Path-specific Causal Fair Prediction via Auxiliary Graph Structure Learning

1 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 et al . ( 2021 ) ; Zhu et al . ( 2018 ) ; Burke ( 2017 ) ; Yao & Huang ( 2017 ) , natural language processing Bolukbasi et al . ( 2016 ) ; Zhao et al . ( 2017 ) ; Gonen & Goldberg ( 2019 ) ; De-Arteaga et al . ( 2019 ) ; Blodgett et al . ( 2020 ) , computer vision Shankar et al . ( 2017 ) ; Nagpal et al . ( 2019 ) ; Raji et al . ( 2020 ) ; Stock & Cisse ( 2018 ) , hiring Hoffman et al . ( 2018 ) , education Brunori et al . ( 2012 ) , banking Mukerjee et al . ( 2002 ) , and crime risk assessment Brennan et al . ( 2009 ) ; Dieterich et al . ( 2016 ) ; Zhang & Bareinboim ( 2018b ) . Algorithm fairness aims to reduce or even eliminate unjustified distinctions of individuals based on their sensitive attributes ( 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 a 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 , pathspecific causal fairness Chiappa ( 2019 ) ; 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 if it 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 . ( 2017 ) shown in Figure 1 , race ( a sensitive attribute R ) is 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 R → Z → Y are unfair paths and path R → Q → Y is a fair path . R Y Q Race Loan Z ZipCode Income unfair unfair fair Figure 1 : Loan Application Example . To fulfill path-specific causal fairness , some existing works directly calculate the path-specific causal effect ( PSE ) Avin et al . ( 2005 ) ; Pearl ( 2001 ) along the unfair pathways , and minimize the effect simultaneously when maximizing prediction accuracy Wu et al . ( 2019b ) ; Nabi & Shpitser ( 2018 ) . Some other works correct the variables located on the unfair pathway by a latent inference-projection method Chiappa ( 2019 ) . However , 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 . ( 2021 ) ; Zhu et al . ( 2018 ) , while this variable is a non-root node as it is affected by the item ’ s characteristics . In light of the above challenges , we propose a Causal Graph based Fairness 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 to 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 is the first work considering such non-root node case . 2 RELATED WORK . Most of the existing works of path-specific causal fairness restrict the unfair pathways by reducing their path-specific effect . In Nabi & Shpitser ( 2018 ) ; 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 . ( 2017 ) 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 response-function variable to bound the path-specific causal fairness . Instead of directly minimizing the path-specific effect , a latent inference-projection based method is proposed in Chiappa ( 2019 ) to correct the variables that are the descent nodes of sensitive attributes . In Helwegen 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 calculate the PSE or to 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 the continuous-optimization based graph structure learning , which can efficiently obtain the causal graph and simplify the PSE calculation . Furthermore , it 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 path-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 fairness 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 . 3 BACKGROUND . Causal Graph . A causal graph is a directed acyclic graph ( DAG ) reflecting the causal relationships between variables . Let G denote a causal graph , and G = 〈V , E〉 , where V is the set of nodes representing all the variables , and E is the set of edges with each edge Vi → Vj describing the causal relation between variable Vi and Vj . The parents nodes of node Vi , denoted as Π ( Vi ) , and Vj ∈ Π ( Vi ) if Vj → Vi . A node is a root node if it has no parent nodes . A path , also named as causal pathway , is defined as a sequence of unique nodes with edges between each consecutive node . The depth of a 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 . 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 definition , 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 is 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-specific causal fairness is equal to demographic parity ( i.e. , removing disparate impact ) Zafar et al . ( 2015 ) . Structure Causal Model ( SCM ) . In Structure Causal Model ( SCM ) , each node in G is associated with a causal mechanism representing the relation between the current node and its parent nodes . It is defined as : F = { fi : Vi = fi ( Π ( Vi ) ) + i } , where Vi ∈ V is the i-th node in the graph , Π ( Vi ) is the set of parent nodes of Vi , and i 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 remaining structure of the observation graph , is the fair graph . Definition 3.3 . ( Model Graph ) . Model graph is the causal graph that the decision model relies on . Figure 2 shows the observational graph , fair graph and model graph of the loan example in Section 1 . 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 it is acceptable , in terms of income , that some people with certain race have a lower loan approval rate because they tend to be underpaid . While , the paths R → Y 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 is 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 is the model graph , which is the graph that the model relies on to 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 . ZipCode Income ZipCode Income ZipCode Income From the above triple-graph perspective , under path-specific causal fairness , the model graph should be consistent with the fair graph , but it is not . Therefore , our objective is to exclude the unfair path ( the red dashed arrow in Figure 2c ) for the decision , and retain the remaining causal pathways .

Summary: 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.