Path-specific Causal Fair Prediction via Auxiliary Graph Structure Learning
Authors: Liuyi Yao
Yaliang Li
Bolin Ding
Jingren Zhou
Jinduo Liu
Mengdi Huai
Jing Gao
Venue: ICLR 2022 Submitted (2022)

Abstract: Algorithm fairness has become a trending topic, and it has a great impact on social welfare. Among different fairness definitions, path-specific causal fairness is a widely adopted one with great potentials, as it distinguishes the fair and unfair effects that the sensitive attributes exert on algorithm predictions. Existing methods based on path-specific causal fairness either require graph structure as the prior knowledge or have high complexity in the calculation of path-specific effect. To tackle these challenges, we propose a novel casual graph based fair prediction framework, which integrates graph structure learning into fair prediction to ensure that unfair pathways are excluded in the causal graph. Furthermore, we generalize the proposed framework to the scenarios where sensitive attributes can be non-root nodes and affected by other variables, which is commonly observed in real-world applications but hardly addressed by existing works. We provide theoretical analysis on the generalization bound for the proposed fair prediction method, and conduct a series of experiments on real-world datasets to demonstrate that the proposed framework can provide better prediction performance and algorithm fairness trade-off. 

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PATH-SPECIFIC CAUSAL FAIR PREDICTION VIA AUXIL- IARY GRAPH
SAPHSTREE STRUCTURE LEARNING
Anonymous authors Paper under double-blind review
ABSTRACT
Algorithm fairness has become a trending topic, and it has great impact on social welfare. Among different fairness definitions, widely adopted one with great potentials, as it distinguishes the fair and unfair effects that the sensitive attributes exert on algorithm predictions. Existing methods based on path-specific causal fairnsss either require graph structure as the prior knowledge or have high complexity in the calculation of path-spa anthsspeifiic effect. To tackle these challenges, framework which integrates graph structure learning into fair prediction to ensure that unfair pathways are excluded in the causal graph. Furthermore, we generalize the proposed framework to the scenarios where sensitive attributes can be non- root nodes and affected by other variables, which is commonly observed in real- world applications but hardly addressed by existing works. We provide theoretical analysis the generalization bound for the proposed fair prediction method, and conduct a series of experiments on real-world datasets to demonstrate that the proposed framework can provide better prediction performance and algorithm fairness trade-off.
causal fairness is a
propose novel casual graph based fair prediction
INTRODUCTION
With the ubiquitous adoption of machine learning algorithms to facilitate decision making, algorithm
fairness has attracted increasingly more attentions, in the areas such as recommendation system Ge
al. (202T); Zhu et al. (2018); Burke (2017); Yao & Huang (2017), natural language processing
Bolukbasi et al. (2016) Zhao al. (2017) 7y: Gonen & Goldberg (2019): De-Arteaga et al. (019); ;
Blodgett et al. (222)) computer vision Shankar et al. (2017): : Nagpal et al. (2019); Raji et al. 2020 ;
Stock & Cisse 2018). hiring Hoffman et al. al..(201) (2018), education Brunori et al. (2012) banking Mukerjee
et al. (2002). and crime risk assessment Brennan et al.] (209); Dieterich et al.) (2016): Zhang &
Bareinboim 2018b). Algorithm fairness aims reduce or even eliminate unjustified distinctions of
individuals based on their sensitive altributes (e.g., gender and race) during the prediction Zhang &
Wu (2017). Unfortunately, machine learning models constructed from the raw data are vulnerable to the unfairness risk due to the historical prejudices in the data. It is crucial for model designers to take algorithm fairness into consideration for long-term social welfare.
In recent years, researchers have developed variety of causal fairness definitions to help machine learning models make fair predictions Zhang et al. (2017); [Huang et al.) (2020); Nabi Shpitser (2018); Kusner et al.) (2017), Russell et al. (2017), Wu et al. (2019a); Zhang & Bareinboim (2018a b); Hu et al. (2020); Xu et al. (2019); Wu et al. (2018): Zhang et al. (2016), and one of them, path- specific causal fairness Chiappa (019); ; Nabi & Shpitser (2018). ; Wu et al. (2019b), is adopted in this paper. Under the definition of path-specific causal fairness, unfairness is viewed as the presence of unfair causal effect through the disallowed causal pathway that the sensitive attributes exert on predictions. In other words, a fair prediction satisfies path-specific causal fairness eliminates the causal effect that the sensitive attributes assert on the prediction through disallowed causal pathways. Such a definition provides the flexibility of tracing the unfairness, because in some scenarios, the sensitive attributes affect the decision along multiple pathways, and not all pathways are unfair. For example, in the loan application Zhang et al. (017)) shown in Figure [ race sensitive attribute R) only allowed to affect the loan application results through the income, since it is reasonable to reject a loan application due to the low income. Under this fairness
rule, paths R + Y and + Z + Y are unfair paths and path R + Q + is a fair path.

