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 study fair prediction subject to Equalized Loss (EL), and they introduce a variety of approaches for exactly and approximately solving the problem of finding the globally optimal predictor that satisfies EL. First, they show how to solve a sequence of convex constrained optimization problems in order to solve the larger non-convex problem. Next, they show how to approximately solve this problem more efficiently by using unconstrained convex optimization. Lastly, they evaluate both of their approaches on two datasets.