Fair Sparse Regression with Clustering: An Invex Relaxation for a Combinatorial Problem

1 Introduction . In modern times , machine learning algorithms are used in a wide variety of applications , many of which are decision making processes such as hiring ( Hoffman et al. , 2018 ) , predicting human behavior ( Subrahmanian & Kumar , 2017 ) , COMPAS ( Correctional Offender Management Profiling for Alternative Sanctions ) risk assessment ( Brennan et al. , 2009 ) , among others . These decisions have large impacts on society ( Kleinberg et al. , 2018 ) . Consequently , researchers have shown interest in developing methods that can mitigate unfair decisions and avoid bias amplification . Several fair algorithms have been proposed for machine learning problems such as regression ( Agarwal , 2019 ; Berk , 2017 ; Calders , 2013 ) , classification ( Agarwal et al. , 2018 ; Donini et al. , 2018 ; Dwork et al. , 2012 ; Feldman et al. , 2015 ; Hardt et al. , 2016 ; Huang & Vishnoi , 2019 ; Pedreshi et al. , 2008 ; Zafar et al. , 2019 ; Zemel et al. , 2013 ) and clustering ( Backurs et al. , 2019 ; Bera et al. , 2019 ; Chen et al. , 2019 ; Chierichetti et al. , 2017 ; Huang et al. , 2019 ) . A common thread in the above literature is that performance is only viewed in terms of risks , e.g. , misclassification rate , false positive rate , false negative rate , mean squared error . In the literature , fairness is discussed in the context of discrimination based on membership to a particular group ( e.g . race , religion , gender ) which is considered a sensitive attribute . Fairness is generally modeled explicitly by adding a fairness constraint or implicitly by incorporating it in the model itself . There have been several notions of fairness studied in linear regression . Berk ( 2017 ) proposed notions of individual fairness and group fairness , and modeled them as penalty functions . Calders ( 2013 ) proposed the fairness notions of equal means and balanced residuals by modeling them as explicit constraints . Agarwal ( 2019 ) , Fitzsimons ( 2019 ) and Chzhen et al . ( 2020 ) studied 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . demographic parity . While Agarwal ( 2019 ) , Fitzsimons ( 2019 ) modeled it as an explicit constraint , Chzhen et al . ( 2020 ) included it implicitly in their proposed model . All the above work assume access to the sensitive attribute in the training samples and provide a framework which are inherently fair . Our work fundamentally differs from these work as we do not assume access to the sensitive attribute . Without knowing the sensitive attribute , it becomes difficult to ascertain bias , even for linear regression . In this work , we focus on identifying unfairly treated members/samples . This adds an extra layer of complexity to linear regression . We solve the linear regression problem while simultaneously solving a clustering problem where we identify two clusters – one which is positively biased and the other which is negatively biased . Table 1 shows a consolidated comparison of our work with the existing literature . Once one identifies bias ( positive or negative ) for each sample , one could perform debiasing which would lead to the fairness notion of equal means ( Calders , 2013 ) among the two groups ( See Figure 1 ) . It should be noted that identifying groups with positive or negative bias may not be same as identifying the sensitive attribute . The reason is that there may be multiple attributes that are highly correlated with the sensitive attribute . In such a situation , these correlated attributes can facilitate indirect discrimination even if the sensitive attribute is identified and removed . This is called the red-lining effect ( Calders , 2010 ) . Our model avoids this by directly identifying biased groups . While the standard algorithms solving the sparse/LASSO problem in this setting do provide an estimate of the regression parameter vector , they do not fit the model accurately as they fail to consider any fairness criteria in their formulations . It is natural then to think about including the hidden attribute in LASSO itself . However , this breaks the convexity of the loss function which makes the problem intractable by the standard LASSO algorithms . The resulting problem is a combinatorial version of sparse linear regression with added clustering according to the hidden attribute . In this work , we propose a novel technique to tackle the combinatorial LASSO problem with a hidden attribute and provide theoretical guarantees about the quality of the solution given a sufficient number of samples . Our method provably detects unfairness in the system . It should be noted that observing unfairness does not always imply that the designer of the system intended for such inequalities to arise . In such cases , our method acts as a check to detect and remove such unintended discrimination . While the current belief is that there is a trade-off between fairness and performance ( Corbett-Davies et al. , 2017 ; Kleinberg et al. , 2017 ; Pleiss et al. , 2017 ; Zliobaite , 2015 ; Zhao & Gordon , 2019 ) , our theoretical and experimental results show evidence on the contrary . Our theoretical results allow for a new understanding of fairness , as an “ enabler ” instead of as a “ constraint ” . Contribution . Broadly , we can categorize our contribution in the following points : • Defining the problem : We formulate a novel combinatorial version of sparse linear regression which takes fairness/bias into the consideration . The addition of clustering comes at no extra cost in terms of the performance . • Invex relaxation : Most of the current methods solve convex optimization problems as it makes the solution tractable . We propose a novel relaxation of the combinatorial problem and formally show that it is invex . To the best of our knowledge , this is the first invex relaxation for a combinatorial problem . • Theoretical Guarantees : Our method can detect bias in the system . In particular , our method recovers the exact hidden attributes for each sample and thus provides an exact measure of bias between two different groups . Our method solves linear regression and clustering simultaneously with theoretical guarantees . To that end , we recover the true clusters ( hidden attributes ) and a regression parameter vector which is correct up to the sign of entries with respect to the true parameter vector . On a more technical side , we provide a primal-dual witness construction for our invex problem and provide theoretical guarantees for recovery . The sample complexity of our method varies logarithmically with respect to dimension of the regression parameter vector . 