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ZipCode
Income
To fulfill path-specifii causal fairness, some existing works directly calcu- late the path-specific causal effect (PSE) Aviin et al. (2005); Pearl (2001) along the unfair pathways, and minimize the effect tannoulyy when maximizing prediction aceuracy Wu et al. (2019b); Nabi & Shpitser
Z
unfair
fair
(2018). Some other works correct the variables located on the unfair
R
pathway by a latent snfferecce-pojecttinn method Chiappa m (2019). How- ever, these existing works still face the following challenges: (1) Most of them require a pre-defined graph as the prior knowledge to calculate PSE. (2) The calculation of the path-specific effect is complex, requiring the sequential ignorability assumption Imai et al. (2010) to ensure the identification. (3) They all assume the sensitive attributes are root nodes in the causal graph. Namely, there are no other variables that affect the sensitive attributes. Few of them consider the case when the sensitive attributes are non-root nodes, which can be widely observed in real-world applications. For example, in the recommendation system, the item popularity is a sensitive attribute Ge et al. (202T): Zhu et al. 2018), while this variable is non-root node as it is affected by the item 's characteristics.
unfair
Race
Loan
Figure 1: Loan Applica-
tion Example.
In light of the above challenges, we propose a Causal Graph based Faimeess Framework, shortened as CGF. To tackle the first challenge about the lack of causal graph information, CGF integrates the causal graph structure learning and fair prediction, which reveals the causal relationships among the observed variables. To simplify the sophisticated PSE calculation, CGF imposes the fairness regularization at the graph level by restricting the existence of unfair edges in the learned causal structure. In this way, fair decisions are made based on the corrected observations reconstructed from the learned graph structure. Furthermore, the proposed CGF framework can straightforwardly generalize the case where sensitive attributes are non-root nodes by introducing the latent variables to divide the fair and unfair effect flow. To the best of our knowledge, the proposed framework CGF the first work considering such non-root node case.
2 RELATED WORK
Most of the existing works of path-specific causal faimess restrict the unfair pathways by reducing their path-specific effect. In Nabi & Shpitser (0188; Nabi et al. (2019), the prediction accuracy and the path-specific effect along with unfair causal pathways are jointly minimized. The work proposed in Zhang et al. (017) designs a two-step algorithm, by first learning the graph structure and then minimizing the prediction error with PSE regularization. In Wu et al. (2019b), the authors adopt the -rpotts-function variable to bound the path-specific causal fairness. Instead of directly minimizing the path-specific effect, latent inference- inference-rrojection based method is proposed in Chiappa (2019) to correct the variables that are the descent nodes of sensitive attributes. In HHelwegen et al. (2020), the CEVAE framework Louizos et al. (2017) is adopted to infer the causal mechanism based on the pre-defined causal graph, and then the auxiliary prediction model is constructed based on the selected causal relation along with the fairness requirement.
Relations to Existing Work. Most of the above existing works require the prior knowledge about causal graph to caleulate the PSE or correct sensitive variables' descent variables, which is hard to be satisfied in real-world applications. Compared with the work in Zhang et al. (2017) that has a separated time-consuming causal structure learning step, our work applies the fairness constraint on based graph structure learning, which can efficiently obtain the causal graph and simplify the PSE calculation. Furthermore, is worth mentioning that all the above existing works assume that the sensitive attributes are root nodes. The proposed framework is the first work that generalizes to the case when sensitive attributes are non-root nodes under aath-specific causal fairness. Additionally, our proposed framework is motivated by the work of utilizing the causal graph discovery to enhance the machine learning generalization ability Kyono et al.) (2020). Compared with Kyono et al. (2020), the proposed CGF framework contains the cascade reconstruction step, which is the major difference. With the cascade reconstruction step, the unfairness contained in the original data can be corrected. Besides, CGF also has the faimess regularization in our proposed method, which reduces the unfair paths in the causal graph and meanwhile assures that the data correction follows the fair graph.
the continuous-optimizzai
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3 BACKGROUND
Causal Graph. A causal graph is a directed acyclic graph (DAG) reflecting the causal relationships between variables. Let G denote causal graph, and G = (V. E), where V is the set of nodes representing all the variables, and E causal relation between variable V2 and Vj. The parents nodes of node Vi. denoted as II(V)), and VG E [I( if J + Vi. A node is root node if it has no parent nodes. A path, also named as causal pathway, is defined depth of node in the graph is the number of arrows in the longest path to the root nodes. In the rest of the paper, we use the term "node", "variable", and "attribute" interchangeably.
set edges with each edge V, - V3 describing the
sequence of unique nodes with edges between each consecutive node. The
Path-specific Causal Fairness. Path-specific causal fairness ensures that the sensitive attributes are not allowed to affect the prediction along the unfair causal pathway. From the dcfinition, path-specific causal fairness distinguishes the causal pathways that start from sensitive variables to predicted variables into fair paths and unfair paths, and the goal of fair prediction is to reduce the unfair paths. Relations to Other Fairness Definition. Path-specific causal fairness closely related to other definitions of fairness. It is equivalent to removing the direct and indirect discrimination Zhang et al. (2017). When all paths starting from the sensitive variables are unfair, achieving path aath-specific causal fairness is equal to demographic parity (i.e., removing disparate impact) Zalar et al. (2015).
Structure Causal Model (SCM). In Structure Causal MSdel(SCM),, each node in 9 is associated with causal mechanism representing the relation between the current node and its parent nodes. It is defined as: F (fi Vi = (VV)) + 6 where V E V is the i-th node in the graph, (V,), is the set of parent nodes of V,, and E; is the random noise.
