beenproposed[1,2,5,6],whichaimtodiscoverclusterswith
tering, fairness, learning analytics, fairlets, knapsack. afairrepresentationregardingsomeprotectedattributes. In
this work, we adopt the cluster fairness of [6], called clus-
1. INTRODUCTION terbalance,accordingtowhichprotectedgroupsmusthave
Machinelearning(ML)playsacrucialroleindecision-making approximately equal representation in every cluster.
in almost all areas of our lives, including areas of high soci-
In a teaching situation, it is obvious that the size of the
etal impact, like healthcare and education. Our work’s mo-
groups should be comparable to allow a fair allocation of
tivationcomesfromtheeducationdomainwhereML-based
work among the students. As traditional clustering algo-
decision-makinghasbeenusedinawidevarietyoftasksfrom
rithms do not consider this requirement, clusters of varying
student dropout prediction [9], forecasting on-time gradua-
sizes might be extracted, reducing the usefulness and ap-
plicability of the partitioning for end-users/teachers. This
leads to the demand for clustering solutions that also take
into account the size of the clusters. The problem is known
as capacitated clustering problem (CCP) [25] which aims to
TaiLeQuy,ArjunRoy,GunnarFriegeandEiriniNtoutsi“Fair-Capacitated extract clusters with a limited capacity while minimizing
Clustering”.2021.In:ProceedingsofThe14thInternationalConferenceon the total dissimilarity in the clusters. Capacitated cluster-
EducationalDataMining(EDM21).InternationalEducationalDataMining ing is useful in quite a few applications such as transfer-
Society,407-414.https://educationaldatamining.org/edm2021/ ring goods/services from the service providers (post office,
EDM’21June29-July022021,Paris,France
stores,etc.),garbagecollectionandsalesforceterritorialde-
Proceedings of The 14th International Conference on Educational Data Mining (EDM 2021) 407
sign [27]. To the best of our knowledge, no solution exists 3. PROBLEMDEFINITION
that considers both fairness and capacity of clusters on the Let X ∈ Rn be a set of instances to be clustered and let
top of the similarity objective. d():X×X →Rbethedistancefunction. Foranintegerk
we use [k] to denote the set {1,2,...,k}. A k-clustering C is
To this end, we propose a new problem, the so-called fair- apartitionofX intokdisjointsubsets,C ={C ,C ,...,C },
1 2 k
capacitated clustering that ensures fairness and balanced calledclusters withS ={s ,s ,...,s }bethecorresponding
1 2 k
cardinalities of the resulting clusters. We decompose the cluster centers. The goal of clustering is to find an assign-
problem into two subproblems: i) the fairness-requirement ment φ:X →[k] that minimizes the objective function:
compliancestepthatpreservesfairnessataminimumthresh-
(cid:88) (cid:88)
old of balance score and ii) the capacity-requirement com- L(X,C)= d(x,s ) (1)
i
pliance step that ensures clusters of comparable sizes. For si∈Sx∈Ci
thefirststep,wegeneratefairlets[6],whichareminimalsets
As shown in Eq. 1, the goal is to find an assignment that
that satisfy fair representation w.r.t. a protected attribute.
minimizes the sum of distances between each point x ∈
Forthesecondstep,weproposetwosolutionsfortwodiffer-
Xand its corresponding cluster center s ∈ S. It is clear
ent clustering types, namely hierarchical and partitioning- i
that such an assignment optimizes the similarity but does
based clustering, that consider the capacity constraint dur-
not consider fairness or capacity of the resulting clusters.
ing the merge step (hierarchical approach)/assignment step
(partitioningapproach). Experimentalresults,onthreereal
Capacitated clustering: The goal of capacitated clustering
datasetsfromtheeducationdomain,showthatourmethods
[25]istodiscoverclustersofgivencapacitieswhilestillmin-
result in fair and capacitated clusters while preserving the
imizing the distance objective L(X,C). The capacity con-
clustering quality.
