On Learning with Fairness Trade-Offs

1 INTRODUCTION . Reducing bias in an algorithm requires a number of steps ; first , specify the fairness definitions ( and thus fairness metrics ) that apply , second , encode them with a penalty so as to measure the discrepancy between outcomes and the perfect fairness scenario , third , choose the trade-off between the original statistical goal and fairness constraints , and , fourth , pick a method to debias ( at least partially ) the model . There are a number of obstacles to this programme . Indeed , debiasing may come at a cost ( Rodolfa et al . ( 2020 ) ) , but there are also theoretical reasons behind this claim . Jointly tackling multiple fairness definitions is usually difficult if not impossible without a decrease in model performance due to a series of impossibility theorems ( Kleinberg et al . ( 2017 ) ; Pleiss et al . ( 2017 ) ; Chouldechova ( 2017 ) ; Kim et al . ( 2020 ) ) . Note that these results apply not only to fairnessperformance but also to fairness-fairness trade-offs . Research has recently been undertaken to tackle specifically learnability and generalisation in fairness-related problems . In particular , Oneto et al . ( 2020 ) ; Oneto et al . ( 2020 ) derive provable probabilistic upper bounds in a fairness setting where one estimates an overall statistical loss and monitors disparities across categories . In Chen et al . ( 2018 ) , the authors have applied bias-variance decomposition techniques to disentangle the sources of unfairness in a classifier . While this is not directly related to this present work , the principle of risk decomposability turns out to be fruitful in both setups . Finally , in Agrawal et al . ( 2020 ) , a limiting distribution is established in the simple case where there is a partial requirement involving a fairness metric with two categories . Our contributions . The novelty of our work consists of the application of known techniques such as PAC inequalities and the central limit theorem to the problem of learning under fairness constraints : • First , we develop a PAC framework for fairness trade-offs , generalising results from Donini et al . ( 2018 ) ; Oneto et al . ( 2020 ) and tackling explicitly partial debiasing ( see Kim et al . ( 2020 ) ; Agrawal et al . ( 2020 ) ) . • Second , we show that certain properties of the probabilistic upper bounds lead to the need for sample-efficient bias mitigation techniques . In particular , we show that a new quantity , ZS , which can be understood as measure of sample concentration , plays a crucial role . We also build on Agrawal et al . ( 2020 ) and develop an asymptotic framework with a known limiting distribution ; this is useful as PAC bounds may not always be very sharp . • Third , in both frameworks , we put forward the decomposability of the learning risk , expressed as an upper bound in the PAC realm and a limiting variance in the asymptotic one . • Last , we illustrate our results on real-life data and show that generalisation is indeed difficult . 2 FAIRNESS METRICS AND LOSS FUNCTIONS . 2.1 SET-UP AND DEFINITIONS . In this article , we define s = 1 , · · · , C to be a categorical ( protected ) attribute , x ∈ X ⊂ Rd to be a set of non-protected features excluding s. y ∈ { − 1 , +1 } is a binary outcome variable and ŷ ∈ { − 1 , +1 } is an estimator for y . Finally , z = ( x , y , s ) . Note that ŷ is derived from a learner h ∈ H , whereH is a given functional space . Furthermore , we define S to be the in-sample ( training ) empirical distribution , and D to refer to the true distribution . Throughout this paper , we consider the case of an overall objective as a statistical performance objective , L0 , plus a fairness loss function φ . The trade-off is tuned by a hyper-parameter λ ≥ 0 . As usual , the aim is to