Individually Fair Gradient Boosting

1 INTRODUCTION . In light of the ubiquity of machine learning ( ML ) methods in high-stakes decision-making and support roles , there is concern about ML models reproducing or even exacerbating the historical biases against certain groups of users . These concerns are valid : there are recent incidents in which algorithmic bias has led to dire consequences . For example , Amazon recently discovered its ML-based resume screening system discriminates against women applying for technical positions ( Dastin , 2018 ) . In response , the ML community has proposed a myriad of formal definitions of algorithmic fairness . Broadly speaking , there are two types of fairness definitions : group fairness and individual fairness ( Chouldechova & Roth , 2018 ) . In this paper , we focus on enforcing individual fairness . At a highlevel , the idea of individual fairness is the requirement that a fair algorithm should treat similar individuals similarly . For a while , individual fairness was overlooked in favor of group fairness because there is often no consensus on which users are similar for many ML tasks . Fortunately , there is a flurry of recent work that addresses this issue ( Ilvento , 2019 ; Wang et al. , 2019 ; Yurochkin et al. , 2020 ; Mukherjee et al. , 2020a ) . In this paper , we assume there is a fair metric for the ML task at hand and consider the task of individually fair gradient boosting . Gradient boosting , especially gradient boosted decision trees ( GBDT ) , is a popular method for tabular data problems ( Chen & Guestrin , 2016 ) . Unfortunately , existing approaches to enforcing individual fairness are either not suitable for training non-smooth ML models ( Yurochkin et al. , 2020 ) or perform poorly with flexible non-parametric ML models . We aim to fill this gap in the literature . Our main contributions are : 1 . We develop a method to enforce individual fairness in gradient boosting . Unlike other methods for enforcing individually fairness , our approach handles non-smooth ML models such as ( boosted ) decision trees . 2 . We show that the method converges globally and leads to ML models that are individually fair . We also show that it is possible to certify the individual fairness of the models a posteriori . 3 . We show empirically that our method preserves the accuracy of gradient boosting while improving widely used group and individual fairness metrics . 2 ENFORCING INDIVIDUAL FAIRNESS IN GRADIENT BOOSTING . Consider a supervised learning problem . Let X ⊂ Rd be the input space and Y be the output space . To keep things simple , we assume Y = { 0 , 1 } , but our method readily extends to multi-class classification problems . Define Z = X × { 0 , 1 } . We equip X with a fair metric dx that measures the similarity between inputs . The fair metric is application specific , and we refer to the literature on fair metric learning ( Ilvento , 2019 ; Wang et al. , 2019 ; Yurochkin et al. , 2020 ) for ways of picking the fair metric . Our goal is to learn an ML model f : X → { 0 , 1 } that is individually fair . Formally , we enforce distributionally robust fairness ( Yurochkin et al. , 2020 ) , which asserts that an ML model has similar accuracy/performance ( measured by the loss function ) on similar samples ( see Definition 2.1 ) . One way to accomplish this is adversarial training ( Yurochkin et al. , 2020 ; Yurochkin & Sun , 2020 ) . Unfortunately , adversarial training relies on the smoothness of the model ( with respect to the inputs ) , so it can not handle non-smooth ML models ( e.g . decision trees ) . We address this issue by considering a restricted adversarial cost function that only searches over the training examples ( instead of the entire input space ) for similar examples that reveal violations of individual fairness . As we shall see , this restricted adversarial cost function is amenable to functional gradient descent , which allows us to develop a gradient boosting algorithm . 