Definition 3.1. (Observed Graph). Observed graph is the causal graph of the observed data.
Definition 3.2. (Fair Graph). The causal graph satisfying the fairness criterion, and meanwhile, preserving the remuining structure of the observation graph, is the fair gruph.
Definition 3.3. (Model Graph). Model graph is the causal graph that the decision model relies on.
Figure shows the observational graph, fair graph and model graph of the loan example in Section Figure 2a is the observational graph, which is the causal graph of the observed data. In the graph, the fair path R - Q + Y represents that is acceptable, in terms of income, that some people with certain race have lower loan approval rate because they tend to be underpaid. While, the paths R + and R + Z + Y are unfair, indicating that it is disallowed that the race affects the loan approval directly or indirectly through ZipCode. Figure 2b the fair graph, which describes the ideal causal relations. Compared with the observational graph, it eliminates the unfair paths. By removing the unfair paths, the fair graph reflects that the difference of loan application results across different race groups is explained by the different income levels among those groups. The rightmost sub-figure the model graph, which is the graph that the model relies on predict. As shown in Figure 2c the model takes R, Z, and Q as input, therefore, they all have directed arrows pointing to prediction Y.
ZipCade
Income
ZipCode
Income
ZipCode
Income
R
Race
Loan
Race
Loan
Race
Loan
(a) Observed graph.
(b) Fair graph.
(c) Model graph.
Figure 2: Causal Graphs of Loan Example.
From the above triple-grapl perspective, under path-specific causal faireess, the model graph should be consistent with the fair graph, but is not. Therefore, our objective is to exclude the unfair path (the red dashed arrow in Figure 2ch for the decision, and retain the remaining causal pathways.
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4 METHODOLOGY
To satisfy the path-specific causal fairness, we propose a causal graph based fairness framework CGF, which imposes faimess at the graph structure level. The key of CGF framework is to make the causal graph and the data that the model relies on close to the ideal fair graph. In detail, the proposed framework contains three components: graph structure learning, fair regularization, and label prediction. The graph learning part aims to reveal the graph structure of the observation data, the fairness restriction targets at reducing the unfair paths, and the label prediction part outputs the fair predictions. These three components influence each other in that: the fairness restriction guides the graph learning by reducing the unfair edges, and in label prediction, the final prediction made based on the values of its parent nodes which is detected by the graph learning component. Overall, the objective function is:
L LGL + LF Lp,
where L.G.. is the graph learning loss, LF is the fairness restriction, I.p is the label prediction loss. In the following sections, we will first present the detailed implementation when the sensitive attributes are root nodes, provide the theoretical analysis about the generalization error, and then generalize the developed method to the case where sensitive attributes non-root nodes.
4.1 CGF FRAMEWORK
Under path-specific fairness criterion, when sensitive attributes are root nodes, such as age, gender race, we need to reduce the unfair edges. In the following three subsections, will present the implementations of the three components in Eqn. (T.
4.1.1 GRAPH STRUCTURE LEARNING
The objective of graph structure learning is to find the optimal causal graph that fits the observed data best. Motivated by the continuous optimization of causal graph structure learning Zheng et al. (2018). the loss of graph structure learning is:
LGL = B D1B2 + m (tr(eWow) - (da + dx + 1))² + 7/2 72llW
where W E exists an edge V, V; with weight wi, indicating the effect strength. da is the dimension of sensitive attributes, and dx is the dimension of other features. D is the observed data, and D is the reconstructed data based on W and D according to the causal graph. @ is the element-wise matrix multiplication operator, WoW denotes the matrix exponential of W @ W,tr tr(.) is the matrix trace. 3, 71, and 722 are the syper-parameters.
is the adjacency matrix, and its element way * 0, there
The first term in Eqn. 2 measures the fitness of the causal graph by calculating the difference between the observed data and the data reconstructed from the graph. The second term is the directed acyelic graph (DAG) constraint, which ensures the learned graph does not contain cycle Zheng et al.) (2018). The third term is the & norm of the adjacency matrix which makes the learned graph to be sparse. The details of data reconstruction (i.e., D) and the DAG constraint are described as follows. Cascade Data Reconstruction. In data reconstruction, each node is reconstructed based its parents' reconstructed value. Eqn. (3) shows the reconstruction of a single node Vi:
V f((((VV [in. il),
(3)
where fi(+) is the causal mechanism of node V., ii((V)) is V'ss parent nodes after reconstruction. in is the index set of Vi's parent nodes. W a] is the elements in the adjacency matrix W whose row indices are in én and column indices are i. It is noticed that the reconstruction of node is based on its parents' reconstructed values, instead of the observed values, because the observed values contain unfair effect if the parent nodes locate in the unfair paths. To satisfy this, the parent nodes should be reconstructed before the child nodes, thereby the data reconstruction follows a cascade reconstruction procedure with ascending order of depth.
We use a graph with five nodes, shown in Figure 3 to illustrate the cascade data recon- struction. In the figure, A denotes the sensitive attribute, X2, X3, X, are regular features, and Y is the class label. The red arrows denote the unfair edges. The reconstruction or- der decided by the depth is: & Xs. X2. Xa. Y and the reconstruction procedure is:
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W4.5
Root Nodes:
A = A; X4 = X4;
X
Depth node:
X2 W1,2A w4,2X4 b2;
Depth node:
X3 = w1,3A + + 32 + b3i
W4.2
X
W2.s
Ws.5
Depth node: Y =1, A W1,3 + 23,3 + by.
