Non-convex Optimization for Learning a Fair Predictor under Equalized Loss Fairness Constraint

Supervised learning models have been increasingly used in various domains such as lending , college admission , natural language processing , face recognition , etc . These models may inherit pre-existing biases from training datasets and exhibit discrimination against protected social groups . Various fairness notions have been introduced to address fairness issues . In general , finding a fair predictor leads to a constrained optimization problem , and depending on the fairness notion , it may be non-convex . In this work , we focus on Equalized Loss ( EL ) , a fairness notion that requires the prediction error/loss to be equalized across different demographic groups . Imposing this constraint to the learning process leads to a non-convex optimization problem even if the loss function is convex . We introduce algorithms that can leverage off-the-shelf convex programming tools and efficiently find the global optimum of this non-convex problem . In particular , we first propose the ELminimizer algorithm , which finds the optimal EL fair predictor by reducing the non-convex optimization problem to a sequence of convex constrained optimizations . We then propose a simple algorithm that is computationally more efficient compared to ELminimizer and finds a sub-optimal EL fair predictor using unconstrained convex programming tools . Experiments on real-world data show the effectiveness of our algorithms . 1 INTRODUCTION . As machine learning ( ML ) algorithms are increasingly being used in applications such as education , lending , recruitment , healthcare , criminal justice , etc. , there is a growing concern that the algorithms may exhibit discrimination against protected population groups . For example , speech recognition products such as Google Home and Amazon Alexa were shown to have accent bias ( Harwell , 2018 ) . The COMPAS recidivism prediction tool , used by courts in the US in parole decisions , has been shown to have a substantially higher false positive rate for African Americans compared to the general population ( Dressel & Farid , 2018 ) . Amazon had been using automated software since 2014 to assess applicants ’ resumes , which were found to be biased against women ( Dastin , 2018 ) . Various fairness notions have been proposed in the literature to measure and remedy the biases in ML systems ; they can be roughly classified into two classes : 1 ) individual fairness focuses on the equity at individual level and it requires the similar individuals to be treated similarly ( Dwork et al. , 2012 ; Biega et al. , 2018 ; Jung et al. , 2019 ; Gupta & Kamble , 2019 ) ; 2 ) group fairness requires certain statistical measures to be ( approximately ) equalized across different groups distinguished by some sensitive attributes . Their suitability for use is often application dependent , and many of them are incompatible with each other ( Zhang et al. , 2019 ; Hardt et al. , 2016 ; Conitzer et al. , 2019 ; Zhang et al. , 2020 ; Khalili et al. , 2020 ) . Extensive approaches have been developed to satisfying a given definition of fairness and they generally fall under three categories : pre-processing , by modifying the original dataset such as removing certain features and reweighing , e.g. , ( Kamiran & Calders , 2012 ; Celis et al. , 2020 ) ; in-processing , by modifying the algorithms such as imposing fairness constraints or changing objective functions , e.g. , ( Zhang et al. , 2018 ; Agarwal et al. , 2018 ; 2019 ; Reimers et al. , 2021 ; Calmon et al. , 2017 ) ; post-processing , by adjusting the output of the algorithms based on sensitive attributes , e.g. , ( Hardt et al. , 2016 ) . In this paper , we focus on group fairness and we aim to mitigate unfairness issues in supervised learning using in-processing approaches . The problem can be cast as a constrained optimization problem where a fair predictor can be found by minimizing the prediction error ( i.e. , loss ) subject to certain group fairness constraint . In Section 2.1 , we present a number of definitions of commonly used group fairness notions , namely , statistical parity ( Dwork et al. , 2012 ) , equal opportunity ( Hardt et al. , 2016 ) , equalized loss ( Zhang et al. , 2019 ) , and bounded