On the Discrimination Risk of Mean Aggregation Feature Imputation in Graphs
Anonymous Author(s)
AffiliationAddress email
Abstract
While machine learning methods have demonstrated success on graph-structureddata, many methods rely on fully-observed node features.
This has led to anincrease in research on imputing unknown or missing node features.
However, inhuman networks, nodes belonging to a marginalized group have a disproportionaterate of unknown features.
Combined with graph structure and known featurebiases, this can cause graph feature imputation algorithms to predict values forunknown features that cause the feature values of the marginalized group to bemore distinct on average from the feature values of the dominant group than theyare in reality.
We call this distinction the discrimination risk.
We prove that ahigher discrimination risk can amplify the unfairness of a machine learning modeltrained on the imputed data.
We then formalize a general graph feature imputationframework called mean aggregation imputation and theoretically and empiricallycharacterize graphs in which applying this framework can yield imputed featureswith a high discrimination risk.
We propose a simple and effective solution toensure mean aggregation-imputed features provably have a low discriminationrisk (while minimally sacrificing utility) and improve the fairness of models.
Weevaluate the fairness and utility of our solution on synthetic and real-world creditnetwork datasets.
Introduction
Machine learning methods [1, 2, 3, 4, 5] have been successfully applied to graph-structured data suchas social networks, product graphs, and molecules [6, 7] to aid in important problems like contentrecommendation, product recommendation and molecular property prediction [8, 9, 6, 10, 11].
However, many methods rely on fully-observed features for each node, which are not availablefor privacy reasons, as a consequence of exclusionary data collection practices, or due to the highexpenses involved in feature annotation [12, 13, 14].
As a result, the development of algorithms toleverage a graph’s structure and known node feature values to impute unknown or missing featureshas emerged as an important research area [15, 16, 17].
In human networks, nodes belonging to a marginalized group (e.g. on the basis of race, gender,disability, etc.) have a disproportionate rate of unknown features compared to the dominant groupbecause marginalized communities are more reluctant to share their data, annotators erase theirdata, or they are sidelined in data collection [18, 19, 20, 21].
Furthermore, graph structure and nodeneighborhoods are correlated with group membership [22, 23], especially in homophilic graphswhere nodes belonging to the same group have a higher likelihood of being connected [24].
This canbe the result of social stratification [25] or the limited collection of inter-links, i.e. edges betweennodes belonging to different groups [24, 6].
Moreover, known feature values are proxies for groupmembership [19, 26].
Hence, even if graph feature imputation algorithms do not have direct access to the group membership of a node, these algorithms are influenced by unknown feature rate disparities,graph structure, and known feature values.
Thus, they can predict values for unknown features thatcause the feature values of the marginalized group to be more distinct on average from the featurevalues of the dominant group than they are in reality.
As an illustrative example, let’s consider an automated candidate screening system based on thejob applicant network shown in Figure 1, where nodes represent applicants and edges betweenapplicants indicate that they have similar past work experiences.
Each node has one feature: thenumber of years that the applicant has previously worked.
Furthermore, each node belongs to oneof two groups, the disabled community (group Q ) or the able-bodied community (group R ).
(Inreality, disability is a fluid and complex identity that cannot be easily reduced to a binary [27].)
Allthe nodes in group R have a known feature value of 5 years, while all the nodes in group Q haveunknown feature values; however, in reality, all the nodes in group Q also have a feature value of5 years.
Additionally, all the nodes in group Q are connected to each other but have few links (ifany) to nodes in group R because of systemic barriers including hiring discrimination and a lack ofaccommodations [28].
Consequently, after applying the graph feature imputation algorithm Feature
Propagation [17], because of the disparate rates of unknown features between the groups and structureof the graph, the nodes in Q will have, on average, feature values that are more distinct from thefeature values of the nodes in R than the ground truth.
Q(?)
Q(?) Q(?)
R(5)
R(5) R(5)
Q(5)
Q(4.33) Q(4.33)
R(5) R(5)
R(5)             values that are more distinct from the feature values of the nodes in R than the ground truth.
We call this distinction in imputed feature values between the marginalized and dominant groupsthe discrimination risk .
In this paper, we present a theoretically-justified formulation of thediscrimination risk of imputed features.
We further prove that a higher discrimination risk can amplifythe unfairness of a machine learning model (regardless of architecture) trained on the imputed data,which is especially dangerous in high-stakes applications like automated candidate screening andloan approval [29, 30, 31].
For instance, in the automated candidate screening example from above, amodel trained on the imputed data may more easily learn to identify and reject disabled job applicantsbecause, on average, they appear to have fewer years of work experience than able-bodied applicants;this would reinforce systemic discrimination against the disabled community [28].
We also formalize a general graph feature imputation framework called mean aggregation imputation that encompasses common, diffusion-based graph feature imputation algorithms in theliterature.
Subsequently, we theoretically and empirically characterize graphs in which applying meanaggregation feature imputation can yield a high discrimination risk, without making any assumptionsabout underlying distributions or models for unknown feature rates or graph structure.
To the bestof our knowledge, we are the first to study the effect of graph feature imputation on the fairness ofmodels.
This is challenging because we must consider biases stemming from graph structure.
Furthermore, we propose a simple and effective solution to ensure mean aggregation-imputed featuresprovably have a low discrimination risk (while minimally sacrificing utility) and improve the fairnessof models.
We do so by viewing mean aggregation imputation through the lens of gradient descentand projecting imputed features into the feasible space of features with low discrimination risk.
Weempirically evaluate the fairness and utility of our solution on synthetic and real-world datasets,finding that it improves fairness without significant loss in utility on the synthetic datasets but doesn’timprove fairness on the real-world datasets.
We close by discussing the limitations of our solution.
Related work
Feature imputation Feature imputation algorithms utilize known feature values to predict unknownfeature values.
For example, unknown feature values may be filled as the mean of known values[17].
However, more intricate feature imputation methods have been proposed in the machine learning, statistics, and epidemiology literature, with popular approaches including matrix completion[32, 33, 34, 35], nearest neighbors [36, 37], multiple imputation via conditional models [38, 39, 40],and causal inference [41, 42].
Notably, while feature imputation may be applied to data with unknownfeature values prior to the data being passed to a machine learning model, feature imputation isdistinct from label prediction with missing data, wherein models work directly with unknown featurevalues [43, 44, 45, 46, 47, 48, 49, 50]; we do not consider the latter paradigm in this paper.
Workshave extended feature imputation approaches for tabular data to incorporate graph structure [51, 52].
Graph feature imputation using deep learning methods is also gaining traction [53, 54, 16].
However,recently, [17] proposed a diffusion-based approach to impute graph features that minimizes thegraph’s Dirichlet energy
Feature Propagation).
In this paper, we focus on a broader class of meanaggregation feature imputation algorithms (formalized in Section 4).
Fairness of missing data and feature imputation Recently, works have theoretically and empirically investigated the impact of missing data [55, 56, 20, 57, 58] and feature imputation on thefairness of machine learning models [56, 59, 60, 61, 31].
These works consistently find that missingdata can amplify algorithmic biases, and some show that in practice, feature imputation can yieldless unfair (relative to excluding missing data) but nevertheless discriminatory model outcomes.
However, these works only study tabular data and do not consider biases that emerge from graphstructure [21, 23, 62].
Furthermore, they either do not provide concrete solutions, or they adaptmodels to directly work with missing data rather than mitigate the unfairness of feature imputationitself.