straint is defined as an upper bound Q on the cardinality
i
of each cluster C :
i
2. RELATEDWORK
|C |≤Q (2)
i i
Chierichettietal. [6]introducedthefairclusteringproblem
with the aim to ensure equal representation for each pro- Clustering fairness: We assume the existence of a binary
tected attribute, such as gender, in every cluster. In their protectedattributeP ={0,1},e.g.,gender={male,female}.
formulation,eachinstanceisassignedwithoneoftwocolors Letψ:X →P denotesthedemographicgrouptowhichthe
(red, blue). They proposed a two-phase approach: clus- point belongs, i.e., male or female.
tering all instances into fairlets - small clusters preserving
the fairness measure, and then applying vanilla clustering Fairness of a cluster is evaluated in terms of the balance
methodsonthosefairlets. Subsequentstudiesfocusongen- score [6] as the minimum ratio between two groups.
eralization and scalability. Backurs et al. [1] presented an (cid:16) (cid:17)
approximatefairletdecompositionalgorithmwhichcanfor- balance(Ci)=min ||{{xx∈∈CCii||ψψ((xx))==01}}||, ||{{xx∈∈CCii||ψψ((xx))==01}}||
mulatethefairletsinnearlylineartimethustacklingtheeffi- (3)
ciencybottleneckof[6]. R¨osnerandSchmidt[29]generalized Fairness of a clustering C equals to the balance of the least
the fair clustering problem to more than two protected at- balanced cluster C ∈C.
i
tributes. A more generalized and tunable notion of fairness
for clustering was introduced in Bera et al. [2]. Anshuman balance(C)= min balance(Ci) (4)
Ci∈C
andPrasant[5]introducedafairhierarchicalagglomerative
clustering method for multiple protected attributes. We now introduce the problem of fair-capacitated cluster-
ing that combines all aforementioned objectives regarding
Thecapacitatedclusteringproblem(CCP),acombinatorial distance, fairness and capacity.
optimization problem, was first introduced by Mulvey and
Definition 1. (Fair-capacitated clustering problem)
Beck [25] who proposed solutions using heuristic and sub-
Wedefinetheproblemof(t,k,q)-fair-capacitatedclustering
gradient algorithms. Several approaches exist to improve
as finding a clustering C ={C ,C ,··· ,C } that partitions
the efficiency of solutions or CCP approaches for different 1 2 k
the data X into |C| = k clusters such that the cardinality
cluster types. Khuller and Sussmann [17], for example, in-
of each cluster C ∈ C does not exceed a threshold q, i.e.,
troduced an approximation algorithm for the capacitated i
|C |≤q(thecapacityconstraint),thebalanceofeachcluster
k-Center problem. Geetha et al. [10] improved k-Means i
is at least t, i.e., balance(C) ≥ t (the fairness constraint),
algorithm for CCP by using a priority measure to assign
and minimizes the objective function L(X,C). Parameters
points to their centroid. Lam and Mittenthal [19] proposed
k,t,q are user defined referring to the number of clusters,
a heuristic hierarchical clustering method for CCP to solve
minimumbalancethresholdandmaximumclustercapacity,
the multi-depot location-routing problem.
respectively.
In this work, we introduce the problem of fair-capacitated
4. FAIR-CAPACITATEDCLUSTERING
clustering which builds upon the notions from fair cluster-
ingandcapacitatedclustering. Inparticular,webuildupon 4.1 Fairletdecomposition
the notion of fairlets [6] to extract the minimal sets that Traditionally, the vanilla versions of clustering algorithms
preserve fairness. Regarding the CCP we follow the formu- arenotcapableofensuringfairnessbecausetheyassignthe
lationof[25]toensurebalancedclustercardinalities. Tothe data points to the closest center without the fairness con-
best of our knowledge, the combined problem has not been sideration. Hence, if we could divide the original data set
studied before and as already discussed, comprises a useful into subsets such that each of them satisfies the balance
tool in quite a few domains like education. thresholdtthengroupingthesesubsetstogeneratethefinal
408 Proceedings of The 14th International Conference on Educational Data Mining (EDM 2021)
clustering would still preserve the fairness constraint. Each Therefore, from Eq. 5 and Eq. 6 we get,
fairsubsetisdefinedasafairlet. Wefollowthedefinitionof
balance(Y)≥t (7)
fairlet decomposition by [6].