group loss ( Agarwal et al. , 2019 ) . Here we are particularly interested in equalized loss which requires the expected loss to be equalized across different groups . Constrained optimization problems for finding a fair predictor have been studied in the iterature . In general , imposing a fairness criterion to the optimization problem may lead to a non-convex optimization problem . Existing works have proposed various approaches to solving such a non-convex optimization in different settings . For example , Komiyama et al . ( 2018 ) studied the non-convex optimization for regression problems under the coefficient of determination constraint . Agarwal et al . ( 2019 ) proposed an approach to finding a fair regression model under bounded group loss and statistical parity fairness constraints . Agarwal et al . ( 2018 ) studied classification problems and aimed at finding fair classifiers under various fairness notions including statistical parity and equal opportunity . In particular , they considered zero-one loss as the objective function and trained a randomized fair classifier over a finite hypothesis space ; this problem was reduced to a problem of finding the saddle point of a linear Lagrangian function in ( Agarwal et al. , 2018 ) . Zhang et al . ( 2018 ) proposed an adversarial debasing technique to find a fair classifier under equalized odd , equal opportunity , and statistical parity . However , there is no guarantee that this technique finds the global optimal solution . The main difference between the present work and the existing in-processing approaches are as follows : 1 ) we consider a non-convex problem for finding a fair predictor satisfying Equalized Loss fairness notion , which has not been studied in the literature to the best of our knowledge . 2 ) We propose algorithms for finding the global optimal solution to this non-convex problem efficiently . 3 ) Our algorithms are easy to implement and are applicable to both regression and classification problems . 4 ) Unlike ( Agarwal et al. , 2018 ) , our algorithms are not limited to finite hypothesis space . Non-convex optimization problems have also been studied in other contexts such as learning overparametrized models . For example , deep neural networks are typically trained by solving unconstrained , non-convex problems , and methods such as gradient descent may not be suitable as they are likely to find saddle points but not optimums . To address this issue , approaches have been proposed in recent works by incorporating the higher order derivatives ( Celis et al. , 2020 ; Anandkumar & Ge , 2016 ) or noisy gradients ( Ge et al. , 2015 ) . However , these methods only find a local minimum ( not a global minimum ) and are not applicable to our problem with a non-convex constraint . In this work , we develop novel algorithms that find the fair ( sub- ) optimal solutions under Equalized Loss fairness constraint efficiently . Note that while our approach and algorithms are presented in the context of fair machine learning , they are applicable to any problem that can be formulated as a constrained optimization problem in the form of minw L0 ( w ) +αL1 ( w ) s.t . |L0 ( w ) −L1 ( w ) | < γ , where α is a constant Our main contributions and findings are as follows . 1 . We study the relationship between Equalized Loss ( EL ) and Bounded Group Loss ( BGL ) fairness notions . We show that given the existence of feasible solutions satisfying ( approximate ) BGL fairness , imposing ( approximate ) EL fairness constraint never increase losses of both groups simultaneously ( Theorems 1 and 2 in Section 2.1 ) . These results help policy makers to have a better understanding of these two fairness notions . 2 . We develop an algorithm ( ELminimizer ) to solve a non-convex constrained optimization problem that finds the optimal ( approximate ) EL fair solution . We show that such non-convex optimization can be reduced to a sequence of convex constrained optimizations and the convergence property of the algorithm is analyzed ( Theorems 3 and 4 , Section 3 ) . 3 . We develop a simple algorithm for finding a sub-optimal ( approximate ) EL fair solution . We show that a sub-optimal solution is a linear combination of optimal solutions to two unconstrained optimizations and it can be found efficiently without solving constrained optimizations ( Theorem 5 , Section 4 ) . 