Despite the prevalence of graph feature imputation methods, to the best of our knowledge,there is no research on their influence on the fairness of models that considers graph structure.
Someworks have explored fairness constraints in semi-supervised settings [63, 64], but they assume thatnode features are entirely available, which is not the case in feature imputation.
Fair graph machine learning We focus on group fairness, which ensures model predictionsexhibit some form of parity between different groups [65, 22, 66, 30, 67, 68].
Works have studiedstatistical parity, wherein models predict the positive outcome at the same rate for different groups, andpredictive parity, in which the accuracy of model predictions is equivalent across different groups [69].
While our theoretical study of discrimination risk is aligned with statistical parity, we empiricallyexplore the effect of mean aggregation imputation with a lower discrimination risk on both thedemographic parity and predictive parity of models.
Mechanisms for improving group fairness havebeen categorized into pre-processing [23], training-time [22, 24, 70], and post-processing [71, 72].
Our work is similar to pre-processing, as we seek to lower the discrimination risk of imputed trainingdata towards improving model fairness.
Recently, there has been a surge of works on modifyinggraph structure to mitigate topology-induced biases [72, 73, 23, 70, 74, 75].
However, because graphsemantics (especially for large graphs) are difficult to interpret, it is unclear if the solutions thatthese papers propose preserve the semantics of the original graph.
On the other hand, our workdoes not modify graph structure and instead optimally transforms imputed features to have a lowdiscrimination risk.
Furthermore, some of these works theoretically characterize graphs with highstructural bias [6, 76], but differ from our paper in that we analyze topology bias in the setting offeature imputation.
[77] investigates the fairness of graph neural networks in the presence of limitedgroup membership information but does not consider imputation; in contrast, our work assumesgroup membership is fully available but that features are not.
Discrimination risk and model unfairness
As machine learning is increasingly used in consequential applications like loan approval andautomated candidate screening [29, 30, 31], ensuring its fairness is critical.
To understand how featureimputation could amplify the unfairness of a machine learning model, we present a theoreticallyjustified metric called the discrimination risk, which applies beyond the setting of graphs and isagnostic to model architecture.
We explore discrimination risk in the context of graphs in Section 5.
Suppose we have a data distribution D .
Furthermore, we have two groups: a marginalized group Qand a dominant group R .
For any data instance ( x, y, s ) ∼ D , let x ∈
X be the d -dimensional featurevalues of the instance (where x i denotes the i -th entry of x ), let y ∈ Y be the label of the instance,and let s ∈ { Q, R } be the group membership of the instance.
We will assume that we can observethe group membership of any data instance and that no instance can belong to both the marginalizedgroup and dominant group.
Definition 1 (Discrimination Risk)
We define the discrimination risk of D as:
[Equation]
where [ d ] is the set { 1 , . .
, d } .
We now explore the relevance of discrimination risk to model unfairness.
Let D ′ be the datadistribution with the ground-truth features and D be the distribution with imputed features.
We willshow that if R D > R D ′ , imputation may amplify model unfairness.
To be concrete, let’s again consider the example of an automated candidate screening system.
Let
S ∈ { Q, R } be the marginal distribution over the group membership of job applicants.
Furthermore,let X ′ ∈
X be the marginal distribution over the ground-truth feature values of applicants (e.g.number of years previously worked, highest degree, etc.)
In contrast, let X ∈ X be the distributionover imputed feature values of applicants.
Let Y ∈ Y be the marginal distribution over applicantlabels (e.g. whether an applicant should be hired, a screening score for an applicant, etc.)
Now,let h ′ be a model trained on samples from D ′ , and h be another model trained on samples from D .
Additionally, let Z
∈
Y be the distribution over the predictions of h ′ on instances sampled from X ,and let Z ∈ Y be the distribution over predictions of h on instances sampled from X .
We have the dependencies S → X ′ and S
→ X because disability affects the number of yearspreviously worked by a job applicant.
S
→ X may be additionally necessary because S can influencethe feature imputation algorithm.
Furthermore, we assume that h ′ and h have access to the featurevalues of an applicant, but not the applicant’s group membership, and that the association of S with
Z ′ through Y can be fully explained by X ′ .
Thus, Z ′ is conditionally independent of S given X ′ .
Similarly, we assume that the association of S with Z through Y can be fully explained by X ,
so Zis conditionally independent of S given X .
Because we are interested in how feature imputation may amplify model unfairness, we investigatewhen the statistical parity or total variation distance d TV ( Z | S = Q, Z | S = R ) >
d TV ( Z ′ | S =
Q, Z ′ | S = R ) [78], where Z | S =
Q is the conditional prediction distribution given groupQ membership.
d TV measures the distance between two probability distributions A and B asd TV ( A, B ) =
sup x ∈F |
A ( x ) − B ( x ) | .
Intuitively, d TV ( Z | S = Q, Z | S = R ) captures how muchthe prediction distribution of h absolutely differs between Q and R , and thus it quantifies the (statistical parity) unfairness of h .
We now discuss the relationships of d TV ( Z ′ | S = Q, Z ′ | S = R ) tod TV
( X ′ | S
= Q, X ′
| S = R ) and d TV ( Z | S = Q, Z | S = R ) to d TV ( X
| S = Q, X | S = R ) .
Webegin with the following lemma from [79].
Lemma 1 (Corollary 17 from [79]) By the Data Processing Inequality, d TV ( Z ′ | S = Q, Z ′ | S =
R ) ≤ d TV ( X ′ | S
= Q, X ′
| S = R ) and d TV ( Z | S = Q,
Z | S = R ) ≤
d TV ( X | S = Q, X | S = R ) .
Please refer to Section A.1 for the proof of Lemma 1.
At a high level, Lemma 1 states that theunfairness of any model is bounded above by the statistical parity distance of the features between thegroups.
We now use Lemma 1 to prove the following theorem that connects d TV ( Z ′ | S
= Q, Z ′
| S =
R ) to R D ′ and d TV ( Z | S = Q, Z | S = R ) to R D .
Theorem 1 Suppose X ′ and X are continuous and bounded on the region X := × di
{+=
+}
[ a i − m i , a +m i ] .
Further suppose WLOG that f X ′ |
S = Q ( x ) − f X ′ | S
= R ( x ) ≥ 0 and f X
| S = Q ( x ) − f X | S
{+= R ( x ) ≥ 0only on the region A := × di =
 [ a i − m i , a i ] , where f A is the density function of A .+}
We then get thefollowing bounds:
[Equation]
Please refer to Section A.2 for the proof of Theorem 1.
This result suggests that minimizing R D canminimize the unfairness of the model trained on the imputed data.
Furthermore, it is possible that R D > R D ′ , in which case feature imputation may amplify the unfairness of a model.
While our definition of discrimination risk applies beyond the setting of graphs and is agnostic tothe model architecture, in practice, it is important to consider model complexity and the context of the task.
We also add that our definition of discrimination risk bears resemblance to the Average
Treatment Effect studied in causality [80].
Furthermore, for clarity, we note that [59] has previouslyproposed a metric also called discrimination risk which quantifies how much the deviation of imputedfeature values from the ground-truth feature values differs across groups; this quantity is related to thepredictive parity of a model trained on the imputed data.
In contrast, our definition of discriminationrisk simply measures how much imputed features differ across groups and is more aligned with thestatistical parity of a model.