Thus,thestatementgiveninTheorem1istrueforanyclus-
Definition 2. (Fairlet decomposition)
terformedbytheunionofanytwofairlets. Nowweassume
Suppose that balance(X) ≥ t with t = f/m for some in-
that the statement holds true for a cluster formed from i
tegers 1 ≤ f ≤ m, such that the greatest common divisor
fairlets, i.e, Y =∪ F , where 1<i<l. Then,
gcd(f,m) = 1. A decomposition F = {F ,F ,...,F} of X j≤i j
1 2 l
is a fairlet decomposition if: i) each point x∈X belongs to (cid:80) f
exactlyonefairletF ∈F;ii)|F |≤f+mforeachF ∈F, balance(Y)= (cid:80)j≤i j ≥t (8)
j j j m
i.e.,thesizeofeachfairletissmall;andiii)foreachF ∈F, j≤i j
j
balance(F )≥t,i.e.,thebalanceofeachfairletsatisfiesthe
j
threshold t. Each Fj is called a fairlet. ConsideranotherfairletFi+1 ∈F whichisnotintheformed
f
By applying fairlet decomposition on the original dataset clusterY,balance(Fi+1)= mi+1 ≥t. Then,byjoiningFi+1
i+1
X, we obtain a set of fairlets F = {F1,F2,...,Fl}. For with the cluster Y we get the new cluster Y(cid:48) such that,
each fairlet F we select randomly a point r ∈ F as the
j j
(cid:80)
caesnttehre. inFdoerxaofptohinetmxap∈peXd,fawireletd.enTotheeγsec:oXnd →step[1,,ils] balance(Y(cid:48))= mfi+1++(cid:80)j≤ifmj (9)
to cluster the set of fairlets F into k final clusters, sub- i+1 j≤i j
ject to the cardinality constraint. The clustering process Following the steps in Eq. 6, we can similarly show that,
is described below for the hierarchical clustering type (Sec- (cid:80)
f + f
tion4.2)andforthepartitioning-basedclusteringtype(Sec- i+1 (cid:80)j≤i j ≥t =⇒ balance(Y(cid:48))≥t (10)
m + m
tion 4.3). Clustering results in an assignment from fairless i+1 j≤i j
to final clusters: δ : F → [k]. The final fair-capacitated Hence, the theoremholds true for cluster formedwithi+1
clustering C can be determined by the overall assignment fairletsifitistrueforifairlets. Sinceiisanyarbitrarynum-
function φ(x) = δ(Fγ(x)), where γ(x) returns the index of beroffairlets,thusthetheoremholdstrueforallcases.
the fairlet to which x is mapped.
Theorem 1 shows that for any cluster formed by union of
4.2 Fair-capacitatedhierarchicalclustering
fairlets, the fairness constraint is always preserved. Hence-
Given the set of fairlets: F = {F1,F2,...,Fl}, let W = forth, we don’t need further interventions w.r.t. fairness.
{w ,w ,...,w} be their corresponding weights, where the
1 2 l
weight wj of a fairlet Fj is defined as its cardinality, i.e., The pseudocode is shown in Algorithm 2 of Appendix B.
number of points in Fj. In each step, the closest pair of clusters is identified and a
new cluster is created only if its capacity does not exceed
Traditional agglomerative clustering approaches merge the the capacity threshold q. Otherwise, the next closest pair
two closest clusters, so rely solely on similarity. We extend is investigated. The procedure continues until k clusters
the merge step by also ensuring that merging does not vio- remain. Theremainingclustersarefairandcapacitatedac-
late the cardinality constraint w.r.t. the cardinality thresh- cordingtothecorrepondingthresholdstandq. Tocompute
old q. theproximitymatrix(line1andline8),weusethedistance
Theorem 1. Thebalancescoreofaclusterformedbythe between centroids of the corresponding clusters. The func-
tion capacity(cluster) in line 5 returns the size of a cluster.
