Generalized Demographic Parity for Group Fairness

1 INTRODUCTION . Fairness problem has attracted increasing attention in many high-stakes applications , such as credit rating , insurance pricing and college admission ( Mehrabi et al. , 2021 ; Du et al. , 2020 ; Bellamy et al. , 2018 ) , the adopted machine learning models encode and even amplify societal biases toward the group with different sensitive attributes . The majority of existing fairness metrics , such as demographic parity ( DP ) ( Feldman et al. , 2015 ) , equal odds ( EO ) ( Hardt et al. , 2016 ) , presumably consider discrete sensitive variables such as gender and race . In many real-world applications including urban studies and mobility predictions ( Tessum et al. , 2021 ) , however , individuals ’ sensitive attributes are unavailable due to privacy constraints . Instead , only aggregated attributes presenting in continuous distributions are available , and thus fairness requires unbiased prediction over neighborhood or region-level objects . Additionally , the sensitive attributes , such as age and weight , are inherently continuous ( Mary et al. , 2019 ; Grari et al. , IJCAI ’ 20 ) . The widely existing continuous sensitive attributes stimulate further fairness metrics definition and bias mitigation methods . Existing fairness metrics on continuous sensitive attributes rely on the statistical measurement of independence , such as Hirschfeld-Gebelein-Renyi ( HGR ) maximal correlation coefficient ( Mary et al. , 2019 ) and mutual information ( Jha et al. , 2021 ; Creager et al. , 2019 ) , which are computationintractable due to the involved functional optimization . Note that the mutual information involves the ratio of probability density function , it is intractable to directly estimate mutual information via probability density function estimation due to the sensitivity over probability density function , especially for the low probability density value . Previous works ( Roh et al. , 2020 ; Lowy et al. , 2021 ; Cho et al. , 2020 ) adopting mutual information or variations as regularizer , however , rely on tractable bound , or computationally complex singular value decomposition operation ( Mary et al. , 2019 ) , or training-needed neural network approximation ( Belghazi et al. , 2018 ) , such as Donsker-Varadhan representation ( Belghazi et al. , 2018 ) , variational bounds ( Poole et al. , 2019 ) . Nevertheless , it is unreliable to adopt the mathematical bound of fairness metric to evaluate different algorithms since lower metrics bound does not necessarily imply lower prediction bias . A question is raised : and generalized demographic parity for continuous sensitive attribute . Markers represent average prediction among specific discrete sensitive attributes , Red dashed line and blue solid line represent prediction average among all data and that with specific sensitive attribute . TV ( · , · ) represents weighted total variation distance . Can we extend DP for continuous attributes while preserving tractable computation ? In this work , we provide positive answers via proposing generalized demographic parity ( GDP ) from regression perspective . Figure 1 provides an illustrative example for DP and GDP . The local prediction average ( blue solid curve ) and the global prediction average ( red dashed line ) represent the average prediction value given sensitive attributes and the whole data samples , respectively . The local and global prediction average should be consistent at any specific continuous sensitive attributes . Therefore , we define GDP , via the weighted total variation distance , to measure the distance between the local and global prediction average , where the weight is the probability density of continuous sensitive attributes . We also theoretically demonstrate the equivalence of GDP and DP for binary sensitive attributes , provide an understanding of GDP from probability perspective , and reveal the bias mitigation methods connection between GDP regularizer and adversarial debiasing . Although GDP is clearly defined on the unknown underlying joint distribution of prediction and sensitive attributes , only data samples , in practice , are available . To this end , we propose two GDP estimation methods , named histogram estimation ( hard group strategy ) and kernel estimation ( soft group