Statistical parity and predictive parity are mutually exclusive, except indegenerate cases [19].
We leave extending our definition of discrimination risk to other notions offairness for future work [69].
Overview of graph feature imputation
Prior to investigating the discrimination risk of graph feature imputation, we first present a generalframework for mean aggregation feature imputation that encompasses common, diffusion-basedgraph feature imputation algorithms in the literature.
Suppose we have an undirected, possibly weighted homogeneous graph G =
( V, E ) .
Each node hasd features, hence the node feature matrix X ∈ R N × d , where N = | V | , the cardinality of the set.
For simplicity, we will assume that d = 1 in our analyses and extend our results to the case d > 1afterwards.
Feature values are unknown for some nodes in G (denoted as the set U ), and knownfor others (denoted as the set K ).
The feature value of each node is either known or unknown, i.e.
U ∪ K = V and U ∩ K = ∅ .
Let X S refer to the feature values of the nodes in set S .
Assume withoutloss of generality that X
= (cid:20)
X K
X U (cid:21) .
Furthermore, let A ∈ R N × N denote the weighted adjacencymatrix of G , where A ij is the nonnegative weight corresponding to the edge from node j to nodei .
Additionally, let A S 1 S 2 denote the submatrix of A with rows belonging to the nodes in set S 1and columns belonging to the nodes in set S 2 .
Let A :
=
(cid:20)
A KK
A KU A UK
A UU (cid:21) .
D is the diagonal degreematrix, i.e. D ii =
(cid:80) Nj =
1 A ij and D := (cid:20) D K
D U (cid:21) .
Definition 2 (Mean Aggregation Feature Imputation)
Denote the feature values at iteration t ofmean aggregation feature imputation as X ( t ) .
Furthermore, let X ( t ) S refer to the feature values of thenodes in set S at iteration t .
Then, at each iteration t :
[Equation]
208 where M : R N → R N is a diagonal invertible map, T ∈ R N × N is a right-stochastic matrix, andβ ∈ [0 , 1] is a regularization hyperparameter.
We can see that mean aggregation feature imputation treats channels independently (i.e. does notsupport channel mixing), and only works with continuous (not discrete) features.
Mean aggregationencompasses many common graph-agnostic and graph-aware feature imputation methods, including
Global Mean (sets the unknown features to the uniform mean of all the known features), Neighbor
Mean (sets the unknown features to the degree-weighted mean of the known features for neighboringnodes), Feature Propagation (predicts the unknown features to minimize the Dirichlet energy of thegraph while preserving the known feature values), and Graph Regularization (predicts the unknownfeatures via a smoothness constraint and a fitting constraint for the known features).
For proofs,refer to Section A.3.
Despite Neighbor Mean, Feature Propagation, and Graph Regularization beingintended for homophilic graphs [17], mean aggregation feature imputation encompasses algorithmsthat could perform well on heterophilic graphs as well, with an appropriate choice of T [17].
We alsonote that T cannot be A , as this might cause feature values to explode over multiple iterations.
Discrimination risk of mean aggregation feature imputation
To better understand how mean aggregation feature imputation amplifies the unfairness of a machinelearning model, we theoretically characterize graphs in which mean aggregation imputation increases the discrimination risk, without making any assumptions about underlying distributions or models forunknown feature rates or graph structure.
We begin by defining new notation.
In particular, we first focus on the case of a single feature (i.e.d = 1 ), and extend our analysis to the case d > 1 in Section A.6.
Let X v be the feature of a node v .
Let Q denote the set of nodes that belong to the marginalized group, and R denote the set of nodesthat belong to the dominant group.
We will assume that Q ∪ R = V and Q ∩ R = ∅ .
We define thediscrimination risk after t iterations of mean aggregation imputation as:
[Equation]
where the expectations are taken uniformly over the nodes in each set.
Now, define ˜ X := MX .
Then,a modified version of the discrimination risk after t iterations of mean aggregation imputation is˜ R ( t ) :=
(cid:12)(cid:12)(cid:12) E q ∼ Q [ ˜ X ( t ) q ] − E r ∼ R [ ˜ X ( t ) r ] (cid:12)(cid:12)(cid:12) .
We now bound the discrimination risk ˜ R ( t ) of the imputedfeatures with respect to ˜ R (0) .
We bound ˜ R ( t ) rather than R ( t ) for simplicity; however, we empiricallyvalidate that the bound properties also hold for R ( t ) in Section 7.
We begin with Theorem 2.
Define ˜ µ ( t ) Q :=
E q ∼ Q [ ˜ X ( t ) q ] and ˜ µ ( t ) R :=
E r ∼ R [ ˜ X ( t ) r ] .
Furthermore, let˜ σ ( t ) denote the maximal deviation of the feature values at iteration t , i.e. ∀ q 1 ∈ Q, | ˜ X ( t ) q 1 − ˜ µ ( t ) Q | ≤
[Equation]
( t ) q 2 | ≤ ˜ σ ( t ) and ∀ r 1 ∈ R, | ˜ X ( t ) r 1 − ˜ µ ( t ) R | ≤ max r 2 ∈ R | ˜ X ( t ) r 1 − ˜ X ( t ) r 2 | ≤ ˜ σ ( t ) .
Additionally, define T S 1 → S 2 := (cid:80) b ∈ S 2 (cid:80) a ∈ S 1 T ba .
Theorem 2 Let the contraction coefficient α :=
(cid:12)(cid:12)(cid:12) 1 − T R → Q ∩ U + βT R → Q ∩ K | Q
|
[Equation]
R | (cid:12)(cid:12)(cid:12) .
Then, α ≤ 1 , and:
[Equation]
Please refer to Section A.4 for a proof of Theorem 2.
Theorem 2 shows that the bounds on thediscrimination risk contract more slowly (with more iterations of mean aggregation feature imputation)with a larger α .
Furthermore, the upper bound on the discrimination risk is larger with a larger α andmay depend on the initial unknown node features.
Theorem 2 allows for interesting interpretations ofhow graph properties like the rate of unknown features, group size, and graph structure affect thediscrimination risk of mean aggregation feature imputation.
In Section A.5, we successively varyeach property (holding all other properties constant) and investigate its impact on α , and in turn thediscrimination risk.
We find that a low unknown feature rate for both groups or disparate unknownfeature rates across the groups and a low inter-link to intra-link ratio can increase α , and thus thediscrimination risk of mean aggregation-imputed features.
Additionally, group size alone may notaffect α or the discrimination risk.
Fair graph feature imputation
We propose a simple and effective solution to ensure mean aggregation feature imputation provablyhas a low discrimination risk while minimally sacrificing utility.
At a high level, we want to constrainthe discrimination risk at every iteration t of imputation to be at most ϵ , i.e. ∀ t ∈ [0 , ∞ ) , R ( t ) ≤ ϵ .
We do this by viewing mean aggregation imputation through a gradient descent lens and projectingimputed features into the feasible space of features with discrimination risk at most ϵ at each iteration[81].
We focus on a single feature i , but our algorithms can be extended to all features by applyingthem to each feature separately.
We begin with the case where the known feature values remain fixed, i.e. β = 0 .
Recall the iterative algorithm for mean aggregation feature imputation when β = 0 :
[Equation]
We see that ∀ t ∈ [0 , ∞ ) , X ( t ) K
= X K .
Furthermore, define ∆ := I N − M − 1 TM .