Wy.
(4)
W1.3
where Wh is the element of the i-th row and the j-th column adjacency matrix W, by is the intercept term, and = [A, X2, X3, X1, V. In this example, we adopt the linear function as the causal mechanism, and can generalize to the more complex functions, such as neural network by changing fi(-) Eqn. (3)
W1.s
Figure 3: Cascade Data Reconstruction Example.
DAG Constraint. The second term in Eqn. (2) is the directed acyclic graph (DAG) constraint, which ensures that there is cycle in the learned graph Zheng al. (2018), node cannot affect itself. The trace of adjacency matrix's 's exponential is adopted in the second term to measure the
graph acyclic. The matrix exponential is given by the power series: ewow = kI (W O W)*
where the k-th term denotes the adjacency after k times walking the graph, and W @ W makes element of adjacency matrix non-negative. The trace the k-th term (k > 0) should be zero if the graph is acyclic, because node cannot go back to itself after times walking. Therefore (oW) = (dA + dx + 1) indicates that the graph is acyclic, where da + dx + is the trace of first term in the power series.
4.1.2 FAIRNESS REGULARIZATION
The goal of fairness regularization is to reduce the unfair edges, that the sensitive attributes pass less effect through the unfair path. As mentioned previously, the element in the adjacency matrix not only represents the causal direction, but also indicates the effect strength along this edge. To reduce the unfair edges, the elements in the adjacency matrix associate with unfair edges should be close to zero. Therefore, we apply the following fairness regularization on the adjacency matrix:
Lp allW Melli.
(5)
where 3 is the element-wise matrix multiplication, W the adjacency matrix, and My is the fairness mask with the same dimension W. The element of the j-th row, i-th column of Mp is set as 1 edge V; + V, is unfair. More details related the construction of the fairness mask are in the appendix. -1- |li is the & norm. The fairness regularization LF minimizes the total strength of effect on the unfair edges, which reduces the effect flow along the unfair paths. By regularizing on the adjacency matrix, faimness regularization is able to climinate the unfair path A - Y and A X2 in Figure Therefore, with the faimness regularization, the reconstructed data D is correction of the original data D with unfair effect reduced.
4.1.3 LABEI. PREDICTION
Since the unfair effect has been reduced in the reconstruction data, the prediction based on the obtained reconstruction is fair. The label Y is predicted as: Y DWy, where Wy is the last column of W, which indicates the existence of the edges and their effect strength starting from other nodes to Y and D is the reconstructed data. Accordingly, the prediction loss is: Lp =|IY YIB.
4.1.4 GENERALIZATION NONLINBAR CAUSAL MECHANISM.
In Eqn. (4 and label prediction, the adopted linear causal mechanism can generalize to more advanced functions by modify the cascade data reconstruction and the adjacency matrix. Assume choose neural network (NN) the causal mechanism, the reconstruction of node V, is: V. J:N (DW,NN), where J.N (-) represents the neural network, WNN (dd (olx+dxxxxol xdNN is the parameter of the first linear layer in f v is the dimension of the first hidden layer. Namely, to use the NN mechanism, replace the linear model in Eqn. 4 with the NN whose first layer is the linear layer and all those
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NNs share the common hidden layers, suggested in Kyono al. (2020). The adjacency matrix is constructed based on the parameter in the first linear layer. Specifically, each element in the adjacency matrix W is calculated as: W I| [N :1122 where W [i is the j-th row in W W.w...
4.2 THEORETICAL ANALYSIS
Here we provide the theoretical analysis about the generalization error of the proposed framework. Theorem 4.1. Suppose the the data D follow the Gaussian distribution, the generalization error of the proposed fair classifier hp the observed dataset, which is denoted as hng satisfies the following inequality with probability - 6. 48 0:
1ath
Dath Cpp
<IDD- [ID-D11froo + VV 220 3
+ C1
C2
+ Css
m
Vin
Vn
(6)
+42D" + m RRday + C.(Ri, + RF) + log( (3)) Css
where Dob is the distribution of the observed data and D is its observed sample. H is the reconstructed data by cascade reconstruction, and its distrubution is denoted DF ch So En(a, x)p (a, r.ddad, is the expected error on the underlying space D, and la (a, Sy L(Y, h((a, )))pp((((())) (app((ooo) x)dY is the expected error on a single point (a,x)- 2he is the empiri- cal error of classifier HF on DF Rdog and Rí, the value of DAG constrain, E regularization, which are the last two terms in Eqn. (2. Ry is the value of fairness regularization defined in Equ. (
T, C Cs are constanis.
Theorem 4.1 give an upper bound of the generalization error of the fair classifier on the observed
dataset. The upper bound shows that the generalization error is related to the qualities of the reconstruction and the classifier trained on the fair dataset, which is exactly the two terms in our objective function. The reconstruction part in Thm 4.1 also represents the fairness level, since the fairer the data is, the fairer the dataset is, the smaller the reconstruction error
4.3 GENERALIZATION TO NON-RoOT NODE CASE
Most of the existing works consider sensitive attributes such age, gen- der, and race that can only the root nodes in the causal graph. How- ever, in some real-world applications, sensitive attributes affected by other variables. For example, in the recommendation system, the item popularity should not affect whether this item to be recommended Ge et al. (2021); Zhu et al.] (2018), for the purpose of recommendation diversity. Figure 4 shows causal graph of the recommendation example where O and represent user and item respectively, P denotes the item popularity and Y denotes whether the user click the item. In this example, the item popularity P is the sensitive attribute, and it is the non-root node as it is affected by the item's 's characteristics.