union of two or more fairlets, is at least t.
balance(Y)≥t, where Y =∪j≤lFj and balance(Fj)≥t 4.3 Fair-capacitatedpartitioningclustering
Partitioning-basedclusteringalgorithms,suchask-Medoids,
Proof. We use the method of induction to derive the
canbeviewedasadistanceminimizationproblem,inwhich,
proof. AssumewehaveasetoffairletsF ={F ,F ,...,F},
1 2 l we try to minimize the objective function in Eq. 1. The
in which, balance(F ) ≥ t, j = 1,...,l. We first con-
j vanillak-Medoidsdoesnotsatisfythecardinalityconstraint
sider the case for any two fairlets {F ,F } ∈ F. We have
1 2 since the allocating points to clusters step is only based on
f f
balance(F ) = 1 ≥ t and balance(F ) = 2 ≥ t. We thedistanceamongthem. Now,ifwechangethegoalofthis
1 m 2 m
1 2 assignmentsteptofindthe“best”datapointswithadefined
denote by Y is the union of two fairlets F and F , then
1 2 capacity for each medoid instead of searching for the most
balance(Y)=balance(F ∪F )= f1+f2 (5) suitablemedoidforeachpoint,wecancontrolthecardinality
1 2 m1+m2 of clusters. We formulate the problem of assigning points
It holds: to clusters subject to a capacity threshold q as a knapsack
problem [23].
f f tm
1 ≥t or, 1 ≥ 1
m m +m m +m
1 1 2 1 2 Let S ={s ,s ,...,s } be the cluster centers, i.e., medoids,
1 2 k
f tm
Similarly, 2 ≥ 2 andC ={C1,C2,...,Ck}betheresultingclusters. Wechange
m1+m2 m1+m2 (6) theassignmentsofpointstoclusters,usingknapsack,inor-
=⇒ f1 + f2 ≥ tm1 + tm2 der to meet the capacity constraint q. In particular, we
m +m m +m m +m m +m define a flag variable y = 1 if x is assigned to cluster C ,
1 2 1 2 1 2 1 2 j j i
=⇒ f1+f2 ≥ t(m1+m2) =t otherwise yj = 0. Now, we define a value vj to data point
m1+m2 m1+m2 xj,whichdependsonthedistanceofxj toCi,withvj being
Proceedings of The 14th International Conference on Educational Data Mining (EDM 2021) 409
maximum if Ci is the best cluster for xj, i.e, the distance Algorithm1: k-Medoids fair-capacitated algorithm
between x and s is minimum. We formulate value v of
j i j Input: F ={F ,F ,...,F}: a set of fairlets
instancex basedonanexponentialdecaydistancefunction: 1 2 l
j W ={w ,w ,...,w}: weights of fairlets
1 2 l
vj =e−λ1∗d(xj,si) (11) qk:: anugmivbeenrmofaxcliumsutemrscapacity of final clusters
where d(x ,s ) is the Euclidean distance between the point Output: A fair-capacitated clustering
j i
xj and the medoid si. The higher λ is the lower the effect 1 Function ClusterAssignment(medoids):
of distance in the value of the points. The point which is 2 clusters←∅;
closer to the medoid will have a higher value. 3 for each medoid s in medoids do
4 candidates ← all fairlets which are not assigned
Then, the objective function for the assignment step is: to any cluster ;
5 p ← length(candidates) ;
n
(cid:88) 6 w ← weights(candidates) ;
maximize vjyj (12) 7 for each fairleti in candidates do
j=1 8 values[i] ← v(fairleti) //Eq. 11 ;
Now, given F ={F1,F2,...,Fl} and W ={w1,w2,...,wl} 9 end
are the set of fairlets and their corresponding weights, i.e., 10 clusters[s]←knapsack(p, values, w, q) ;
the number of instances in the fairlet, respectively; q is the 11 end
maximum capacity of the final clusters. Our target is to 12 return clusters;
cluster the set of fairlets F into k clusters centered by k 13 Function main():
medoids. WeapplytheformulasinEq. 11andEq.12onthe 14 medoids← select k of the l fairlets arbitrarily ;