In the case of
Feature Propagation, ∆ =
I N − D 12 ( D − 1 A ) D −
=
I N − D − 12 AD − 12 is the symmetric normalized
Laplacian of G [17].
We see that X ( t +1) U
=
( I N − ∆) UU X
( t ) U − ∆ UK X K .
As discussed in [17], wecan view the update
X ( t +1)
U :=
X ( t ) U − γ (∆ UU X ( t ) U + ∆ UK X K ) as an iteration of gradient descent(with step size γ = 1 ) on the objective function ℓ ( x ) = 12 x T ∆ UU x + X TK ∆ KU x + 12 X TK ∆ KK X K ,where X K is constant.
For Feature Propagation, ℓ is the Dirichlet energy of G [17].
We now presenta theorem that shows how to perform mean aggregation feature imputation with a discrimination riskof at most ϵ when the known feature values remain fixed.
Theorem 3 ( ϵ -Fair Imputation, β = 0 )
Vanilla mean aggregation feature imputation updates
X ( t +1)
U :=
X ( t ) U − γ (∆ UU X ( t ) U + ∆ UK X K ) =
Z ( t ) U , where
γ = 1 .
Let ϵ -fair mean aggregation feature imputation instead update X ( t +1) U :=
P W Z ( t ) U + P B , where:
[Equation]
0 , otherwise
[Equation]
Then, assuming 0 ≤ λ min (∆ UU ) ≤ λ max (∆ UU ) < 1 (where λ min and λ max ) are the minimumand maximum eigenvalues, respectively): 1) a unique optimal (with respect to ℓ ) feasible solution
X ∗ U exists; 2) for fixed step size γ = 1 λ max (∆ UU ) , ϵ -fair imputation converges as ∥ X ( t ) U − X ∗ U ∥ 22 ≤(cid:16) 1 − λ min (∆ UU ) λ max (∆ UU ) (cid:17)
t
∥ X (0) U − X ∗ U ∥ 22 ; 3) for fixed step size γ = 1 , ϵ -fair imputation converges to X ∗ U .
Please refer to Section A.7 for a proof of Theorem 3.
The convergence of our solution to the uniqueoptimal (with respect to ℓ ) feasible solution implies that our solution provably has a discriminationrisk of at most ϵ while minimally sacrificing utility.
Furthermore, because our solution simplyinterleaves projections into the mean aggregation imputation framework, it preserves the framework’sspeed and scalability [17].
We similarly have a solution when β > 0 , i.e. when the known nodefeature values do not remain fixed, which we present in Section A.8.
Experimental results and discussion
We empirically evaluate the fairness and utility of our solution on various mean aggregation feature imputation algorithms and synthetic and real-world datasets.
We find that while our solutionyields improved fairness without significant loss in utility on the synthetic datasets, there is not animprovement in fairness on the real-world datasets.
Datasets We construct undirected two-block synthetic networks ( SBM ) using
StochasticBlockModelDataset from PyTorch Geometric [82] (where one block corresponds to the marginalized group Q and the other block to the dominant group R ) with various(relative) group sizes and inter- and intra-link rates (more information in Section B.1).
SBM does nothave a corresponding task, i.e. the nodes do not have labels.
We also use the real-world Creditdefaulter and German credit networks from [30].
The Credit defaulter network consistsof 30,000 nodes representing individuals, with edges between them indicating similar spending andpayment patterns.
The corresponding task is to predict whether an individual will default on theircredit card payment or not, and the groups are those 25 years old or younger and those above theage of 25 (more information in Section B.1).
The German credit network comprises 1,000 nodesrepresenting clients in a German bank who are connected if they have similar credit accounts.
Thecorresponding task is to predict whether a client has good or bad credit risk, and the groups are menand women (more information in Section B.1).
(In reality, gender is neither binary nor static, andtreating it as such can produce harms [83].)
We refrain from using the Recidivism graph from[30] so as not to support the development of carceral technology [84].
Protocols and performance evaluation By default, none of the datasets contain unknown ormissing features.
To simulate diverse scenarios with unknown features, for each group, we independently at random mark node feature values as unknown with a different probability.
We experimentwith all 25 combinations of unknown feature rates of { 0 . 1 , 0 . 3 , 0 . 5 , 0 . 7 , 0 . 9 } for the groups.
Nodes,even within the same group, may have different unknown features.
We choose this scheme tostudy the effect of disparate unknown feature rates across the groups, which reflects the real world[18, 19, 20, 21].
We encourage future empirical work on other unknown feature schemes, includingmissingness based on node degree or marking all or none of the features for each node as unknown.
To impute unknown features, we use the regular mean aggregation imputation algorithms overviewedin Section 4: Global Mean (GM), Neighbor Mean (NM), Feature Propagation (FP), and Graph
Regularization (GR) (with β = 0 . 25 ), and their ϵ -fair counterparts (for ϵ ∈ { 0 . 0 , 0 .
Formodels, we use a simple linear classifier ( linear ), two-layer MLP ( mlp ), and Graph Convolutional
Network ( gcn ) [2] (more information in Section B.3).
We train all models on the imputed data, butvalidate and test on fully-observed data.
Because there are no previous works (to the best of ourknowledge) that directly address the unfairness of graph feature imputation, we do not have baselinesagainst which to compare.
To evaluate the utility of imputed features for SBM , since we don’t have labels, we employ relativereconstruction error RE [17].
For the real-world datasets, we consider the test accuracy ( Acc ) ofmodels trained on the imputed data.
To evaluate group fairness, we compute the discriminationrisk ( DR ) of the imputed data.
For SBM , we also train models on the imputed data to predict groupmembership and calculate the test accuracy of the models on identifying group membership (whichwe refer to as MI ) [85, 22].
To evaluate group fairness for the real-world datasets, we use the teststatistical parity ( SP ) of the models, defined as | P ( Z = 1 | S = Q ) − P ( Z = 1 | S = R ) | [78], andtest predictive parity ( PP ), defined as | P ( Z = 1 | S
= Q, Y = 1) − P ( Z = 1 | S
= R, Y = 1) | [86].
Please refer to Section B.4 for more details on the metrics.
For all metrics, we report the mean andstandard error over 5 runs using different random seeds.
On each run, a new dataset (in the case of SBM ) is generated, new splits are created, and a new model is trained.
Q1.
Does the contraction coefficient α align with the discrimination risk of mean aggregationimputation across graphs with different properties?
Figures 2 to 12 in the Appendix show that for
SBM , discrimination risk and α generally have a strong positive association over unknown featurerates, group sizes, and inter-link rate to intra-link rate ratios, which substantiates Theorem 2.
Thisassociation is weaker for Global Mean and Neighbor Mean.
Refer to Section B.5 for more details.
Q2.
Does ϵ -fair mean aggregation imputation (compared to regular mean aggregation imputation) improve the group fairness of a model trained on the imputed data?
Table 1 shows that,for SBM , ϵ -fair mean aggregation imputation achieves comparable reconstruction error to its regularcounterpart while greatly reducing the discrimination risk and test group membership identificationaccuracy for all models.
As expected, we see that the discrimination risk of ϵ -fair imputation is at mostϵ , and discrimination risk and test group membership identification accuracy are positively correlated,which substantiates Theorem 1.