Figure 4:
Sensitive At- tributes as Non-root Nodes.
Challenge. When the sensitive attributes A are non-root nodes, their parent nodes II(AA) also contain the information about those sensitive nodes. If the parent nodes have other causal pathways to the label node Y not passing the sensitive node, the information related to the sensitive attributes can still reach the label node through those paths. Therefore, the proposed framework in Section 4.1 requires slight modification handle this case.
Effect Diversion. To tackle the above challenge, we add two latent nodes between the parent nodes and the sensitive nodes to divert the effect flow, that the fairness regularization proposed in the previous section can be applied to the unfair flow. One latent node, denoted as Z., controls the effect from sensitive attributes' parent nodes to label node passing through sensitive attributes. The other latent node Zy controls the effect from parent node to label node not passing through sensitive attributes. In other words, the paths from parent nodes I1(AA) to label node Y are divided into two categories, one only contains latent node Zy. and the other only contains Z,. By disentangling Z, and Z,, the faimess regularization now can be directly applied to the paths containing Zg, as no other paths face information leakage risk. When there are multiple non-root sensitive attributes, the above procedure is applied to each one of them.

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Figure5 shows the causal graph with effect diversion in the recom- mendation example, where the latent node Zu is user embedding, Z3 is the item popularity related embedding and Zy is the clicking variable Y related embedding. The parent node affects item popu- larity only through Zs and other paths from to Y all pass through Zy. The red arrows indicate the unfair paths where the fairness regularization is applied on.
Z
Z.
Objective Function. Eqn. 22. the reconstruction part requires the variables' observed values, while in this case latent nodes lack that. To address this issue, we notice that to the graph with effect diversion, Zy and Z, should have less overlapped information. regularizations on Zy and Z. are adopted in the graph structure learning part to ensure the separation of the effect flow. Overall, the objective function
Figure 5: Causal Graph with Effect Diversion.
To fulfill this, the orthogonal
I
L = =|||Y - PI BIID D1E B2 E cos ( =
+7i (tr(eWWW wow (dA + dx 1)) + 72 [ allW o Mellis
where Z is the number of total items. cos denotes cosine similarity, which ensures the orthog- onality between - and -,(², and other types of correlation measure such as HSIC (2007) can be adopted. Compared with Eqn. (I). the graph structure learning part is modified to satisfy the effect diversion design. In the recommendation system example, ||Y Y is the clicking prediction error, and || D112 is the error of predicting the item popularity.
al.
EXPERIMENT
We experiment on both synthetic and real-world datasets confirm: (1) The proposed CGF framework works on both cases where sensitive attributes are root or non-root nodes. (2) Our proposed framework provides a better trade-off between utility and fairness. More details regarding CGF implementations, experiment settings and additional experiment on synthetic dataset are listed in the Appendix.
5.1 EXPERIMENT ON ADULT DATASET
Dataset. Adult datase [ is commonly adopted dataset for faimess evaluation. In this dataset, there are total 48842 individuals and each has 14 attributes regarding their demographic information, jobs, and level of education. The class label is binary indicating whether an individual's 's income is above or below 50k. The objective to predict the income class given an individual's attributes.
Unfair Paths. As suggest in Nabi & Shpitser (2018), the direct path "Gender" + "Income", and the all unfair. Namely, the gender should not direct affect the income and meanwhile it is not allowed to affect income through marital status.
paths containing edge "Gender" + "Married"
Baselines. The logistic regression model (LR) and neural network (NN) constructed from raw data is adopted as the baseline. We also adopt the Fair Inference (F1O) Nabi & Shpitser (2018) and PSE-DR Zhang et al. (017)) as the baselines. Our proposed models are denoted as LR-CGF and NN-CGF, which take the linear logistic regression model and neural network as the causal mechanism function, respectively.
Evaluation Metrics. Due to label imbalance, Area Under the ROC Curve (AUC) is adopted as the utility metric. The higher the AUC value, the better the utility. Following Nabi & Shpitser (20T8), adopt the path-specific causal effect (PSE) Pearl (2001); Shpitser (2013) to measure the faimess.. The PSE value may have a negative value indicating the negative effect. The closer the PSE value is the fairer the model is.
5.1.1 RESULT ANALYSIS
Table I summarizes the results of different methods on Adult dataset with 5-fold cross validation. It observed that compared with baseline methods, our proposed methods are fairer and meanwhile
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have better utility. We also notice that the method with nonlinear causal mechanism performs best in terms of both utility and fairness. The reason is that compared with linear causal mechanism, the neural network reconstruct the data better while ensuring fairness. This observation also confirms Theorem 4.1 that the better the reconstruction is, the better the performance is.