strategy ) methods , where kernel estimation is provable with faster estimation error convergence rate w.r.t . data sample size . Specifically , histogram estimation manually divides continuous sensitive attributes into several disjointed and complementary sensitive attributes bins , and then each sample only belongs to one specific bin containing the sensitive attribute of the sample . In other words , the group indicator of the sample is one-hot . As for kernel estimation , instead of quantizing continuous sensitive attributes as several groups , the group indicator of the sample is treated as a kernel function . In other words , to calculate the mean prediction value given specific sensitive attributes , group smoothing strategy is adopted via a soft indicator determined by the sensitive attribute distance between the sample sensitive attribute with target-specific sensitive attribute . In short , the contributions of this paper are : • We develop a tractable group fairness metric GDP for continuous sensitive attributes . We theoretically justify GDP via demonstrating the equivalence with DP for binary sensitive attributes , providing GDP understanding from probability perspective , and revealing the connection between GDP regularizer and adversarial debiasing . • We propose histogram and kernel GDP estimation and provably demonstrate the superiority of kernel GDP estimation method with faster estimation error convergence rate w.r.t . sample size . • We experimentally evaluate the effectiveness and expansibility of GDP on different domain benchmarks ( e.g. , tabular , graph , and temporal graph data ) , tasks ( e.g , classification and regression tasks ) , and compositional sensitive attributes . 2 RELATED WORK . Machine Learning Fairness Fair machine learning targets bias mitigation for automated decisionmaking systems . Various fairness definitions , such as group fairness and individual fairness , have been proposed ( Zemel et al. , 2013 ) . Group metrics , such as DP and EO , measure prediction difference between the groups with different sensitive attributes such as gender , age ( Louizos et al. , 2016 ; Hardt et al. , 2016 ) . While pre- and post-processing methods have been proposed for fairness boosting , these methods can still lead to higher prediction bias ( Barocas et al. , 2017 ) compared with in-processing methods , such as adding regularizer , adversarial debiasing and data augmentation . For example , the covariance between the predictions and sensitive attributes regularization are imposed to boost the independence in ( Woodworth et al. , 2017 ) . ( Zafar et al. , 2017 ) constrains the decision boundaries of classifier to minimize prediction disparity between different groups . Adversarial training has been originally proposed for deep generative modeling ( Goodfellow et al. , 2014 ) and has been introduced for prediction debias in representation learning ( Zhao et al. , 2020 ; Beutel et al. , 2017 ; Louppe et al. , 2017 ) and transfer learning ( Madras et al. , 2018 ) . Data augmentation , such as fair mixup ( Chuang & Mroueh , 2020 ) , can improve the generalization ability for fairness . Representation neutralization is proposed to boost fairness without sensitive attribute ( Du et al. , 2021 ) . Kernel Density Estimation and Kernel Regression Kernel density estimation ( KDE ) is a nonparametric method to estimate the continuous probability density function of a random variable ( Davis et al. , 2011 ; Parzen , 1962 ) . Given finite data samples , KDE smoothly estimate the probability function via weighted summation , where the weight is determined via kernel function ( Epanechnikov , 1969 ) . Kernel regression is a non-parametric technique to estimate the conditional expectation of a random variable ( Nadaraya , 1964 ) . Nadaraya-Watson Kernel regression function estimator is proposed for regression via locally normalized weighted average in ( Bierens , 1988 ) , where the sample weight is determined by kernel function . 