Furthermore, the reconstruction error for ϵ -fair FP and GR (whichleverage graph structure and are thus more susceptible to graph structural bias) are much lower thanthat of the naïve ϵ -fair GM (which is graph-agnostic), but fair FP and GR reduce the discriminationrisk and test group membership identification accuracy for all models to similar levels as fair GM.
However, the test group membership identification accuracy generally decreases less as ϵ decreasesfor mlp and gcn than it does for linear , which suggests that minimizing the discrimination risk ofimputed features is more effective at removing linearly-encoded group membership information thannon-linearly encoded information.
Furthermore, ϵ -fair feature imputation does not guard againstgroup membership information that gcn learns via graph structure during training.
We have similarfindings when averaging over different relative group sizes (refer to Table 3) and combinations ofinter- and intra-link rates (refer to Table 4).
In contrast, Tables 2 and 6 (in the Appendix) show that, for the real-world datasets, regular meanaggregation imputation and its ϵ -fair counterpart yield comparable test accuracy and statistical parityfairness for all models.
In the case of Credit defaulter , our solution even appears to exacerbatethe unfairness of gcn .
We suspect this is because the datasets already produce models with lowfairness, upon which it is difficult for our ϵ -fairness solution to improve.
We were unable to reproducesimilar unfairness results to those in [30], even when all features are known.
We find similar resultsfor predictive parity, as shown in Tables 5 and 7.
[Table] [Table]
Conclusion
We prove that a higher discrimination risk can amplify the unfairness of a machine learning modeltrained on imputed data.
We formalize a general graph feature imputation framework called meanaggregation imputation and theoretically and empirically characterize graphs in which applyingthe framework can yield a high discrimination risk.
We propose a simple and effective solution toensure mean aggregation-imputed features provably have a low discrimination risk (while minimallysacrificing utility) and improve the fairness of models.
Our analysis and solution, like many fair machine learning algorithms, assume that groups are discreteand that group membership is known and static, which is not true in reality [81, 83, 27].
Furthermore,we don’t consider fairness at the intersections of different groups [87, 88], and operationalizationsof fairness beyond parity between two groups [89]; we encourage future work in these directions.
Furthermore, while fairness is often framed as sufficient for the creation of ethical systems, this is notnecessarily the case.
For instance, ϵ -fair mean aggregation feature imputation may be used to train amore fair model that diversifies news recommendations to social media users [6], but if this modelrecommends hostile or intolerant news sources to LGBTQIA+ users, this could lead to psychologicalharm [81].
We remind readers to carefully consider the ethics of downstream model use cases.
Checklist
For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper’scontributions and scope?
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[Yes] (We transparently discuss thelimitations of theoretical results, formalizations, and experimental results throughoutthe paper, directly before or after they are presented.)
(c) Did you discuss any potential negative societal impacts of your work?
[Yes] (We
refrain from using a recidivism prediction dataset and inferring identity from data for
ethical reasons in Sections 7 and B.4, and discuss potential negative societal impacts ofour work in Section 8.)
(d) Have you read the ethics review guidelines and ensured that your paper conforms to
them?
[Yes]
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(a) Did you state the full set of assumptions of all theoretical results?
[Yes] (We enumerate
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theorems.)
(b) Did you include complete proofs of all theoretical results?
[Yes] (We do so in SectionA.)
If you ran experiments...
(a) Did you include the code, data, and instructions needed to reproduce the main experi- mental results (either in the supplemental material or as a URL)?
[Yes] (All experiment code and relevant dependencies and instructions are included in the supplemental
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(b) Did you specify all the training details (e
g., data splits, hyperparameters, how they were chosen)?
[Yes] (All training details are specified in Sections 7, B.1, B.2, andB.3.)
(c) Did you report error bars (e g., with respect to the random seed after running experiments multiple times)?
[Yes] (We report all metrics with their mean and standard error
computed over 5 runs with different random seeds, as discussed in Section 7.)
(d) Did you include the total amount of compute and the type of resources used (e.g., typeof GPUs, internal cluster, or cloud provider)?
[Yes] (We do so in Section B.3.)4.
If you are using existing assets (e.g., code, data, models ) or curating/releasing new assets...
(a) If your work uses existing assets, did you cite the creators?
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We do not create any new datasets.)
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using/curating?
[Yes] (The real-world datasets are used in accordance with their MIT
license, as explained in Section B.1.)
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information or offensive content?
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If you used crowdsourcing or conducted research with human subjects...
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A Proofs
A.1 Proof of Lemma 1 (special case of total variation distance)
We assume Z and X are continuous probability distributions, but this proof can be easily altered forother kinds of probability distributions and the result still holds.
Let P ( S = Q ) = p .
By definition,the total variation information:
[Equation]
[Equation]
S =
R ( z ) − ( pf Z | S = Q ( z )
+ (1 − p )
f Z | S = R ( z )) (cid:12)(cid:12) d z (cid:105) =
(cid:104) p (cid:90)
Y (cid:12)(cid:12) (1 − p )
f Z | S = Q ( z ) − (1 − p )
f Z | S
=
R ( z ) (cid:12)(cid:12)
d z + (1 − p ) (cid:90)
Y (cid:12)(cid:12) pf Z | S =
R ( z )
[Equation]
pf Z | S = Q ( z ) (cid:12)(cid:12)
d z (cid:105) =
p (
[Equation]
p ) (cid:20)(cid:90)
Y (cid:12)(cid:12)
f Z | S =
[Equation]
( z )
[Equation]
f Z | S =
[Equation]
( z ) (cid:12) (cid:12) d
z + (cid:90) Y (cid:12)(cid:12) f Z | S = R ( z )
[Equation]
f Z | S =
[Equation]
( z ) (cid:12) (cid:12) d
z (cid:21) =
p (1 − p ) (cid:90)
Y (cid:12)(cid:12)
f Z | S =
[Equation]
( z ) − f Z | S
= R ( z ) (cid:12)(cid:12)
d z = 2 p (1 − p )
d TV ( Z | S
[Equation]
Q, Z | S = R )
It can be similarly shown that I TV ( X ; S )
= 2 p (1 − p )
d TV ( X | S = Q, X | S = R ) .
Now, by the Data Processing Inequality, I TV ( Z ; S ) ≤ I TV ( X ; S ) .
Hence:
[Equation]
Similarly:
[Equation]
A.2 Proof of Theorem
By definition: [Equation] This can be decomposed as:
[Equation]
We can bound the re-expression of d TV ( X | S = Q, X | S = R ) by applying Markov’s inequality:(cid:90) [Equation] =
[Equation]
A [Equation] ) = [Equation]
a )) = [Equation] [Equation] m
[Equation]
m ) [Equation] [Equation] a
[Equation] m ) [Equation] ( a − m ))
−≤ min i ∈ [ d ] (cid:26) ( a i + m i ) − E [ X i | S = Q ] m i + E
[Equation] X i | S = R ] − ( a i − m i ) m i (cid:27)
−= min i ∈
[ d ] (cid:26) 2 m i + ( E [ X i | S = R ] − E [ X i | S = Q ]) m i (cid:27) −= min i ∈
[ d ] E [ X i | S = R ] − E [ X i | S = Q ] m i +f X
| S = Q ( x ) ≥ f X | S
= R ( x ) only on A implies E [ X i | S = R ] ≥ E [ X i | S = Q ] . Hence, E [ X i | S =
R ] − E [ X i | S = Q ] = | E [ X i | S = Q ] − E [ X i | S =
R ] | = R D .