We further experimentally explore the rela- tionship between the reconstruction and fair- ness regularization. We fix one part's hyper- parameter and tune the other one. Figure 6 reports the results of NN-CGF. From Fig- ure [6a) and [bb] it is observed that, with the increasing strength of reconstruction, im- proves the accuracy but reduces the fairness. The fairness regularization has an opposite effect with reconstruction part. As shown in Figure 6c and 6d the fairness regularization improves the model fairness but reduces utility. Overall, the reconstruction part and the fairness regularization, together, control the trade-off between model utility and faimesss
Method
AUC (11)
PSE (= 0)
LR
0.712 = 0.005
3.508 + 0.005
NN
0.721 + 0.012
2.068 + 0.223
FIO
0.505 = 0.007
1.048 0.003
PSE-DR
0.686 = 0.018
0.450 + 0.151
LR-CGF
0.507 = 0.099
0.925 + 0.073
NN-CGF
0.689 = 0.012
0.198 0.109
Table 1: Results on Adult Dataset. ft: the higher the better, and 0: the closer to 0, the better.
0.54
0.70
00664
2-03
-.3
0.64
3..00
0.62
0.6
(a) AUC (1)) V.S.
(b) PSE (= 0) V.s. 3.
(c) AUC (1) V.S. a
(d) PSE (= 0) v.s. a.
Figure 6: Effects of Reconstruction and Fairness Regularization.
5.2 EXPERIMENT ON RECOMMENDATION DATASET
Dataset. The MovieLens dataset Harper & Konstan (2155) adopted validate the performance of CGF. Following the settings in Ge et al. (2021), the sensitive attribute item popularity is added to each item, and for each item, the value of item popularity The item popularity is non-root node since it is affected by item characteristics. For each user, sort their interactions according to the timestamp, and the last interaction is put into the test set, and others are in the training set. The validation set is the last interaction of each user in the training set. Baselines. We adopt Matrix Factorization (MF) Koren et al. (2009), Generalized Matrix Factorization (GMF) and Multiple Layer Perceptron (MLP) He et al. (2017)
if its total exposure is top 20%, otherwise
baselines. GMF and MLP are also the base models of our proposed framework, and we name our methods as GMF-CGF and MLP-CGF accordingly.
Evaluation Metrics. In terms of utility/accuracy measure, Top-k ranking metrics hit rate (HR) and normalized discounted cumulative gain (NDCG) are adopted to measure the recommendation performance. Following Ge al. (202T), , the Gini Index and Popularity Rate (PR) are also adopted to measure the fairness. The details of Gini Index and Popularity Rate (PR) are in the Appendix. For HR and NDCG, the higher the value is, the better the performance is. For Gini Index and PR, the lower the value is, the fairer the model is.
5.2.1 RESULTS ANALYSIS
Table 2 shows the results of baselines and our proposed methods with Top 10 rankings metrics. We also list the results of our methods' variants. +-CGF w/o rec denotes the CGF without the orthogonal regularization part, i.e., 32 = 0 in Eqn. 7 =-CGF w/o fairness and denotes CGF without the fairness regularization part (a: 0). +-CGF w/o ord fairness CGF without both of these two parts.
In terms of fairness, our proposed methods recommend more diverse items, and meanwhile, have the comparable recommendation accuracy to the baselines. This observation verifies that our proposed method makes the base model to be fair without scarifying too much utility. is worth to mention that the results measured by GINI and PR are also the indirect indicator of how good the disentanglement of the effect from sensitive attributes' parent nodes label node. The better it disentangles, the fairer the model. Furthermore, the ablation results shown the table aableeinde indicate that the orthogonal regularization and the fairness regularization both contribute the model fairness.
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Method
HR (1t)
NDCG (1)
GINT (U)
PR (V)
MF
0.197
0.103
0.878
0.861
GMF
0.195
0.098
0,875
0,857
MLP
0.149
0.077
0.919
0.925
GMF-CGF (ours)
0.166
0.084
0.837
0.773
GMF-CGF w/o ord
0.174
0.089
0.853
0.820
GMF-CGF w/o faimesss
0.166
0.089
0.857
0.808
GMF-CGF w/o ord & fairness
0.195
0.102
0.881
0.860
MLP-CGF (ours)
0.116
0.059
0.882
0.844
MLP-CGF w/o ord
0.112
0.054
0.902
0,891
MLP-CGF w/o fairness
0.141
0.066
0.923
0.932
MLP-CGF w/o ord faimess
0.139
0.066
0.903
0.863
Table 2:
Results on the Movielens Dataset. | indicates the lower, the better.
To further analyze the effect of orthogonal regularization and fairness regularization. in Figure plot the four metrics with respect to different regularization strengths by tuning one hyper- hyper-parameter and fixing the others. From this figure, is, the fairer the model is, and the more utility is sacrificed. Furthermore, The utility and fairness trade-off can be controlled by tuning the values of two regularizations' hyper-parameters. We also notice that MLP-CGF performs slightly different in terms of HR and NDCG: The stronger the orthogonal regularization and the fairness regularization, the better the performance. The reason is that compared with GMF-CGG CGF, MLP-CGR has more learnable parameters in the neural network, and adding those regularizations would prevent the over-fitting.
observe that the stronger the regularization strength
Mix m
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NDCB
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0.85
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4:
8.
(a) HR (1t) v.S. 0g-
(b) NDCG (1t) v.s. 82.
(c) Gini Index (1) v.s. Bg.