2 Notation and Problem Definition . In this section , we collect all the notations used throughout the paper . We also formally introduce our novel problem . We consider a problem where we have a binary hidden attribute , and where fairness depends upon the hidden attribute . Let y P R be the response variable and X P Rd be the observed attributes . Let z˚ P t´1 , 1u be the hidden attribute and γ P Rą0 be the amount of bias due to the hidden attribute . The response y is generated using the following mechanism : y “ X⊺w˚ ` γz˚ ` e ( 1 ) where e is an independent noise term . For example , y could represent the market salary of a new candidate , X could represent the candidate ’ s skills and z could represent the population group the candidate belongs to ( e.g. , majority or minority ) . While the group of the candidate is not public knowledge , a bias associated with the candidate ’ s group may be present in the underlying data . For our problem , we will assume that an estimate of the bias γ P Rą0 is available . In practice , even a rough estimate ( ˘25 % ) of γ also works well ( See Appendix J ) . Let rds denote the set t1 , 2 , ¨ ¨ ¨ , du . We assume X P Rd to be a zero mean sub-Gaussian random vector ( Hsu et al. , 2012 ) with covariance Σ P Sd ` , i.e. , there exists a ρ ą 0 , such that for all α P Rd the following holds : Epexppα⊺Xqq ď expp } α } 2 2ρ 2 2 q . By simply taking αi “ r and αk “ 0 , @ k ‰ i , it follows that each entry of X is sub-Gaussian with parameter ρ . In particular , we will assume that @ i P rds , Xi ? Σii is a sub-Gaussian random variable with parameter σ ą 0 . It follows trivially that maxiPrds ? Σiiσ ď ρ . We will further assume that e is zero mean independent sub-Gaussian noise with variance σe . We assume that as the number of samples increases , the noise in the model gently decreases . We model this by taking σe “ k ? logn for some k ą 0 . Our setting works with a variety of random variables as the class of sub-Gaussian random variable includes for instance Gaussian variables , any bounded random variable ( e.g. , Bernoulli , multinomial , uniform ) , any random variable with strictly log-concave density , and any finite mixture of sub-Gaussian variables . Notice that for the group with z “ ` 1 , Epyq “ γ and for the group with z “ ´1 , Epyq “ ´γ . This means that after correctly identifying groups , one could perform debiasing by subtracting or adding γ for z “ ` 1 and ´1 respectively . After debiasing , the expected value of both groups would match ( and be equal to 0 ) . This complies with the notion of equal mean fairness proposed by Calders ( 2013 ) . The parameter vector w˚ P Rd is s-sparse , i.e. , at most s entries of w˚ are non-zero . We receive n i.i.d . samples of X P Rd and y P R and collect them in X P Rnˆd and y P Rn respectively . Thus , in the finite-sample setting , y “ Xw˚ ` γz˚ ` e , ( 2 ) where z˚ P t´1 , 1un and e P Rn both collect n independent realizations of z˚ P t´1 , 1u and e P R. Our goal is to recover w˚ and z˚ using the samples pX , yq . We denote a matrix A P Rpˆq restricted to the columns and rows in P Ď rps and Q Ď rqs respectively as APQ . Similarly , a vector v P Rp restricted to entries in P is denoted as vP . We use eigipAq to denote the i-th eigenvalue ( 1st being the smallest ) of matrix A . Similarly , eigmaxpAq denotes the maximum eigenvalue of matrix A . We use diagpAq to denote a vector containing the diagonal element of matrix A . By overriding the same notation , we use diagpvq to denote a diagonal matrix with its diagonal being the entries in vector v. We denote the inner product between two matrices A and B by xA , By , i.e. , xA , By “ tracepA⊺Bq , where trace denotes the trace of a matrix . The notation A ľ B denotes that A ´ B is a positive semidefinite matrix . Similarly , A ą B denotes that A ´ B is a positive definite matrix . For vectors , } v } p denotes the ℓp-vector norm of vector v P Rd , i.e. , } v } p “ p řd i “ 1 |vi|pq 1 p . If p “ 8 , then we define } v } 8 “ maxdi “ 1 |vi| . For matrices , } A } p denotes the induced ℓp-matrix norm for matrix A P Rpˆq . In particular , } A } 2 denotes the spectral norm of A and } A } 8 fi maxiPrps řq j “ 1 |Aij | . A function fpxq is of order Ωpgpxqq and denoted by fpxq “ Ωpgpxqq , if there exists a constant C ą 0 such that for big enough x0 , fpxq ě Cgpxq , @ x ě x0 . Similarly , a function fpxq is of order Opgpxqq and denoted by fpxq “ Opgpxqq , if there exists a constant C ą 0 such that for big enough x0 , fpxq ď Cgpxq , @ x ě x0 . For brevity in our notations , we treat any quantity independent of d , s and n as constant . Detailed proofs for lemmas and theorems are available in the supplementary material .

Summary: A problem of fair sparse linear regression is addressed in this paper, where bias depends upon a hidden binary random variable. This hidden r.v. can be thought of as a clustering assignment combining sparse regression. An optimization problem is formulated by the authors as a mixed-integer linear programming originally, then, a continuous relaxation is used to reformulate it into an invex optimization problem. The theoretical results mainly focus on the solution's uniqueness of the invex problem by verifying Karush-Kuhn-Tucker conditions and primal-dual certificates. In the experiments, the proposed method shows the ability to correctly recover the support of the regression parameter vector, and the correct clusters.