2.1 ENFORCING INDIVIDUAL FAIRNESS WITH A RESTRICTED ADVERSARIAL COST FUNCTION . We first review how to train individually fair ML models with adversarial training ( see Yurochkin et al . ( 2020 ) for more details ) to set up notation and provide intuition on how adversarial training leads to individual fairness . Let F be a set of ML models , let ` : Y × Y → R be a smooth loss function ( see Section 3 for concrete assumptions on ` ) that measures the performance of an ML model , and let D = { ( xi , yi ) } ni=1 be a set of training data . Define the transport cost function c ( ( x1 , y1 ) , ( x2 , y2 ) ) , d 2 x ( x1 , x2 ) +∞ · 1 { y1 6=y2 } . ( 2.1 ) We see that c ( ( x1 , y1 ) , ( x2 , y2 ) ) is small iff x1 and x2 are similar ( in the fair metric dx ) and y1 = y2 . In other words , c is small iff two similar examples are assigned the same output . Define the optimal transport distance W ( with transport cost c ) on probability distributions on Z : W ( P1 , P2 ) , infΠ∈C ( P1 , P2 ) ∫ Z×Z c ( z1 , z2 ) dΠ ( z1 , z2 ) , where C ( P1 , P2 ) is the set of couplings between P1 and P2 ( distributions on Z ×Z whose marginals are P1 and P2 ) . This optimal transport distance lifts the fair metric on ( points in ) the sample space to distributions on the sample space . Two distributions are close in this optimal transport distance iff they assign mass to similar areas of the sample space Z . Finally , define the adversarial risk function Lr ( f ) , supP : W ( P , P∗ ) ≤ EP [ ` ( f ( X ) , Y ) ] , ( 2.2 ) where P∗ is the data generating distribution and > 0 is a small tolerance parameter . The adversarial risk function looks for distributions on the sample space that ( i ) are similar to the data generating distribution and ( ii ) increases the risk of the ML model f . This reveals differential performance of the ML model on similar samples . This search for differential performance is captured by the notion of distributionally robust fairness : Definition 2.1 ( distributionally robust fairness ( DRF ) ( Yurochkin et al. , 2020 ) ) . An ML model h : X → Y is ( , δ ) -distributionally robustly fair ( DRF ) WRT the fair metric dx iff supP : W ( P , Pn ) ≤ ∫ Z ` ( z , h ) dP ( z ) ≤ δ . ( 2.3 ) In light of the preceding developments , a natural cost function for training individually fair ML models is the adversarial cost function : Le ( f ) , supP : W ( P , Pn ) ≤ EP [ ` ( f ( X ) , Y ) ] , ( 2.4 ) where Pn is the empirical distribution of the training data . This is the empirical counterpart of ( 2.2 ) , and it works well for training smooth ML models ( Yurochkin et al. , 2020 ) . Unfortunately , ( 2.4 ) is hard to evaluate for non-smooth ML models : it is defined as the optimal value of an optimization problem , but the gradient ∂x ` ( f ( x ) , y ) is not available because the ML model f is non-smooth . To circumvent this issue , we augment the support of the training set and restrict the supremum in ( 2.4 ) to the augmented support . Define the augmented support set D0 , { ( xi , yi ) , ( xi , 1− yi ) } ni=1 and the restricted optimal transport distance WD between distributions supported on D0 : WD ( P1 , P2 ) , infΠ∈C0 ( P1 , P2 ) ∫ Z×Z c ( z1 , z2 ) dΠ ( z1 , z2 ) , where C0 ( P1 , P2 ) is the set of distributions supported on D0 ×D0 whose marginals are P1 and P2 . We consider the restricted adversarial cost function L ( f ) , supP : WD ( P , Pn ) ≤ EP [ ` ( f ( X ) , Y ) ] , ( 2.5 ) where > 0 is a small tolerance parameter . The interpretation of ( 2.5 ) is identical to that of ( 2.4 ) : it searches for perturbations to the training examples that reveal differential performance in the ML model . On the other hand , compared to ( 2.4 ) , the supremum in ( 2.5 ) is restricted to distributions supported on D0 . This allows us to evaluate ( 2.5 ) by solving a ( finite-dimensional ) linear program ( LP ) . As we shall see , this LP depends only on the loss values ` ( f ( xi ) , yi ) and ` ( f ( xi ) , 1− yi ) , so it is possible to solve the LP efficiently even if the ML model f is non-smooth . This is the key idea in this paper . Before delving into the details , we note that the main drawback of restricting the search to distributions supported on D0 is reduced power to detect differential performance . If the ML model exhibits differential performance between two ( similar ) areas of the input space but only one area is represented in the training set , then ( 2.4 ) will detect differential performance but ( 2.5 ) will not . Augmenting the support set with the points { ( xi , 1 − yi ) } ni=1 partially alleviates this issue ( but the power remains reduced compared to ( 2.4 ) ) . This is the price we pay for the broader applicability of ( 2.5 ) . 