Fair Representation Learning through Implicit Path Alignment

1 INTRODUCTION . Machine learning has been widely adopted in the real world decision-making practice such as job candidate screening ( Raghavan et al. , 2020 ) and credit application . However , it has been observed that learning algorithms treated some groups of population unfavorably , for example , denying credit on the grounds of gender , age or ethnicity ( Hardt et al. , 2016 ) . To this end , algorithmic fairness that is to mitigate the prediction bias for different subgroups has recently received tremendous attentions . With the rapid advancement of representation learning ( LeCun et al. , 2015 ) , learning a fair embedding ( Zemel et al. , 2013 ) has been recently highlighted . Specifically , the learned fair representation can easily transfer the unbiased prior knowledge to the downstream tasks , with various successful applications in computer vision ( Kim et al. , 2019 ; Kehrenberg et al. , 2020 ) , language understanding ( Chang et al. , 2019 ; Ethayarajh , 2020 ) and artificial intelligence for health ( Fletcher et al. , 2021 ) . Typically , the fair representation learning is achieved by adding various statistical fair metrics during the training process . Based on this , most existing fair representation approaches in classification or regression principally aim to meet the independence or separation rule , e.g. , ( Madras et al. , 2018 ; Song et al. , 2019 ; Chzhen et al. , 2020 ) . However , in various realworld scenarios , the sufficiency rule is preferable . For example , health systems rely on commercial algorithms to identify and help patients with complex health needs . The algorithm outputs a healthcare need score , where a higher score indicates the patient is sicker and requires more healthcare . Obermeyer et al . ( 2019 ) revealed that a widely used algorithm , typical of this industry-wide approach and affecting millions of patients , exhibits significant racial bias . At a given predicted healthcare need score Ŷ = t , Black patients are considerably sicker than White patients ( Eblack [ Y |Ŷ = t ] > Ewhite [ Y |Ŷ = t ] ) . Obermeyer et al . ( 2019 ) also pointed out that remedying the disparity would increase the percentage of Black patients receiv- ing additional healthcare from 17.7 to 46.5 % . Moreover , it has been theoretically justified ( Barocas et al. , 2019 ) that the Sufficiency rule is generally not compatible with Independence and Separation . Thus learning the fair representation w.r.t . the sufficiency rule is promising in both the algorithmic design and real-world applications . In this paper , we address the sufficiency rule by considering the following intuition : given a fixed representation function , if the optimal predictor that learned on the embedding space are invariant from different sub-groups , then the corresponding representation function is fair . Fig . 1 provides an illustrative example . when the representation function λ : X → Z is unfair and we adopt gradient descent to learn the predictor h : Z → R. The optimal predictors of different subgroups ( blue , red ) are not invariant , resulting in biased predictions . We will later demonstrate such an intuition ensures the learned representation satisfying the sufficiency rule ( Liu et al. , 2019 ; Chouldechova , 2017 ) . The aforementioned intuition can be naturally formulated as a bi-level optimization problem , where we aim to adjust the representation λ ( in the outer-level ) to satisfy the invariant optimal predictor h ( in the inner-level ) . Thus , when we adopt the gradient-based approach in solving the bi-level objective , a straightforward solution is to learn the representation λ to fulfill the identical explicit gradient-descent directions in learning predictor h ? of different groups , shown in Fig . 2 ( a ) . Intuitively , if the inner gradient descent step of each sub-group is identical , their final predictors ( as the approximation of h ? ) will be invariant . However , the corresponding algorithmic realization is challenging in deep learning : 1 ) It requires storing the whole gradient steps , which induces a high memory burden . 2 ) the embedding function λ is optimized via backpropagation from the whole gradient optimization path , which induces a high computational complexity . To this end , we propose an implicit path alignment , shown in Fig . 2 ( b ) . Notably , we only consider the final ( t-th ) update of the predictor h ( t ) , then we update representation function λ by approximating its gradient at point h ( t ) through the implicit function ( Bengio , 2000 ) . By using the gradient approximation , it is no more required to store the whole gradient step and conduct the backpropagation through the entire path . Overall , the highlights in this paper are as follows : Fair-representation learning to satisfy the sufficiency rule Instead of enforcing the independence or separation rule , the considered fair-representation criteria is proved to satisfy the sufficiency rule in both classification and regression . We also find such a criteria is intrinsically consistent with the recent Invariant Risk Minimization ( IRM ) ( Arjovsky et al. , 2019 ; Bühlmann , 2020 ) , which aims to eliminate suspicious correlations while keeping robust correlations that are invariant across different environments . Intuitively , reducing the correlation w.r.t . the protected attributes enables the fair representation . Principled and efficient algorithm We proposed a novel implicit path alignment algorithm to learn the fair representation , which addressed the prohibitive memory and computational cost in the original bi-level objective . Besides , we analyzed the approximation error gap of the proposed implicit algorithm , which induces a trade-off between the correct gradient estimation and fairness measures . Improved fairness in classification and regression We evaluated the implicit algorithm in both classification and regression with tabular , computer vision and NLP datasets . Compared to the baselines , the implicit algorithm effectively improved the fairness with a smaller sufficiency gap . 