minimise the overall objective , i.e. , minimise the statistical loss and the lack of fairness . This set-up is typical for partial debiasing and can be found in Kim et al . ( 2020 ) . Definition 1 . The overall objective or fairness trade-off is defined as LD ( h ) = L0D ( h ) + λφ ( L+D,1 ( h ) , L − D,1 ( h ) , · · · , L + D , C ( h ) , L − D , C ( h ) ) = L0D ( h ) + λφ ( L ± D ( h ) ) , where we have used the standard notations ( see Shalev-Shwartz & Ben-David ( 2014 ) ) : L0D ( h ) = E [ ` 0 ( h , z ) ] L+D , a ( h ) = E [ ` + ( h , z ) |s = a , y = 1 ] L−D , a ( h ) = E [ ` − ( h , z ) |s = a , y = −1 ] and the vector notation L±D ( h ) = [ L+D,1 ( h ) , L − D,1 ( h ) , · · · , L + D , C ( h ) , L − D , C ( h ) ] T , for some functions ` 0 , ` + and ` − . Remark 1 . In the whole paper , we assume that there exists a uniform bound B > 0 on all functions ` , i.e. , | ` ( h , z ) | ≤ B for all h and z . ` refers to any such function in what follows . Let us detail two particular cases . If λ = 0 , then the overall objective boils down to the usual risk minimisation problem . If ` 0 = 0 or λ→ +∞ , then the trade-off becomes a fairness constraint . We are now in a position to define the empirical counterpart by considering the empirical distribution rather than the true distribution . With the additional notationsN±a = { i ∈ { 1 , · · · , n } ; si = a , yi = ±1 } , |N±a | = n±a , the corresponding sample versions can be defined as L0S ( h ) = 1n ∑n i=1 ` 0 ( h , zi ) , L+S , a ( h ) = 1 n+a ∑ i∈N+a ` + ( h , zi ) and L−S , a ( h ) = 1 n−a ∑ i∈N−a ` − ( h , zi ) , where na = n+a + n − a , n = ∑C a=1 na , leading to LS ( h ) = L0S ( h ) + λφ ( L±S ( h ) ) ( 1 ) For the sake of simplicity , we will generally refer to φ ( L±T ( h ) ) as φT ( h ) for T = D , S . 2.2 FAIRNESS DEFINITIONS AND METRICS . One can find multiple technical definitions of fairness in the literature ; they have been reviewed in various papers ( Narayanan , 2018 ; Verma & Rubin , 2018 ; Berk et al. , 2018 ; Kim et al. , 2020 ; Agrawal et al. , 2020 ) . We offer an overview of the most frequent metrics in Table 1 . We note that all fairness metrics considered here can be expressed as an equality requirement on probabilities , hence on the expectation of indicator variables . These fall within our framework . However , this framework can also handle other types of functions . 2.3 FAIRNESS LOSS FUNCTIONS . In addition to picking one or multiple fairness definitions , we need to specify φ to measure the discrepancy from perfect fairness , the ideal case . A first approach , similar to the Calders-Verwer gap ( Calders & Verwer , 2010 ) , consists of looking at the discrepancy between two categories a and a′ : ∆L , ±T , a , a′ ( h ) = L ± T , a ( h ) − L ± T , a′ ( h ) , for T ∈ { D , S } . It is worth 0 under perfect fairness , but is asymmetric and thus requires defining an advantaged ( or benchmark ) category a . To avoid the issue of asymmetry , one can follow Oneto et al . ( 2020 ) , and define φ as the sum of all absolute discrepancies across categories : φ ( L±T ( h ) ) = ∑ a6=a′ { ∣∣∣L+T , a − L+T , a′ ∣∣∣+ ∣∣∣L−T , a − L−T , a′∣∣∣ } , ( 2 ) for T = D , S . Notice that , thanks to the reverse triangle inequality , the function φ is Lipschitz continuous . One could naturally weigh the various contributions to this fairness function and use another norm than L1 . Other functions involving ratios or relative differences , are also possible . Finally , in Kim et al . ( 2020 ) , the authors consider a convex combination of multiple loss functions φ ( 1 ) , · · · , φ ( M ) : φ ( L±T ( h ) ) = M∑ j=1 λjφ ( j ) ( L±T ( h ) ) . Note that if the φ ( j ) ’ s are Lipschitz-continuous ( with respect to the loss vector L±T ( h ) ) , with respective Lipschitz constant K ( j ) , then the overall loss φ is itself Lipschitz-continuous with constant Kφ = ∑M j=1 λjK ( j ) . 