4 . We conduct sample complexity analysis and provide the guarantee on generalization performance ( Theorem 7 , Section 5 ) . 5 . We validate the theoretical results by conducting experiments on real-world data ( Section 6 ) . 2 PROBLEM FORMULATION . Consider a supervised learning problem where the training dataset consists of triples ( X , A , Y ) from two social groups . Random variableX ∈ X ⊂ Rdx is the feature vector ( in form of a column vector ) , A ∈ { 0 , 1 } is the sensitive attribute ( e.g. , race , gender ) indicating the group membership , and Y ∈ Y ⊂ R is the label . The feature vector X may or may not include sensitive attribute A. Label Y can be either discrete or continuous depending on the given problem : if Y is discrete ( resp . continuous ) , then the problem is a classification ( resp . regression ) problem . Let F be a set of predictors fw : X → R parameterized by weight vector w ∈ Rdw .1 Consider loss function l : Y×X → Rwhere l ( Y , fw ( X ) ) measures the error of fw in predicting label Y . Denote the expected loss with respect to the joint probability distribution of ( X , Y ) by L ( w ) : = E { l ( Y , fw ( X ) ) } . Then , La ( w ) : = E { l ( Y , fw ( X ) ) |A = a } denotes the expected loss of the group with attribute A = a . A predictor that minimizes the total expected loss , i.e. , argminw L ( w ) , can be biased against certain groups . To mitigate the risk of unfairness , various fairness notions have been proposed in the literature . Some of the most commonly used notions of group fairness are as follows : 1 ) Statistical Parity ( SP ) ( Dwork et al. , 2012 ) implies that the predictor and the sensitive attribute should be independent , i.e. , fw ( X ) ⊥ A ; 2 ) Equal Opportunity ( EqOpt ) ( Hardt et al. , 2016 ) requires that conditional on Y = 1 , prediction and sensitive attribute are independent , i.e. , fw ( X ) ⊥ A|Y = 1 ; 3 ) Equalized Odds ( EO ) ( Hardt et al. , 2016 ) requires the conditional independence between prediction and sensitive attribute given Y , i.e. , fw ( X ) ⊥ A|Y ; 4 ) Equalized Loss ( EL ) ( Zhang et al. , 2019 ; Berk et al. , 2021 ) requires that the losses experienced by different groups are equalized , i.e. , L0 ( w ) = L1 ( w ) ; 5 ) Bounded Group Loss ( BGL ) ( Agarwal et al. , 2019 ) requires that the loss experienced by each group is bounded . With fairness consideration , the goal is to find weight vectorw that minimizes total expected loss in predicting Y givenX , subject to certain fairness condition , i.e. , minw L ( w ) s.t . fairness constraint . This is a typical formulation in fair machine learning literature , and above method of finding a fair predictor belongs to in-processing approaches . Because such constrained optimization can be nonconvex , finding the optimal solution efficiently can be challenging . In this work , we develop novel algorithms that solves such an optimization problem udder EL fairness constraint . 2.1 EQUALIZED LOSS ( EL ) AND BOUNDED GROUP LOSS ( BGL ) As mentioned in Section 2 , various fairness notions have been introduced in the literature . Among them , Statistical Parity ( SP ) , Equal Opportunity ( EqOpt ) , Equalized Odds ( EO ) , and Bounded Group Loss ( BGL ) have been studied extensively in the literature , and both in-processing and postprocessing approaches have been developed to satisfy these constraints ( Dwork et al. , 2012 ; Agarwal et al. , 2018 ; Hardt et al. , 2016 ; Zafar et al. , 2019 ; Fitzsimons et al. , 2019 ) . Note that different fairness notions may be conflict with each other and which one to adopt is application and context dependent . In this work , we are interested in Equalized Loss ( EL ) fairness notion ( Zhang et al. , 2019 ; Berk et al. , 2021 ) which implies that the prediction error should be the same across different groups,2 and Group Bounded Loss ( BGL ) fairness notion ( Agarwal et al. , 2019 ) which requires the prediction error of every group to be bounded . We consider a relaxed version of EL fairness defined as follows . Definition 1 ( γ-EL ) A predictor f satisfies γ-EL if the expected losses experienced by different demographic groups satisfy the following , − γ ≤ L0 ( w ) − L1 ( w ) ≤ γ . ( 1 ) Parameter γ controls the degree of fairness ; the smaller γ implies the stronger fairness . When γ = 0 , the exact EL fairness is attained . We say a group is disadvantaged if it experiences a larger loss . Similarly , Group Bounded Loss ( BGL ) fairness notion is formally defined as follows . Definition 2 ( γ-BGL ) A predictor f satisfies γ-BGL if the expected loss of each demographic group is bounded by γ , i.e. , La ( w ) ≤ γ , ∀a ∈ { 0 , 1 } . ( 2 ) 1Predictive models such as logistic regression , linear regression , deep learning models , etc. , are parameter- ized by a weight vector . 