setoffairletsF, i.e, eachfairletFj hasthesameroleasxj. 15 ClusterAssignment(medoids) ;
Then,theproblemofassigningthefairletstoeachmedoidin 16 costbest ← current clustering cost;
the clusterassignmentstepbecomesfindingasetof fairlets 17 sbest ←null ;
withtotalweightslessthanorequaltoqandthetotalvalue 18 obest ←null ;
is maximized. In other words, we can formulate the cluster 19 repeat
assignmentstepinthepartitioning-basedclusteringasa0-1 20 for each medoid s in medoids do
knapsackproblem. 21 for each non-medoid o in F do
22 consider the swap of s and o, compute
l
(cid:88) the current clustering cost;
maximize v y
j j 23 if current clustering cost < costbest then
j=1
(13) 24 sbest ←s;
(cid:88)l 25 obest ←o;
subject to w y ≤q and y ∈{0,1}
j j j 26 costbest ←current clustering cost;
j=1 27 end
Inwhich,yjistheflagvariableforFj,yj =1ifFjisassigned 28 end
toacluster,otherwiseyj =0;vj isthevalueofFj whichis 29 end
computedbytheEq. 11;qisthedesiredmaximumcapacity. 30 update medoids by the swap of sbest and obest ;
31 ClusterAssignment(medoids)
Thepseudocodeofourk-Medoidsfair-capacitatedalgorithm 32 until no improvements can be achieved by any
is described in Algorithm 2. In which, for each medoid we replacement;
wouldsearchfortheadequatepoints(line3)byusingfunc- 33 return clusters;
tion knapsack(p,values,w,q) (line 10) implemented using
dynamic programming, which returns a list of items with a
maximum total value and the total weight not exceeding q.
Inthemainfunction,line12,weoptimizetheclusteringcost
2005-2006.“Gender”isselectedastheprotectedattribute,
by replacing medoids with non-medoid instances when the
i.e., we aim to balance gender in the resulting clusters.
clustering cost is decreased. This optimization procedure
will stop when there is no improvement in the clustering Open University Learning Analytics (OULAD). This
cost is found (lines 19 to 32). is the dataset from the OU Analyse project [18] of Open
University,Englandin2013-2014. Informationofstudents
5. EXPERIMENTS includestheirdemographics,courses,theirinteractionswith
the virtual learning environment, and final outcome. We
Inthissection,wedescribeourexperimentsandtheperfor-
aim to balance the“Gender”attribute in the results.
manceofourproposedmethodsonthreeeducationaldatasets.
MOOC. The data covers students who enrolled in the 16
edXcoursesofferedbythetwoinstitutions(HarvardUniver-
5.1 Experimentalsetup sity and the Massachusetts Institute of Technology) during
Datasets. The datasets are summarized in Table 1. 2012 - 2013 [13]. The dataset contains aggregated records
whichrepresentstudents’activitiesandtheirfinalgradesof
UCI student performance. Thisdatasetincludesthede- the courses.“Gender”is the protected attribute.
mographics,grades,socialandschool-relatedfeaturesofstu-
dentsinsecondaryeducationoftwoPortugueseschools[7]in Baselines. We compare against well-known fairness-aware
410 Proceedings of The 14th International Conference on Educational Data Mining (EDM 2021)
Table1: Anoverviewofthedatasets
Dataset #instances #attributes Protected attribute Balance score
UCI student performance 649 33 Gender (F: 383; M: 266) 0.695
OULAD 4,000 12 Gender (F: 2,000; M: 2,000) 1
MOOC 4,000 21 Gender (F: 2,000; M: 2,000) 1
clustering methods and a vanilla clustering method. Center method. Figure 1-b depicts the clustering fairness.