3 LEARNING FAIRNESS TRADE-OFFS . In this section , we consider the learning problem from different angles . We start by considering the statistical loss L0 and derive a bound based on the Rademacher complexity observed in each category . To do so , we borrow from the statistical learning toolbox and briefly review Rademacher complexities . We then move on to learning the fairness part per se , namely study the generalisation properties of φS ( h ) . Finally , we bring the two together and derive probabilistic upper bounds on fairness trade-off generalisation . 3.1 LEARNING THE STATISTICAL LOSS . Let us first focus on the statistical component of the loss , i.e. , L0T for T ∈ { S , D } . Much of this paper is based on theory of bounds derived from Rademacher complexities , introduced in Bartlett & Mendelson ( 2003 ) and surveyed in Boucheron , Stéphane et al . ( 2005 ) . Recent textbooks such as Shalev-Shwartz & Ben-David ( 2014 ) ; Mohri et al . ( 2012 ) provide excellent introductions to the topic . We start by recalling the definition of Rademacher complexities , as indicated in Boucheron , Stéphane et al . ( 2005 ) : Definition 2 . The Rademacher complexity ( or average ) of a function ` is given by R ( ` ◦ H ◦ S ) = E [ sup h∈H 1 n ∣∣∣∣∣ n∑ i=1 σi ` ( h , zi ) ∣∣∣∣∣ ] . ( 3 ) Note that definitions can slightly vary ( depending for instance on the presence or absence of the absolute value in the definition ) , but all downstream results are qualitatively similar , up to some multiplicative constants . To make this more concrete , we can consider a simple ( but usual ) case ( as describe in ShalevShwartz & Ben-David ( 2014 ) ) . We suppose that almost surely ‖x‖2 ≤ R. In addition , if we let H = { w ; ‖w‖2 ≤ R′ } and assume that the loss function ` is of the type ` ( w , ( x , y ) ) = ρ ( wTx , y ) , where |ρ| is bounded by B and is Lρ-Lipschitz , then , almost surely , R ( ` ◦ H ◦ S ) ≤ L ρRR′√ n . Rademacher complexities are key to establishing learnability bounds and lead to fundamental results in statistical learning theory . In particular , we will make use of a standard result ( see Boucheron , Stéphane et al . ( 2005 ) ) . Proposition 1 . With probability at least 1− δ , it holds sup h∈H |LD ( h ) − LS ( h ) | ≤ 2R ( ` ◦ H ◦ S ) +B √ 2 log 2δ n . ( 4 ) It will also be useful to introduce conditional Rademacher complexities that will be used throughout this paper . In particular , since we have a partition of the sample in terms of categories S = ⋃C a=1Na , we can consider the Rademacher complexity of that particular sample : R ( ` ◦ H ◦ Na ) = E [ sup h∈H 1 na ∣∣∣∣∣∑ i∈Na σi ` ( h , zi ) ∣∣∣∣∣ ] . ( 5 ) Lemma 1 . The sample Rademacher complexity can be bounded from above by the weighted sum of conditional Rademacher complexities : R ( ` ◦ H ◦ S ) ≤ C∑ a=1 na n R ( ` ◦ H ◦ Na ) , ( 6 ) where R ( ` ◦ H ◦ Na ) = E [ suph∈H 1 na ∣∣∑ i∈Na σi ` ( h , zi ) ∣∣ ] . Proof . This comes directly from the sub-additivity of the absolute value and the supremum . One can further interpret the non-negative gap ∑C a=1 na n R ( ` ◦ H ◦ Na ) − R ( ` ◦ H ◦ S ) as a diversification benefit amongst categories . Finally , this leads us to a proposition leveraging our results so far : Proposition 2 . With probability at least 1− δ , sup h∈H ∣∣L0D ( h ) − L0S ( h ) ∣∣ ≤ 2 C∑ a=1 na n R ( ` 0 ◦ H ◦ Na ) +B √ 2 log 2δ n . ( 7 )

Summary: The paper provides an analysis of fairness via regularizing a loss function with a suitably selected fairness measure / metric. In particular, the paper analysis this under a PAC learning setting and a sample limit (asymptotic) setting. A majority of the paper focuses on the PAC setting, where Rademacher complexity bounds are made and analysed. Some experiments are included.