3 GENERALIZED DEMOGRAPHIC PARITY . Without loss of generality , we consider a binary classification task to predict the output variable Y given the input variable X , while avoiding prediction bias for sensitive attribute S. Define the input X ∈ X ⊂ Rd , labels Y ∈ { 0 , 1 } , and machine learning model f : Rd → [ 0 , 1 ] provides prediction score Ŷ = f ( X ) . Fairness requires predictor Ŷ to be independent of sensitive attribute S , regardless of continuous or discrete , i.e. , P ( Ŷ = ŷ ) = P ( Ŷ = ŷ|S = s ) for any support value y and s ( Beutel et al. , 2017 ) . Since the independent constraint is difficult to optimize , the relaxed demographic parity ( DP ) ( Madras et al. , 2018 ) metrics are proposed to quantitatively measure the predictor bias for binary sensitive attribute S ∈ { 0 , 1 } . Formally , the demographic parity is defined as ∆DP = |EŶ [ Ŷ |S = 0 ] − EŶ [ Ŷ |S = 1 ] | , where E [ · ] represents variable expectation . For categorical sensitive attribute S ∈ S , work ( Cho et al. , 2020 ) introduces a fairness metric , named difference w.r.t . demographic parity ( DDP ) ∆DDP = ∑ s∈S |EŶ [ Ŷ |S = s ] − EŶ [ Ŷ ] | . Although DP has been widely used to evaluate the prediction bias , it is still inapplicable for continuous sensitive attributes since the data samples can not be directly divided into several distinctive groups based on the sensitive attributes . Without loss of generality , we assume continuous sensitive attributes S ∈ [ 0 , 1 ] and propose GDP to extend tractable DP for continuous sensitive attributes . Assume the joint distribution of tuple ( S , Ŷ ) is PS , Ŷ ( s , ŷ ) , the local prediction average and global prediction average are defined as the prediction expectation given sensitive attribute S = s and without any sensitive attribute condition , i.e. , local prediction average m ( s ) △ = E [ Ŷ |S = s ] and global prediction average mavg △ = ES [ m ( S ) ] = E [ Ŷ ] , respectively . Then , we adopt weighted total variation distance on local prediction average and global prediction average , where the weight is specified by the probability density function of the sensitive attribute . The formal definition of the discrepancy demographic parity for continuous sensitive attributes is as follows : ∆GDP = ∫ 1 0 ∣∣∣m ( s ) −mavg∣∣∣PS ( S = s ) ds = ES [ |m ( S ) −mavg| ] , ( 1 ) We also provide the connection of GDP and DP for binary sensitive attributes , which implies that GDP is equivalent to DP for binary sensitive attributes , as follows : Theorem 1 ( Connection between DP and GDP ) . For binary sensitive attribute S ∈ { 0 , 1 } , GPD and DP are equivalent except the coefficient only dependent on datasets . Specifically , the relation of ∆GDP and ∆DP satisfies ∆GDP = 2PS ( S = 1 ) · PS ( S = 0 ) ·∆DP . The proof of Theorem 1 is presented in Appendix A . For categorical sensitive attributes , it is easy to obtain that ∆GDP = ∑ s∈S PS ( S = s ) |EŶ [ Ŷ |S = s ] − EŶ [ Ŷ ] | , i.e. , GDP is weighted DDP for categorical sensitive attributes . In a nutshell , GDP is a natural fairness metric extension for binary and categorical sensitive attributes . Since the independence between prediction Ŷ and sensitive attribute S implies that the joint distribution PS , Ŷ ( s , ŷ ) and product marginal distribution PS ( s ) PŶ ( ŷ ) are the same , the bias can be measured by the distance of the joint distribution and product marginal distribution . Subsequently , we show the connection of GDP and prediction-weighted total variation distance between these two distributions as follows : Theorem 2 ( Probability View of GDP ) . Assume the joint distribution of ( Ŷ , S ) with support [ 0 , 1 ] 2 is PS , Ŷ ( s , ŷ ) . Define the prediction-weighted total variation distance as TVpred ( P 1 , P 2 ) △ =∫ 1 0 ∫ 1 0 ŷ|P 1 ( ŷ , s ) − P 2 ( ŷ , s ) |dŷds . Then the proposed fairness for continuous senstive attribute is upper bounded by prediction- weighted total variation distance between the joint distribution and product marginal distribution : ∆GDP = ∫ 1 0 ∫ 1 0 ∣∣∣ŷ [ PS , Ŷ ( s , ŷ ) − PS ( s ) PŶ ( ŷ ) ] ∣∣∣dŷds ≤ TVpred ( PS , Ŷ ( s , ŷ ) , PS ( s ) PŶ ( ŷ ) ) . The proof of Theorem 2 is presented in Appendix B. Theorem 2 demonstrates that GDP is actually a lower bound for prediction-weighted total variation distance between these two distributions and implies the necessity of GDP for bias measurement .

Summary: This paper proposes a new extension of demographic parity when the sensitive attribute is continuous. It proposes two techniques to estimate this quantity from data. The authors also consider adding this quantity as a regularization term to control discrimination while designing machine learning models.