As we can observe, in terms of fairness, vanilla fairlet hier-
k-Medoids [16]: a traditional partitioning clustering tech- archicalfair-capacitated hasthebestperformancewhenkis
nique that divides the dataset into k clusters so as to mini- less than 10. Contrary to that, by selecting each point for
mize clustering cost. Cluster centers are actual instances. each cluster in the cluster assignment step, the k-Medoids
fair-capacitated method can maintain well the fairness in
Vanillafairlet[6]: afairness-awareclusteringapproachthat manycases. Regardingthecardinality,asillustratedinFig-
i)decomposesthedatasetintofairletsandii)appliesavanilla ure 1-c, our approaches outperform the competitors when
k-centerclusteringalgorithm[12]toformthefinalkclusters. they can keep the number of instances for each cluster un-
der the specified thresholds.
MCF fairlet [6]: Similar to Vanilla fairlet but the fairlet
OULAD. Our MCF fairlet k-Medoids fair-capacitated ap-
decompositionpartistransformedintoaminimumcostflow
proach outperforms other methods in terms of clustering
(MCF) problem, by which an optimized version of fairlet
cost, although there is an increase compared to the vanilla
decomposition in terms of cost value is computed.
k-Medoids algorithm, as we can see in Figure 2-a. Con-
cerning fairness, in Figure 2-b, k-Medoids is the weakest
Evaluationmeasures. Wereportonclusteringquality(mea-
method while others can achieve the highest balance. The
sured as clustering cost, see Eq. 1), cluster fairness (ex-
balanceofGender featureinthedatasetisthemainreason
pressed as cluster balance, see Eq. 4) and cluster capacity
for this result. All fairlets are fully fair; this is a prerequi-
(expressed as cluster cardinality).
site for our methods of being able to maintain the perfect
balance. Regardingcardinality,ourapproachesdemonstrate
Parameterselection. Regardingfairness,aminimumthresh-
theirstrengthinensuringthecapacityofclusters(Figure2-
old of balance t is set to 0.5 for all datasets in our experi-
c). The difference in the size of the clusters generated by
ments. It means that the proportion of the minority group
our methods is tiny. This is in stark contrast to the trend
is at least 50% in each resulting cluster. Regarding the λ
of competitors.
factor in Eq. 11, a value λ = 0.3 is chosen for our experi-
mentsfromarangeof[0.1,1.0]viagrid-search. Weevaluate MOOC. The results of clustering quality are described in
the clustering cost and balance score on a small dataset, Figure3-a(AppendixA). Althoughanincreaseintheclus-
i.e., UCI student performance dataset - Mathematics sub- tering cost is the main trend, our methods outperform the
ject w.r.t λ. Theoretically, the ideal capacity of clusters is vanilla fairlet and MCF fairlets methods. Regarding clus-
(cid:108)|X|(cid:109)
where |X| is the population of dataset X, k is the tering fairness, as depicted in Figure 3-b, our approaches
k can maintain the perfect balance for all experiments. This
number of desired clusters. However, in many cases, the
is the result of an actual balance in the dataset and the
clustering models cannot satisfy this constraint, especially
fairlets. Theemphasisisourmethodscandividealltheex-
thehierarchicalclusteringmodel. Hence,wesettheformula
perimented instances into capacitated clusters, as shown in
inEq. 14tocomputethemaximumcapacityqofclusters;εis
Figure 3-c, which proves their superiority in presenting the
aparameterchoseninexperimentsforeachfair-capacitated
results over the competitors regarding clusters’ cardinality.
clustering approach.
Summary of the results. In general, fairness is well main-
(cid:108)|X|∗ε(cid:109)
q= (14) tained in all of our experiments. When the data is fair, in
k
case of OULAD and MOOC datasets, our methods achieve
To find the appropriate value of ε, we set a range of [1.0, a perfect fairness. In terms of cardinality, our methods are
1.3] to ensure all the generated clusters have members and able to maintain the cardinality of resulting clusters within
evaluatethecardinalityofresultingclustersontheUCIstu- the maximum capacity threshold, which is significantly su-
dentperformance(Mathematicssubject)dataset. εissetto perior to competitive methods. The fair-capacitated par-
1.01and1.2,fork-Medoidsfair-capacitatedandhierarchical titioning based method is better than the hierarchical ap-
fair-capacitated methods, respectively. proachsinceitcandeterminethecapacitythresholdclosest
to the ideal capacity. Regarding the clustering cost, the hi-
5.2 Experimentalresults erarchicalapproachhasanadvantageoverothermethodsby
outperforming its competitors in most experiments.
