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: The paper is motivated by the need to account for continuous attributes in fair Machine Learning. In particular, it proposes a Generalized Demographic Parity metric (GDP), i.e., a group fairness metric that can work with both continuous and discrete variables. A key challenge in doing so is preserving tractable computation.   Theoretically, GDP is defined via the weighted total variation distance and measures the distance between the local and global prediction average, where the weight corresponds to the pdf of the continuous sensitive attributes. Since the joint distribution between model prediction and sensitive attributes might not be available in practice, the paper also proposes two estimation methods: histogram estimation and kernel estimation. The former quantizes the continuous sensitive attributes into bins, whereas in the latter method the group indicator of the sample is treated as a kernel function. The kernel method leads to faster estimation error convergence rate in terms of sample size. The paper provides an extensive evaluation with tabular, graph and temporal graph data, synthetic experiments, and classification and regression tasks. It is demonstrated experimentally that the kernel method has better bias mitigation performance.