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: The paper considers the learning of fair representations (and classifiers). In particular, the paper considers the (group) sufficiency rule for fairness and optimization path alignment to achieve the fairness requirement. The paper considers a representation fair if the optimization path (gradient steps) of subgroups are equivalent / invariant, with respect to optimal subgroup classifiers. To this end, the paper introduces a bi-level optimization problem for a representation function, which is solved by approximating gradients with the implicit function theorem. The approximation error of the gradients and the convergence of representation function (appendix) are analysed. Their algorithm is then experimentally tested under various datasets and compared to various baseline approaches.