UCI student performance. When k is less than 4, as shown
in Figure 1-a, the clustering quality of our models can be 6. CONCLUSIONANDOUTLOOK
close to that of the vanilla k-Medoids method. However, In this work, we introduced the fair-capacitated clustering
the clustering cost is fluctuated thereafter due to the ef- problem that extends traditional clustering, solely focusing
fort to maintain the fairness and cardinality of methods. onsimilarity,byalsoaimingatabalancedcardinalityamong
Our vanilla fairlet hierarchical fair-capacitated outperforms the clusters and a fair-representation of instances in each
other competitors in most cases. Vanilla fairlet and MCF cluster according to some protected attributes like gender
fairlet show the worst clustering cost as an effect of the k- or race. Our solutions work on the fairlets derived from
Proceedings of The 14th International Conference on Educational Data Mining (EDM 2021) 411
a) Clustering quality (lower is better) b) Clustering fairness (higher is better)
0.7
2600
0.6
2400 0.5
Clustering cost2200 kV-aMneildlao ifdasirlet Balance00..34 kVVV-aaaMnnneiiillldlllaaao ifffdaaasiiirrrllleeettt hk-ieMreadrcohidicsa fla fiar-irc-acpaapcaicaittaatteedd
2000 Vanilla fairlet hierarchical fair-capacitated 0.2 MCF fairlet
Vanilla fairlet k-Medoids fair-capaciatated MCF fairlet hierarchical fair-capacitated
MCF fairlet 0.1 MCF fairlet k-Medoids fair-capacitated
1800 MCF fairlet hierarchical fair-capacitated Minimum balance
MCF fairlet k-Medoids fair-capacitated 0.0 Dataset's balance
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Number of clusters Number of clusters
c) Clustering cardinality
300 k-Medoids
Vanilla fairlet
Vanilla fairlet hierarchical fair-capacitated
250
Number of instances112050000 VMMMMMaCCCaanxxFFFiii lmmffflaaaauu iiifrrrmmallleeei rtttccl ehkaat-ippe Mkaare-accMdiirttceoyyhdi dooiocsffia dhkflas -ifeM iafrra-iearcid-rrac-copcahiaadpicpcsaai actflacai tfitiaatrea-tidrcet-eadcdpaapcaictaittaetded
50
0
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Number of clusters
Figure1: PerformanceofdifferentmethodsonUCIstudentperformancedataset
a) Clustering quality (lower is better) b) Clustering fairness (higher is better)
1.0
12500
12000 0.8 k-Medoids
Vanilla fairlet
Clustering cost111011505000000 kVVV-aaaMnnneiiillldlllaaao ifffdaaasiiirrrllleeettt hk-ieMreadrcohidicsa fla fiar-irc-acpaapcaicaittaatteedd Balance00..46 VVMMMMDaaaCCCinnntFFFiaiim llsfffllaaaaaeu iiitmffrrr'aalllseee ii rrbtttbll aeeahklttl-iaae Mhknnre-iacceMdreerceoahdirdicocshia diflcas fa iafrla- ifrcia-rac-iprca-aapcpcaaiapctaciatitacaeitttdeaadtteedd
MCF fairlet
0.2
10000 MCF fairlet hierarchical fair-capacitated
MCF fairlet k-Medoids fair-capacitated
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Number of clusters Number of clusters
c) Clustering cardinality
k-Medoids
2000 Vanilla fairlet
Vanilla fairlet hierarchical fair-capacitated
mber of instances11050000 VMMMMMaCCCaanxxFFFiii lmmffflaaaauu iiifrrrmmallleeei rtttccl ehkaat-ippe Mkaare-accMdiirttceoyyhdi dooiocsffia dhkflas -ifeM iafrra-iearcid-rrac-copcahiaadpicpcsaai actflacai tfitiaatrea-tidrcet-eadcdpaapcaictaittaetded
Nu
500
0
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Number of clusters
Figure2: PerformanceofdifferentmethodsonOULADdataset
theoriginalinstances: thehierarchical-basedapproachtakes Acknowledgements
intoaccountthecardinalityrequirementduringthemerging The work of the first author is supported by the Ministry
step,whereasthepartitioning-basedapproachtakesintoac- ofScienceandEducationofLowerSaxony,Germany,within
count the cardinality of the final clusters during the assign- thePhDprogram“LernMINT:Data-assistedteachinginthe
ment step which is formulated as a knapsack problem. Our MINT subjects”. The work of the second author is sup-
experimentsshowthatourmethodsareeffectiveintermsof ported by the Volkswagen Foundation under the call“Arti-
fairness and cardinality while maintaining clustering qual- ficial Intelligence and the Society of the Future”.