2 PRELIMINARIES . We suppose the input X ∈ X , the ground truth label Y ∈ Y , and the algorithmic output Ŷ ∈ Y . Throughout the paper , we only consider binary sensitive attribute ( i.e , two sub-groups ) with distributions D0 and D1 . Then based on ( Liu et al. , 2019 ) , the sufficiency rule is defined as : ED0 [ Y |Ŷ = t ] = ED1 [ Y |Ŷ = t ] , ∀t ∈ Y ( 1 ) To measure the fairness w.r.t . the sufficiency rule , we propose the sufficiency gap as the metric . Since we aim to evaluate the fairness in both binary classification ( Y ∈ { −1 , 1 } ) and regression ( Y ∈ R ) , the metric is separately defined on these two scenarios . Sufficiency gap in binary classification Based on the sufficiency rule , the sufficiency gap in binary classification is naturally defined as : ∆SufC = ∑ y∈ { −1,1 } |D0 ( Y = y|Ŷ = y ) −D1 ( Y = y|Ŷ = y ) | ( 2 ) ∆SufC encourages the two subgroups with identical Positive predicted value ( PPV ) and Negative predicted value ( NPV ) . On the practical side , considering the healthcare evaluation system outputs either High Risk or Low Risk , Obermeyer et al . ( 2019 ) essentially revealed Dblack ( Y = High Risk|Ŷ = Low Risk ) > Dwhite ( Y = High Risk|Ŷ = Low Risk ) : the severity of Black patients is actually underestimated . Thus if ∆SufC is small , the racial discrimination can be remedied . Sufficiency gap in regression Based on the sufficiency rule and ( Kuleshov et al. , 2018 ) , the sufficiency gap in regression is defined as : ∆SufR = ∫ t∈Y |D0 ( Y ≤ t|Ŷ ≤ t ) −D1 ( Y ≤ t|Ŷ ≤ t ) |dt ( 3 ) ∆SufR ∈ [ 0 , 1 ] is an approximation of |D0 ( Y = y|Ŷ = y ) −D1 ( Y = y|Ŷ = y ) | , ∀y ∈ R , since the latter is difficult to estimate . From the practical aspect , assuming the health system outputs a realvalue healthcare score Ŷ = t ( higher indicates sicker ) , Obermeyer et al . ( 2019 ) ; Sjoding et al . ( 2020 ) observed Dblack ( Y > t|Ŷ ≤ t ) > Dwhite ( Y > t|Ŷ ≤ t ) : for the patients whose predicted healthcare score is less than t , the actual proportion of sicker ( Y > t ) in Black patients is considerably higher than White patients . Therefore a small ∆SufR suggests an improved disparity . 3 PROBLEM SETUP . We denote the representation function λ that maps the input X into the latent variable Z , the prediction function h such that h : Z → R for regression and h : Z → { −1 , 1 } for binary classification . We then denote the prediction loss as ` , the prediction loss on subgroup D0 , D1 is expressed as : L0 ( h , λ ) = E ( x , y ) ∼D0 ` ( h ◦ λ ( x ) , y ) , L1 ( h , λ ) = E ( x , y ) ∼D1 ` ( h ◦ λ ( x ) , y ) According to the intuition , we aim to solve the following bi-level objective : min λ L0 ( h ? 0 , λ ) + L1 ( h ? 1 , λ ) ( Outer level ) s.t . h ? 0 = h ? 1 , h ? 0 ∈ argmin h L0 ( h , λ ) , h ? 1 ∈ argmin h L1 ( h , λ ) . ( Inner level ) Specifically , in the outer level , we aim to find a representation λ for minimizing the prediction error , given the optimal predictor ( h ? 0 , h ? 1 ) on the embedding space Z . As for the inner level , given a fixed representation λ , h ? 0 , h ? 1 are the optimal predictor for each sub-group . The constraints h ? 0 = h ? 1 additionally encourage the invariant optimal predictors from D0 , D1 . Relation to the explicit path alignment In deep learning we adopt the gradient-based approach to minimize the loss , therefore h ? in the inner level is approximated as h ( t+1 ) , the t-th update in the gradient descent : h ? 0 ≈ h ( 0 ) − ∑ t∇hL0 ( h ( t ) , λ ) , h ? 1 ≈ h ( 0 ) − ∑ t∇hL1 ( h ( t ) , λ ) , where h ( 0 ) is the common initialization . Thus the invariant optimal predictor is equivalent to : ∑ t ∇hL0 ( h ( t ) , λ ) = ∑ t ∇hL1 ( h ( t ) , λ ) . The aforementioned equation suggests learning a representation λ that ensures the identical optimization path w.r.t . h for each sub-group , which recovers the explicit path alignment . Relation to the Sufficiency rule We further demonstrate the relation between the bi-level objective and Sufficiency rule . Proposition 1 . If we specify the prediction loss ` as logistic regression loss in the classification log ( 1 + exp ( −yh ( z ) ) ) with Y = { −1 , 1 } and the square loss in the regression ( h ( z ) − y ) 2 with Y ⊂ R. Then minimizing the inner-level loss is equivalent to : ED0 [ Y |Z = z ] = ED1 [ Y |Z = z ] , ED0 [ Y |Ŷ = h ? ( z ) ] = ED1 [ Y |Ŷ = h ? ( z ) ] , where h ? = h ? 0 = h ? 1 and z = λ ( x ) . Proposition 1 reveals that the objective of inner-level loss is to encourage the sufficiency rule .

Summary: This paper proposes a methodology for fair representation learning. The core novelty of the proposal is the satisfaction of a sufficient rule. Experimental results claim good performance with respect to other approaches in terms of an accuracy-fairness tradeoff, in which a sufficient gap is measured.