Therefore, d TV ( X | S = Q, X | S =
R ) ≤ min i ∈ [ d ] R D
{+m i + 
+}
Similarly:
[Equation]
Then, the theorem is proved by application of Lemma 1.
A.3 Example Mean Aggregation Imputation Algorithms
Global Mean This method sets the unknown features to the uniform mean of all the known features.
To achieve this, we can choose M :
= I N , T := (cid:20)
I | K | 0| K | 1 | U |×| K | (cid:21) (where 1 is the all-ones matrix),β := 0 , X (0) K := X K , and X (0) U := 0 .
We only need to complete one iteration.
Neighbor Mean This method sets the unknown features to the degree-weighted mean of the knownfeatures for neighboring nodes.
We can choose M :
= I N , T :
= D − 1 A , β := 0 , X (0) K :=
X K , and
X (0) U := 0 .
We only need to complete one iteration.
Feature
Propagation This method proposed by [17] predicts the unknown features to minimizethe Dirichlet energy of the graph while preserving the known feature values.
[17] shows that this isequivalent to iteratively computing until convergence:
[Equation] [Equation]
Multiplying both sides by D − 12 U , we can re-express the second update rule as:
[Equation]
703 Therefore, to achieve Feature Propagation, we can choose M := D − 12 , T :
=
D − 1 A , β := 0 , and
X (0) K :=
X K .
Per [17], we can choose X (0) U arbitrarily, and we need to iterate till convergence.
Graph Regularization This method inspired by [90] predicts the unknown features via a smoothness constraint and a fitting constraint for the known features.
[90] shows that this is equivalent to iteratively computing until convergence:
[Equation]
where the hyperparameter β ∈ [0 , 1] .
Therefore, similar to Feature Propagation, to achieve Graph
Regularization, we can choose M := D − 12 , T :
= D − 1 A , and X (0) K :=
X K .
Per [90], we can choose
X (0) U arbitrarily, and we need to iterate till convergence.
A.4 Proof of Theorem
The following proof is partially inspired by the proof of Theorem 4.1 in [6].
Fix t to be an arbitrary iteration of feature imputation.
Recall we use the following iterative update rule to impute features:
[Equation]
For a node q ∈ Q ∩ U , after one iteration of feature imputation:
[Equation]
Similarly, for a node r ∈ R ∩ U , after one iteration of feature imputation:
[Equation]
In contrast, for a node q ∈ Q ∩ K , after one iteration of feature imputation:
[Equation]
Similarly, for a node r ∈ R ∩ K , after one iteration of feature imputation:
[Equation]
We say v ∈ [ µ ± σ ] ⇐⇒ µ − σ ⪯ v ⪯ µ + σ .
Then, for a node q ∈ Q ∩ U , by the right-stochasticnature of T :
[Equation]
713 Similarly, for a node r ∈ R ∩ U :
[Equation]
In contrast, for a node q ∈ Q ∩ K :
[Equation]
Similarly, for a node r ∈ R ∩ K :
[Equation]
  717 By the
[Equation]
Similarly, E r ∼ R [ ˜ X
[Equation]
t +
[Equation]
r ] :
[Equation]
Thus, the gap in expectation of the features of the nodes in Q and R after one iteration of featureimputation is:
[Equation]
Define the contraction coefficient:
[Equation]
Because 0 ≤ T R → Q ∩ U + βT R → Q ∩ K | Q
[Equation]
≤
[Equation]
= [Equation] < T V → Q | Q | = 1 , and similarly,0 ≤ T Q → R ∩ U + βT Q → R ∩ K | R | < 1 , it must be that 0 ≤ α ≤ 1 {+.
+}
Then:
[Equation] [Equation]
Inductively, the discrimination risk ˜ R ( t ) after t iterations of feature imputation is bounded by:
[Equation]
∀ v ∈ V , ˜ X ( t +1) v is a convex combination of (cid:83) u ∈ V { ˜ X ( t ) u } .
This is because each row of T and βT + (1 − β )
I | K | contains nonnegative entries that sum to 1.
Therefore, ˜ X ( t +1) v must be in the (closed) convex hull of (cid:83) u ∈ V { ˜ X ( t ) u } .
Thus, (cid:83) u ∈ V { ˜ X ( t ) u } inductively must be contained within the (closed) convex hull of (cid:83) u ∈ V { ˜ X (0) u } , which has extreme points ⊆
(cid:83) u ∈ V { ˜ X (0) u } .
Consequently, ∀ t ∈ [0 , ∞ ) , ˜ σ ( t ) ≤ ˜ σ (0) .
Hence:
[Equation]
If α < 1 :
[Equation]
Moreover, upon convergence:
[Equation]
A.5 Analysis of Theorem 2
We focus on Feature Propagation [17], but our analysis may be easily extended to other meanaggregation feature imputation algorithms.
We assume for simplicity that all edges have a weight of1.
Unknown feature rates Suppose the intra-link rate P (( u, v ) ∈ E | u ∈ Q, v ∈ Q ) =
P (( u, v ) ∈
E | u ∈ R, v ∈ R ) = 12 and inter-link rate P (( u, v )
∈
E | u ∈ Q, v ∈ R )
=
P (( u, v ) ∈
E | u ∈
R, v ∈ Q ) = 12 .
Furthermore, assume equal (relative) group sizes, i.e. | Q | N =
| R | N = 12 .
[Equation]
By decomposition, D qq =
(cid:80)
u ∈ Q
A qu + (cid:80) v ∈ R A qv = 12 | Q | + 12 | R | =
N 2 .
Therefore, T R → Q ∩ U + βT R → Q ∩ K | Q | = [Equation] = 12 ( | Q [Equation]
× [Equation] N [Equation]
= 12 ( P ( v [Equation] U |[-) N 2 /= 
-]
[Equation] Q ) ·| Q |×| R | ) N [Equation] = 12 P ( v ∈ U | v ∈ Q ) , where
P ( v ∈ U | v ∈ Q ) is the unknown feature rate for group Q .
Similarly, T Q → R ∩ U + βT Q
→ R ∩ K | R | =12 P ( v ∈ U | v ∈ R ) .
Thus, α = | 1 − 12 P ( v ∈ U | v ∈ Q ) − 12 P ( v ∈ U | v ∈ R ) | .
This reveals that alow unknown feature rate for both groups or disparate unknown feature rates across the groups canincrease α , and thus the discrimination risk of mean aggregation-imputed features.
This aligns with[60]’s finding that imputation fairness can be influenced by the imbalance of feature missingnessacross groups, although [60] studies predictive parity rather than statistical parity fairness.
Group sizes Now, suppose the intra-link rate P (( u, v )
∈
E | u ∈ Q, v ∈ Q ) =
P (( u, v ) ∈
E | u ∈
R, v ∈ R ) = 12 and inter-link rate P (( u, v ) ∈
E | u ∈ Q, v ∈ R )
=
P (( u, v ) ∈
E | u ∈ R, v ∈
Q )
= 12 .
Furthermore, assume equal unknown feature rates, i.e. P ( v ∈ U | v ∈ Q ) =
P ( v ∈ U | v ∈
R )
= 12 .
[Equation]
D qq
=
N
Therefore,
[Equation]
= 12 ( 12 |
[Equation]
= 12 · | R | N .
Similarly,
[Equation]
= 12 · | Q | N .
