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: The paper considers the task of sparse regression on a biased dataset where the bias depends on a hidden binary attribute. A new optimization problem is formulated which is then relaxed into an invex optimization problem. The authors then prove that their invex fair lasso formulation correctly recovers the hidden attributes and finds a regression parameter vector with the same support as the true parameter vector. This is done by constructing a pair of primal and dual variables that have the desired properties and fulfill the KKT conditions. In the experimental section, the theoretical results are validated on synthetic data. Moreover, the method is evaluated on two real world datasets where the hidden attributes are recovered in both cases.