(d) PR (U) v.s. 83.
0.95
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6 0.85
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(c) HR (f))
(f) NDCG (1)) v.s.
(g) Gini Index (1) v.s.
(h) PR (())
Figure Effects of the Orthogonal Regularization and Fairness Regularization.
CONCLUSIONS AND FUTURE WORK
In this work, propose novel causal graph based fair prediction framework under path-specific causal fairness. The core of the proposed framework is to ensure that the graph adopted by the prediction model should be close to the fair graph. To fulfill this, we integrate the graph structure learning and the fairness regularization in an interactive way. The learned graph structure reveals the causal graph of the original observations with unfair edges eliminated, and the data reconstructed from the learned graph is close to the original observations with unfair effect corrected. Based on the corrected causal graph and its associated data, the prediction model achieves the path-specific causal fairness. Experimental results on the real-world dataset confirm that the proposed framework ensures fair predictions and meanwhile retains the comparable utility. We also generalize the proposed framework to the case of sensitive attributes being non-root nodes by effect redividing, which is further validated by experiments on a real-world recommendation dataset.
In this paper, assume that there are no latent confounders in the dataset. When this assumption is not satisfied, the causal graph may not be identified from the observation data. Recently, some causal discovery works that target recover the causal graph in the presence of latent confounders Xie et al. (2020); Cai et al. (2019) have been developed. Relaxing the no latent confounders assumption and generalizing work to the latent confounders case will be the future work.

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Under review as a conference paper at ICLR 2022
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12


Reviews:
[{"review_id": "o-iGvblm274", "reviewer": "Reviewer_ZYeg", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "This is an interesting paper which tackles an important problem. I agree that the approach is interesting. But I feel the technical novelty is limited. I noticed that the authors used L_{GL} from Zhang et al. 2018 in Eq. (2) directly and added fairness aware loss L_F and label prediction loss L_P.  These losses are nothing new (I understand the fairness aware loss is blocking some path, but there is no technical novelty in that aspect).\n\nI found the generalisation error analysis is a bit decorative. There is no comment of a wide variety of parameters like $\\mathcal{R}_{\\cdot}$. How do they look like? What are the typical values of these parameters? While for examples in euclidean space, these generalisation bounds are easy to understand, for learning with graphs, they are difficult and the authors should make a great effort to elaborate on this.      \n\nI have several reservations about the loss function L_{GL} *in the current context* :  First, why do you need to enforce sparsity constraint? Why is it being assumed that the underlying graph is sparse? Such constraints can block some necessary fair paths, right? Second, sparsity constraint in terms of L_1 regularizer can work well for large graphs.  I believe the number of the nodes in the graph is very small-- right? To my understanding, $|V|=14 $ for Audit dataset. Hence, I am not sure why do we need such constraint.  \nIn contrast, what I would prefer is to have a dissimilarity regularizer between causal graph and the reconstructed graph in terms of Kernel measures for example. In fact, I am not sure how the euclidean distance $||D-\\bar{D}||$ will enforce causality in reconstructed graph--- this is the observed data from the graph but not the graph itself and therefore, I am not sure it will be free from confounding effect.\n\nFinally, I do think that the experiments should be improved. I did not found any ablation study on different loss components. Maybe I have overlooked?\n\nThe statement \"Due to label imbalance, Area Under the ROC Curve (AUC) is adopted as the utility metric.\" It is misleading. For label imbalance AUC is rather high and over optimistic. For example, consider a ranked list of 1000 items with 100 positive and 900 negative items. Now if the first positive item appears in 100th place, AUC can be as high 0.88 which is extremely high for an algorithm which assigns the first positive item in 100th position. AUC often helps in training with the pairwise surrogate ranking loss. But for evaluation with imbalance data, average precision on positive and negative items are preferred.\n\nTypo: Page 5, second last line: \"linear layer in f_i..\":  Please correct the typo on f_i\n", "summary_of_the_paper": "This paper introduces fair prediction framework which combines graph structure learning into fair prediction to ensure  that unfair pathways are discouraged in the causal graph. It  shows generalization bound and proves the efficacy of the underlying method using several experiments.", "summary_of_the_review": "Interesting direction, but lack of novelty. Still lot of work is needed to get a good score.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "ECCpaE8X2HD", "reviewer": "Reviewer_V6if", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "I have several concerns in terms of its technical contributions, clarity and experiments. Please see them as follows.\n\nConcerns\n- The technical contributions seem a bit limited. This work is largely relied on the existing work by Kyono et al., including a majority part of the loss function as well as a majority part of the generalization error analysis. The major contribution lies in the cascade data reconstruction.\n\n- Since I am not expert in causality, please clarify if this concern makes sense. If we have a large causal graph, let's say n nodes in the causal graph, then we need n different causal mechanisms. I am afraid it won't scale up, especially when we are using a NN as a causal mechanism. \n\n- In section 4.1.4, there is a typo regarding the first linear layer. Is there any intuition of setting w_ij as the squared l2 norm? And why are we only using the parameter in the first linear layer?\n\n- I have several questions in terms of the experiment: (1) Why cannot we have a method for MF (like MF-CGF)? (2) Since you used GMF and MLP in He et al., why is its final model NeuMF not considered as a baseline? (3) I think it is a bit overclaim about \"This observation verifies that our proposed\nmethod makes the base model to be fair without scarifying too much utility.