Thus, α = | 1 − 12 · | R | N − 12 · | Q | N | = 12 .
This demonstrates that groupsize alone may not affect α or the discrimination risk of mean aggregation-imputed features.
Graph structure Finally, suppose we have equal unknown feature rates, i.e. P ( v ∈ U | v ∈
Q ) =
P ( v ∈ U | v ∈ R )
= 12 .
Furthermore, assume equal (relative) group sizes,i.e.
[Equation]
[Equation] [Equation]
P (( u,
[Equation]
) ∈ E | u ∈ Q, v ∈ Q ) | Q | + P (( u, v ) ∈
E | u ∈ R, v ∈ Q ) | R | =
N 2 [ P (( u, v ) ∈
E | u ∈ Q, v ∈ Q ) + P (( u, v )
∈ E | u ∈
R, v ∈ Q )] .
Therefore, T R → Q ∩ U + βT R → Q ∩ K | Q | = P [Equation] ( u
[Equation]
[Equation]
| u ∈ R,
[Equation]
Q )( |
[Equation]
×
[Equation]
R | )
[ P (( u,v )
[Equation]
E | u
[Equation]
Q, [Equation] ∈ Q ) + P (( u,v )
∈ E | u ∈ R,v ∈ Q )] · N 2 / 4 =12 | Q |×| R | [ 1+ P (( u
[Equation]
v ) ∈ E | u ∈ R
[Equation]
v
[Equation] Q ) [Equation] [Equation]
u
[Equation]
v ) ∈ E | u ∈ Q [Equation] v ∈ Q ) ] · N 2 / 4 = 12 · 1 1+
[Equation]
E | u
[Equation]
Q
[Equation]
Q ) .
Similarly, T
[Equation]
= 12 · [Figure] [Equation] [Figure]
[Equation]
v ) ∈ E | u ∈ Q,v ∈ R ) P (( u,v )
∈ E | u ∈ R,v ∈ R ) .
Thus, α = (cid:12)(cid:12) (cid:12)(cid:12) 1 − 1 2 · 1 1+ P
( u,v ) ∈ E | u ∈ R,v ∈ Q ) P (( u,v ) ∈
E | u ∈ Q,v ∈ Q ) − 12 · 1 1+ P (( u,v ) ∈
E | u ∈ Q,v ∈ R ) P (( u,v )
∈ E | u ∈ R,v ∈ R ) (cid:12)(cid:12) (cid:12)(cid:12) .
Because G is undirected, P (( u, v )
∈ E | u ∈ R, v ∈ Q ) =
P (( u, v ) ∈
E | u ∈ Q, v ∈ R ) and 758 P (( u, v ) ∈ E | u ∈ Q, v ∈ Q ) =
P (( u, v ) ∈
E | u ∈ R, v ∈ R ) .
Therefore, a low inter-link tointra-link ratio can increase α and the discrimination risk of mean aggregation-imputed features.
A.6 Extending Theorem 2
We can extend Theorem 2 to the case the number of features d > 1 .
By Theorem 1, the modified discrimination risk at iteration t (including all features) is:
[Equation]
Moreover, assuming ∀ i ∈ [ d ] , α i < 1 , upon convergence, the discrimination risk is:
[Equation]
[Table]
A.7 Proof of Theorem 3
We want to constrain the discrimination risk of mean aggregation feature imputation to be at most ϵ .
To this end, we can modify mean aggregation feature imputation to update X ( t +1)
U :=
P W Z ( t ) U + P B such that X ( t +1) has a discrimination risk of at most ϵ for all t .
| E q ∼ Q [ X ( t +1) q ] − E r ∼ R
[ X ( t +1) r ] | = | 1 | Q | (cid:80) q ∈ Q ∩ K X q + 1 | Q | (cid:80)
q ∈ Q ∩ U Z ( t ) q − ( 1 | R | (cid:80) r ∈ R ∩ K X
r + 1 | R | (cid:80)
r ∈ R ∩ U Z ( t ) r ) | .
Hence, we have a closed convex polytope wherein unknown feature values yield discrimination risk of at most ϵ :
[Equation]
If R K − ϵ ≤ c T Z ( t ) U ≤ R K + ϵ , then P W = I | U | and P B = 0 .
Otherwise, we must projectonto the closer of the two boundaries of the polytope.
In this case, P W = I | U | − cc T c T c and P B =cc T c T
[Equation]
The affine projection we perform at each step is closed and convex.
Furthermore, ℓ is λ max (∆ UU ) -smooth for the Euclidean norm (where λ max is the maximum eigenvalue) because for x 1 , x 2 ∈ R | U | :
[Equation]
In the case of Feature Propagation, λ max (∆ UU ) < 1 due to properties of the symmetric normalized
Laplacian [17].
Additionally, for m ≥ 0 , when m ≤ λ min (∆ UU ) , ℓ ( x ) − m 2 x T x is convex because:
[Equation]
This expression is convex if and only if its Hessian ∆ UU − mI has nonnegative eigenvalues.
Therefore,m can be at most λ min (∆ UU ) .
Then, by [91] and [92]:
a unique optimal (with respect to ℓ ) feasible solution X ∗ U exists2.
for fixed step size γ = 1 λ max (∆ UU ) , ϵ -fair imputation converges as ∥ X ( t ) U − X ∗ U ∥ 22 ≤(cid:16)
[Equation] ∥
for fixed step size γ = 1 < 1 λ max (∆ UU ) , ϵ -fair imputation converges to X ∗ U
A.8 Theorem
We have a solution when β > 0 , i.e. when the known node feature values do not remain fixed.
We canview the update of X ( t +1) :
= (cid:20) βI | K |
[Equation] 1 − β ) I | K |   (cid:21)
X as an iterationof gradient descent (with step size γ = 1 ) for the objective function ℓ ( x ) = 12 x T ∆ x + 12 ( 1 − ββ )
∥ x K −
X K ∥ 22 [90].
Theorem 4 ( ϵ -Fair Imputation, β > 0 ) Vanilla mean aggregation feature imputation updates
[Equation]
X =
Z ( t ) .
Let ϵ -fair mean aggregation feature imputation instead update X ( t +1) :
= P W Z ( t ) + P B , where:
[Equation]
Then, assuming 0 ≤ λ min (∆) + 1 − ββ ≤ λ max (∆) + 1 − ββ < 1 : 1) a unique optimal (with respect toℓ ) feasible solution X ∗ exists; 2) for fixed step size γ = 1 λ max (∆)+ 1 − ββ , ϵ -fair imputation convergesas ∥ X ( t ) − X ∗ ∥ 22 ≤ (cid:18) 1 − λ min (∆)+ 1 − ββ λ max (∆)+ 1 − ββ (cid:19) t
∥ X (0) − X ∗ ∥ 22 ; 3) for fixed step size γ = 1 , ϵ -fairimputation converges to X ∗ .
Proof of Theorem 4 We want to constrain the discrimination risk of mean aggregation featureimputation to be at most ϵ .
To this end, we can modify mean aggregation feature imputation to update
X ( t +1) :
= P W Z ( t ) + P B such that X
( t +1) has a discrimination risk of at most ϵ for all t .|
E q ∼ Q [ X ( t +1) q ] − E r ∼ R [ X ( t +1) r ] | = 1 | Q | (cid:80) q ∈ Q Z ( t )
q − 1 | R | (cid:80) r ∈ R Z ( t ) r .