ity. In the future, we plan to extend our approach for more
thanoneprotectedattributesaswellastofurtherinvestigate
what fair group assignments means in educational settings.
412 Proceedings of The 14th International Conference on Educational Data Mining (EDM 2021)
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Proceedings of The 14th International Conference on Educational Data Mining (EDM 2021) 413
APPENDIX
A. MOOCDATASET
a) Clustering quality (lower is better) b) Clustering fairness (higher is better)
1.0
8000
7000 0.8 k-Medoids
Vanilla fairlet
Clustering cost56000000 kVV-aaMnneiilldllaao iffdaasiirrlleett hierarchical fair-capacitated Balance00..46 VVMMMMaaCCCinnnFFFiiim llfffllaaaaau iiimffrrraallleee iirrbtttll aeehkltt-iae Mhknre-iaceMdrerceoahdirdicocshia diflcas fa iafrla- ifrcia-rac-iprca-aapcpcaaiapctaciatitacaeitttdeaadtteedd
Vanilla fairlet k-Medoids fair-capaciatated 0.2 Dataset's balance
4000 MCF fairlet
MCF fairlet hierarchical fair-capacitated
MCF fairlet k-Medoids fair-capacitated 0.0
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Number of clusters Number of clusters
c) Clustering cardinality
k-Medoids
3000 Vanilla fairlet
Vanilla fairlet hierarchical fair-capacitated
mber of instances122505000000 VMMMMMaCCCaanxxFFFiii lmmffflaaaauu iiifrrrmmallleeei rtttccl ehkaat-ippe Mkaare-accMdiirttceoyyhdi dooiocsffia dhkflas -ifeM iafrra-iearcid-rrac-copcahiaadpicpcsaai actflacai tfitiaatrea-tidrcet-eadcdpaapcaictaittaetded
Nu1000
500
0
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Number of clusters
Figure3: PerformanceofdifferentmethodsonMOOCdataset
B. HIERARCHICAL FAIR-CAPACITATED
ALGORITHM
Algorithm2: Hierarchical fair-capacitated algorithm
Input: F ={F ,F ,...,F}: a set of fairlets
1 2 l
q: a given maximum capacity of final clusters
W ={w ,w ,...,w}: weights of fairlets
1 2 l
k: number of clusters
Output: A fair-capacitated clustering
1 compute the proximity matrix ;
2 clusters←F //each fairlet Fj is considered as cluster ;
3 repeat
4 cluster1,cluster2 ← the closest pair of clusters ;
5 if capacity(cluster1)+capacity(cluster2)≤q then
6 newcluster← merge(cluster1,cluster2);
7 update clusters with newcluster;
8 update the proximity matrix ;
9 else
10 continue;
11 end
12 until k clusters remain;
13 return clusters;
414 Proceedings of The 14th International Conference on Educational Data Mining (EDM 2021)