2EL has also been referred to as Overall Accuracy Equality in ( Berk et al. , 2021 ; Agarwal et al. , 2019 ) . 2.2 RELATIONS BETWEEN γ-EL AND γ-BGL In this section , we formally study the relations between γ-EL and γ-BGL fairness notions . Under γ-EL fairness constraint , finding a fair predictor is equivalent to solving the following constrained optimization problem : minw L ( w ) s.t . |L0 ( w ) − L1 ( w ) | ≤ γ . ( 3 ) Letw∗ be denoted as the solution to ( 3 ) and fw∗ is the optimal γ-EL fair predictor . Theorem 1 below shows that given the existence of a feasible point satisfying γ-BGL fairness , it ’ s impossible for both groups experiencing loss larger than γ from the optimal γ-EL fair predictor . Theorem 1 Consider the following optimization for finding the optimal γ-BGL fair predictor , minw L ( w ) s.t . La ( w ) ≤ γ , ∀a ∈ { 0 , 1 } . ( 4 ) If L0 ( w∗ ) > γ and L1 ( w∗ ) > γ , then optimization problem ( 4 ) does not have a feasible point . Proof 1 We prove by contradiction . Assume w̃ is a feasible point of optimization ( 4 ) . Note that w̃ is a feasible point for optimization problem ( 3 ) as well . Since both L0 ( w∗ ) and L1 ( w∗ ) are larger than γ , we have , E { l ( Y , fw∗ ) } = Pr { A = 0 } L0 ( w∗ ) + Pr { A = 1 } L1 ( w∗ ) > γ , E { l ( Y , fw̃ ) } = Pr { A = 0 } L0 ( w̃ ) + Pr { A = 1 } L1 ( w̃ ) ≤ γ. Therefore , w∗ can not be the solution to ( 3 ) . This contradiction proves that the optimization problem ( 4 ) can not have a feasible point . Theorem 1 implies that if γ-EL notion leads to an increase of the loss of every demographic group , then there is no optimal predictor under γ-BGL.3 The next theorem further shows that for any predictor satisfying γ-EL , it must satisfy 2γ-BGL . Theorem 2 Assume optimization problem ( 4 ) has at least one feasible point . Then , we have , min { L0 ( w∗ ) , L1 ( w∗ ) } ≤ γ and max { L0 ( w∗ ) , L1 ( w∗ ) } ≤ 2γ . Proof 2 Let w̃ be a feasible point of optimization problem ( 4 ) , then w̃ is also a feasible point to ( 3 ) . If min { L0 ( w∗ ) , L1 ( w∗ ) } > γ , then L ( w∗ ) > γ ≥ L ( w̃ ) must hold . This is a contradiction because it implies thatw∗ is not an optimal solution to ( 3 ) . Therefore , min { L0 ( w∗ ) , L1 ( w∗ ) } ≤ γ . Similarly , we can prove max { L0 ( w∗ ) , L1 ( w∗ ) } ≤ 2γ by contradiction . Assume max { L0 ( w∗ ) , L1 ( w∗ ) } > 2γ . Then , max { L0 ( w∗ ) , L1 ( w∗ ) } − min { L0 ( w∗ ) , L1 ( w∗ ) } > γ which shows that w∗ is not a feasible point for ( 3 ) . This is a contradiction . Therefore , max { L0 ( w∗ ) , L1 ( w∗ ) } ≤ 2γ . Theorems 1 and 2 investigated the relations between EL and BGL fairness notions . Since γ-EL implies 2γ-BGL and it additionally requires the approximate equality across different groups , we will focus on γ-EL fairness notion in the rest of the paper . Because optimization problem ( 3 ) is a non-convex optimization , finding the optimal fair γ-EL solution efficiently can be challenging . In the next sections , we propose a number of algorithms that are easy to implement and can solve the optimization ( 3 ) efficiently .

Summary: The authors consider minimization of convex losses constrained by either bounded loss on each group, or bounded difference of losses over two groups. The second formulation is non-convex, whereas the first formulation is convex.   When the losses are strictly convex on both the demographic groups, so that their optima are distinct (I think this is the condition they need, but they use a more restrictive condition in the paper), they can find the "EL" fair predictor by solving a sequence of convex constrained optimizations, by exploiting a monotonicity property. They next give a more computationally efficient approximate algorithm for finding the EL fair predictor.