Hence, we have a closed convex polytope wherein feature values have discrimination risk of at most ϵ :
[Equation]
If − ϵ ≤ c T Z ( t ) ≤ ϵ , then P W = I N and P B = 0 .
Otherwise, we must project onto the closer of thetwo boundaries of the polytope.
In this case, P W = I N − cc T c T c and P B = cc T c T c (cid:26) −
[Equation]
, c
[Equation]
Z
[Equation]
t
[Equation]
< − ϵ ϵ, c T Z ( t ) > ϵ .
The affine projection we perform at each step is closed and convex.
Furthermore, ℓ is 794 (cid:16) λ max
[Equation]
∆) + 1 − ββ (cid:17) -smooth for the Euclidean norm because for x 1 , x 2 ∈ R N :
[Equation]
[Equation] [Equation]
Additionally, for m ≥ 0 , when m ≤ λ min (∆) + 1 − ββ , ℓ ( x ) − m 2 x T x is convex because:
[Equation]
This expression is convex if and only if its Hessian ∆ − mI + 1 − ββ
(cid:20)
I | K |   (cid:21) has nonnegativeeigenvalues.
Therefore, m can be at most λ min (∆) + 1 − ββ .
Then, by [91] and [92]:
a unique optimal (with respect to ℓ ) feasible solution X ∗ exists
for fixed step size γ = 1 λ max (∆)+ 1 − ββ , ϵ -fair imputation converges as ∥ X ( t ) − X ∗ ∥ 22 ≤(cid:18) 1 − λ min (∆)+ 1 − ββ λ max (∆)+ 1 − ββ (cid:19) t
∥ X (0) − X ∗ ∥ 223.
for fixed step size γ = 1 < 1 λ max (∆)+ 1 − ββ , ϵ -fair imputation converges to X ∗
B Additional experimental results
B.1 Datasets
SBM synthetic datasets Each network has 500 train nodes, 250 validation nodes, and 250 testnodes, split uniformly at random.
Each node has a 10-dimensional feature vector sampled as describedin the documentation 1 .
All edges have a weight of 1.
PyTorch Geometric is used in accordance withits MIT license.
Real-world datasets In the Credit defaulter dataset, each node has 13 features (e.g. education,credit history, etc.), with an average degree of 95 .
79 ± 85 . 88 [30].
In the German credit dataset,each node has 27 features (e.g. loan amount, account-related features, etc.), with an average degreeof 44 .
48 ± 26 . 51 .
For both datasets, we use a 50/25/25 train/validation/test split, with each splitcomprising an equal portion of each label, and we do not include group membership as a feature.
Tothe best of our knowledge (via manual sampling and inspection), neither dataset contains personallyidentifiable information or offensive content.
We use [30]’s data and data loading code 2 in accordancewith the MIT license.
B.2 Imputation algorithms
We run GM and NM for 1 iteration each, and FP and GR for 40 iterations.
We adapted the code fordata utilities, Feature Propagation, and model training from [17] 3 in accordance with its Apache-2.0license.
We state all changes in this paper.
We implement all algorithms using PyTorch, in accordancewith its license [93].
B.3 Models and training
For mlp and gcn , we use a hidden dimension of 64.
We train all models full-batch using the Adamoptimizer with a learning rate of 0.005 and Dropout rate of 0.5 [94, 95].
We also use early stoppingwith a patience of 200 epochs, i.e. we stop training when the best validation accuracy has not changedfor 200 epochs, and train for a maximum of 10000 epochs.
We do not do any hyperparameter tuning.
We implement and train all models using PyTorch and PyTorch Geometric [93, 82].
We train allmodels on a single tesla v100-sxm2-16gb GPU on an internal cluster.
B.4 Performance Evaluation
To evaluate the utility of imputed features for SBM , since we don’t have labels, we employ relativereconstruction error RE (calculated as ∥ X true − X pred ∥ 2 / ∥
X true ∥ 2 , where X true and X pred arethe ground-truth and imputed features, respectively [17].
A lower reconstruction error is better, andwe would like regular mean aggregation imputation and its ϵ -fair counterparts to have comparablereconstruction errors.
To measure utility for the real-world datasets, we consider the test accuracy( Acc ) of models trained on the imputed data.
A higher test accuracy is preferable, and we againwould like comparable accuracies for regular and ϵ -fair imputation.
To evaluate group fairness, we compute the discrimination risk ( DR ) of the imputed data.
Alower discrimination risk is preferable.
For the SBM synthetic datasets, we also measure how much [Table] information the imputed features contain about group membership.
We do this by training the [Table] models on the imputed data to predict group membership and calculate the test accuracy of themodels on identifying group membership (which we refer to as MI ) [85, 22].
The models may beconceptualized as adversaries attempting to recover group membership from the imputed features.
Thus, we would like MI to be closer to 0.5 (i.e. the imputed features contain no informationabout group membership).
We do not compute MI for the real-world datasets, as inferring groupmembership or identity from real-world data is invasive, invalid, and can be weaponized againstmarginalized communities (e.g. to find and incarcerate LGBTQIA+ individuals) [96].
To evaluategroup fairness for the real-world datasets, we use the test statistical parity ( SP ) of the models, definedas | P ( Z = 1 | S = Q ) − P ( Z = 1 | S = R ) | (disparity in positive outcome rate for the groups)
[78],and test predictive parity ( PP ), defined as | P ( Z = 1 | S = Q, Y = 1) − P ( Z = 1 | S
= R, Y = 1) |(disparity in accuracy of predicting positive outcome for the groups) [86].
B.5 Contraction coefficient
As we analyzed, Figures 2 to 12 show that, for SBM : 1) a low unknown feature rate for both groups ordisparate unknown feature rates across the groups increases α and the discrimination risk (Figures 2,5, 8, 11); 2) group size alone does not affect α or the discrimination risk (Figures 3, 6, 9, 12); 3) alower inter-link to intra-link ratio increases α and the discrimination risk (Figures 4, 7, 10).
[Figure]
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Q Group Size 0.
D i s c
r
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Method
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0 . 025 -Fair GR
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0 . 05 -Fair GR
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Regular
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r
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Rate / Intra
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Regular NM 0 .
031 ± 0 .0 .
041 ± 0 .0 .
005 ± 0 .0 .
-Fair FP 0 .
035 ± 0 .0 .
059 ± 0 .0 .
013 ± 0 .0 .
025 -Fair FP 0 .
031 ± 0 .0 .
062 ± 0 .0 .
013 ± 0 .0 .
-Fair FP 0 .
043 ± 0 .0 .
057 ± 0 .0 .
014 ± 0 .
Regular FP 0 .
028 ± 0 .0 .
05 ± 0 .0 .
004 ± 0 .0 .
-Fair GR 0 .
051 ± 0 .0 .
07 ± 0 .0 .
008 ± 0 .0 .
025 -Fair GR 0 .
03 ± 0 .0 . 06 ± 0 .0 .
012 ± 0 .0 .
05 -Fair GR 0 .
048 ± 0 .0 .
067 ± 0 .0 .
015 ± 0 .
Regular GR 0 .
03 ± 0 .0 .
051 ± 0 .0 .
007 ± 0 .
Inter-Link Rate / Intra-Link Rate
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Inter-Link Rate / Intra-Link Rate 0.
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h a
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Neighbor Mean 0.
Q Group Size
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r i m i n a t i o n
R i s k 0.
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