2.2 FUNCTIONAL GRADIENT DESCENT ON THE RESTRICTED ADVERSARIAL COST FUNCTION . Gradient boosting is functional gradient descent ( Friedman , 2001 ) , so a key step in gradient boosting is evaluating ∂L∂ŷ , where the components of ŷ ∈ R n are ŷi , f ( xi ) . By Danskin ’ s theorem , we have ∂L ∂ŷi = ∂∂f ( xi ) [ supP : WD ( P , Pn ) ≤ EP [ ` ( f ( xi ) , yi ) ] ] = ∑ y∈Y ∂ ∂f ( xi ) [ ` ( f ( xi ) , y ) ) ] P ∗ ( xi , y ) , ( 2.6 ) where P ∗ is a distribution that attains the supremum in ( 2.5 ) . We note that there is no need to differentiate through the ML model f in ( 2.6 ) , so it is possible to evaluate the functional gradient for non-smooth ML models . It remains to find P ∗ . We devise a way of finding P ∗ by solving a linear program . We start with a simplifying observation : if c ( zi , zj ) = ∞ for any zi ∈ D0 and zj ∈ D , then any weight at zj can not be transported to zi . Thus , we will only focus on the pairs ( zi , zj ) ∈ D0×D with c ( zi , zj ) < ∞ . Let C ∈ Rn×n be the matrix with entries given by Ci , j = c ( ( xi , yj ) , ( xj , yj ) ) = d2x ( xi , xj ) . We also define the class indicator vectors y 1 , y0 ∈ { 0 , 1 } n by y1j = { 1 : yj = 1 0 : yj = 0 and y0 = 1n − y1 . ( 2.7 ) For any distribution P on D0 , let Pi , k = P ( { ( xi , k ) } ) for k ∈ { 0 , 1 } . Then , the condition that WD ( P , Pn ) ≤ is implied by the existence of a matrix Π such that 1 . Π ∈ Γ with Γ = { Π|Π ∈ Rn×n+ , 〈C , Π〉 ≤ , ΠT · 1n = 1n1n } . 2 . Π · y1 = ( P1,1 , . . . , Pn,1 ) , and Π · y0 = ( P1,0 , . . . , Pn,0 ) . Further define the matrix R ∈ Rn×n by Rij = ` ( f ( xi ) , yj ) - this is the loss incurred if point j with label yj is transported to point i . With this setup , given the current predictor f , we can obtain a solution Π∗ to the optimization as the solution to the linear program ( in n2 variables ) Π∗ ∈ arg max Π∈Γ 〈R , Π〉 . ( 2.8 ) Then the optimal distribution P ∗ on D0 is given by P ∗ ( { ( xi , k ) } ) = ( Π∗ · yk ) i . An outline of the full gradient boosting procedure is provided in Algorithm 1 . It is important to note that we have made no assumptions about the class of candidate predictors F in finding the optimal transport map Π∗ in ( 2.8 ) . In particular , F can contain discontinuous functions - for example , decision trees or sums of decision trees . This allows us to apply this fair gradient boosting algorithm to any class F of base classifiers . Algorithm 1 Fair gradient boosting 1 : Input : Labeled training data { ( xi , yi ) } ni=1 ; class of weak learnersH ; initial predictor f0 ; search radius ; number of steps T ; sequence of step sizes α ( t ) ; fair metric dx on X 2 : Define the matrix C by Ci , j ← d2x ( xi , xj ) . 3 : for t = 0 , 1 , . . . , T − 1 do 4 : Define the matrix Rt by ( Rt ) ij = ` ( ft ( xi ) , yj ) 5 : Find Π∗t ∈ arg maxΠ∈Γ 〈Rt , Π〉 ; and set Pt+1 ( xi , k ) ← ( Π∗t · yk ) i 6 : Fit a base learner ht ∈ H to the set of pseudo-residuals { ∂L∂ft ( xi ) } n i=1 ( see ( 2.6 ) ) . 7 : Let ft+1 = ft + αtht . 8 : end for 9 : return fT

Summary: The authors presented in the submission a thorough study on enforcing the aggregated individual fairness with non-differentiable ML models. The proposed method generates individually fair and robust ML models in a minimax fashion among all possible samples that are close to the true distribution w.r.t. a given fair metric. They introduce the augmented support and transfer the standard gradient descent to a gradient descent in the functional space anchored by the augmented support to optimize the adversarial risk function with non-smooth ML models, e.g. decision trees. Solid theoretical guarantees and convincing empirical study results are provided to support their claims. The paper is highly completed, well-structured (though a bit dense given the page limit) and well-written - a clear accept.