\" because MLP-CGF has a much worse utility compared with MLP.\n\n- In appendix about adjacency matrix construction, it is unclear why the entry in adjacency matrix is calculated in this way. Please explain the motivation or intuition here. ", "summary_of_the_paper": "This paper aims to improve path-specific causal fairness by removing unfair causal pathways. Basically, the framework leverages the existing work by Kyono et al. and adds additional regularization terms. A big advantage of the proposed method is that it can handle the case where sensitive attribute is not root node. Based on experimental results, it seems CGF achieves a good trade-off in fairness and quality.", "summary_of_the_review": "I think this work largely relies on an existing work, which limits its technical contribution. The paper clarity can be further improved by giving more intuitions or explanations to some tricks (or definitions maybe). As such, I think this paper could be much further improved.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "5XLVxazh4ij", "reviewer": "Reviewer_nUyn", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Strength:\n\n1. This paper targets challenging problems in causal fair machine learning: the causal graph is required, and the computation of PSE is complex.\n2. The idea of combining causal structure learning and causal fairness regularization is novel.\n3. The experimental results show the effectiveness of the proposed method, compared with the baseline approaches.\n\nWeakness:\n\n1. Some summaries of the existing research are inaccurate. The ignorability assumption or identification requirement is not necessary. [1] provides a bounded method for unidentification cases. The sensitive attributes are not required as the root nodes. Path-specific effects are generic as defined by Avin et al. 2005. Zhang and Bareinboim 2018b define spurious discrimination which covers the effect from the sensitive attribute to the decision via a confounder.\n\n2. The definition of the Structure causal model: the function is not necessarily an additive-noise style.\n\n3. Eq (5): there is a gap between fair edges and fair paths, further small path-specific effects. Intuitively, the small elements in an adjacent matrix lead to small effects but it is not necessary. In addition, the mask construction described in the Appendix is not clear. The path-specific effect is defined via paths. But the mask matrix is edge-based. There is a gap between the path and the edge-based definitions. For example, the mask considers paths from R to Y are fair iff the first edge is R->Q along the paths. It considers R->Q->Y or R->Q->O->Y as fair but R->O->R->Y as unfair, which is not convincing.\n\n4. Ch. 4.3: the spurious effect from P<-I->Y is not unfair based on the definition of path-specific fairness as changing P doesn't affect Y. In addition, if there is no causal graph, it is unclear how to distinguish the root-node cases and non-root-node cases. In the experiments, the root/non-root cases are separated based on domain knowledge but causal discovery is able to find confounders beyond the domain knowledge. \n\n5. It is not clear how the proposed method addresses the challenge of unidentification.\n\n6. Fig 6 shows alpha is tuned based on the PSE value. It is expected that the path-specific effect is not used as CGF is proposed to address the challenge of complex computation of PSE and simply the PSE calculation. \n\n[1]L. Zhang, Y. Wu, and X. Wu, “Causal modeling-based discrimination discovery and removal: Criteria, bounds, and algorithms,” IEEE Trans. Knowl. Data Eng, 2019, doi: 10.1109/TKDE.2018.2872988.\n \n", "summary_of_the_paper": "This paper proposes a framework to integrate graph structure learning and path-specific fairness constraints into prediction tasks. The loss function consists of three parts: the graph structure learning loss through cascade data reconstruction, the causal fairness regularization through an adjacency matrix, and the label prediction.", "summary_of_the_review": "The idea of combining structure learning and path-specific effect regularization is very interesting. But the gaps/incorrectness concerns me most.\n\nBased on my understanding, the three challenges are not well addressed:\n\n1. The requirement on causal graphs: The proposed framework does not require the causal graph but it cannot correctly define the path-specific effect or readily recognize the root/non-root cases. The proposed approach faces the challenges of unidentidication.\n\n2. The complex calculation of PSE: PSE is calculated for parameter tuning.\n\n3. The root assumption is not necessary for PSE. In addition, the proposed method requires domain knowledge to recognize the root/non-root case.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}]

Author rebuttal:

Meta-review: In this paper, the authors aim to work within the path-specific framework to implement fair predictions by learning a causal graph in such a way that some path-specific effect is removed.

Generally, the paper was not received very well by reviewers, with the primary concern being lack of novelty in particular in comparison with (Kyono et al.)

One additional comment I wanted to make is this: any prediction task that is a part of a pipeline that contains a graphical model selection step is properly a "post-selection inference problem."  Such problems are very challenging because:

(a) Learning a graph from data is known to lack consistency at any rate (meaning that the algorithm is only pointwise consistent, but not uniformly consistent).  This issue propagates to "downstream" tasks in the pipeline, including prediction problems.  Probably, the way this issue would manifest in this work is unless sample sizes were very large, there would be no particular reason to assume the correct causal path is removed.

(b) Even if uniformly consistent modifications of structure learning algorithms were used, the uncertainty in learning the graph with error must be propagated to all subsequent steps in the pipeline.  Doing so appropriately is very challenging.

When revising the paper, in addition to taking reviewer comments into account, please consider how your method deals with post-selection inference issues -- I think this is a very interesting but challenging question that is likely to come in peer review.
Decision: Reject