{"title": "Noise-tolerant fair classification", "book": "Advances in Neural Information Processing Systems", "page_first": 294, "page_last": 306, "abstract": "Fairness-aware learning involves designing algorithms that do not discriminate with respect to some sensitive feature (e.g., race or gender). Existing work on the problem operates under the assumption that the sensitive feature available in one's training sample is perfectly reliable. This assumption may be violated in many real-world cases: for example, respondents to a survey may choose to conceal or obfuscate their group identity out of fear of potential discrimination. This poses the question of whether one can still learn fair classifiers given noisy sensitive features. In this paper, we answer the question in the affirmative: we show that if one measures fairness using the mean-difference score, and sensitive features are subject to noise from the mutually contaminated learning model, then owing to a simple identity we only need to change the desired fairness-tolerance. The requisite tolerance can be estimated by leveraging existing noise-rate estimators from the label noise literature. We finally show that our procedure is empirically effective on two case-studies involving sensitive feature censoring.", "full_text": "Noise-tolerant fair classi\ufb01cation\n\nAlexandre Lamy\u21e4\nColumbia University\n\na.lamy@columbia.edu\n\nZiyuan Zhong\u21e4\n\nColumbia University\n\nziyuan.zhong@columbia.edu\n\nNakul Verma\n\nColumbia University\n\nverma@cs.columbia.edu\n\nAditya Krishna Menon\n\nGoogle\n\nadityakmenon@google.com\n\nAbstract\n\nFairness-aware learning involves designing algorithms that do not discriminate\nwith respect to some sensitive feature (e.g., race or gender). Existing work on the\nproblem operates under the assumption that the sensitive feature available in one\u2019s\ntraining sample is perfectly reliable. This assumption may be violated in many\nreal-world cases: for example, respondents to a survey may choose to conceal or\nobfuscate their group identity out of fear of potential discrimination. This poses\nthe question of whether one can still learn fair classi\ufb01ers given noisy sensitive\nfeatures. In this paper, we answer the question in the af\ufb01rmative: we show that\nif one measures fairness using the mean-difference score, and sensitive features\nare subject to noise from the mutually contaminated learning model, then owing\nto a simple identity we only need to change the desired fairness-tolerance. The\nrequisite tolerance can be estimated by leveraging existing noise-rate estimators\nfrom the label noise literature. We \ufb01nally show that our procedure is empirically\neffective on two case-studies involving sensitive feature censoring.\n\n1\n\nIntroduction\n\nClassi\ufb01cation is concerned with maximally discriminating between a number of pre-de\ufb01ned groups.\nFairness-aware classi\ufb01cation concerns the analysis and design of classi\ufb01ers that do not discriminate\nwith respect to some sensitive feature (e.g., race, gender, age, income). Recently, much progress\nhas been made on devising appropriate measures of fairness (Calders et al., 2009; Dwork et al.,\n2011; Feldman, 2015; Hardt et al., 2016; Zafar et al., 2017b,a; Kusner et al., 2017; Kim et al., 2018;\nSpeicher et al., 2018; Heidari et al., 2019), and means of achieving them (Zemel et al., 2013; Zafar\net al., 2017b; Calmon et al., 2017; Dwork et al., 2018; Agarwal et al., 2018; Donini et al., 2018;\nCotter et al., 2018; Williamson & Menon, 2019; Mohri et al., 2019).\nTypically, fairness is achieved by adding constraints which depend on the sensitive feature, and then\ncorrecting one\u2019s learning procedure to achieve these fairness constraints. For example, suppose the\ndata comprises of pairs of individuals and their loan repay status, and the sensitive feature is gender.\nThen, we may add a constraint that we should predict equal loan repayment for both men and women\n(see \u00a73.2 for a more precise statement). However, this and similar approaches assume that we are able\nto correctly measure or obtain the sensitive feature. In many real-world cases, one may only observe\nnoisy versions of the sensitive feature. For example, survey respondents may choose to conceal or\nobfuscate their group identity out of concerns of potential mistreatment or outright discrimination.\nOne is then brought to ask whether fair classi\ufb01cation in the presence of such noisy sensitive features\nis still possible. Indeed, if the noise is high enough and all original information about the sensitive\n\n\u21e4Equal contribution\n\n33rd Conference on Neural Information Processing Systems (NeurIPS 2019), Vancouver, Canada.\n\n\ffeatures is lost, then it is as if the sensitive feature was not provided. Standard learners can then be\nunfair on such data (Dwork et al., 2011; Pedreshi et al., 2008). Recently, Hashimoto et al. (2018)\nshowed that progress is possible, albeit for speci\ufb01c fairness measures. The question of what can be\ndone under a smaller amount of noise is thus both interesting and non-trivial.\nIn this paper, we consider two practical scenarios where we may only observe noisy sensitive features:\n(1) suppose we are releasing data involving human participants. Even if noise-free sensitive features\nare available, we may wish to add noise so as to obfuscate sensitive attributes, and thus protect\nparticipant data from potential misuse. Thus, being able to learn fair classi\ufb01ers under sensitive\nfeature noise is a way to achieve both privacy and fairness.\n\n(2) suppose we wish to analyse data where the presence of the sensitive feature is only known for\na subset of individuals, while for others the feature value is unknown. For example, patients\n\ufb01lling out a form may feel comfortable disclosing that they do not have a pre-existing medical\ncondition; however, some who do have this condition may wish to refrain from responding. This\ncan be seen as a variant of the positive and unlabelled (PU) setting (Denis, 1998), where the\nsensitive feature is present (positive) for some individuals, but absent (unlabelled) for others.\n\nBy considering popular measures of fairness and a general model of noise, we show that fair\nclassi\ufb01cation is possible under many settings, including the above. Our precise contributions are:\n(C1) we show that if the sensitive features are subject to noise as per the mutually contaminated\nlearning model (Scott et al., 2013a), and one measures fairness using the mean-difference\nscore (Calders & Verwer, 2010), then a simple identity (Theorem 2) yields that we only need to\nchange the desired fairness-tolerance. The requisite tolerance can be estimated by leveraging\nexisting noise-rate estimators from the label noise literature, yielding a reduction (Algorithm 1)\nto regular noiseless fair classi\ufb01cation.\n\n(C2) we show that our procedure is empirically effective on both case-studies mentioned above.\nIn what follows, we review the existing literature on learning fair and noise-tolerant classi\ufb01ers in \u00a72,\nand introduce the novel problem formulation of noise-tolerant fair learning in \u00a73. We then detail how\nto address this problem in \u00a74, and empirically con\ufb01rm the ef\ufb01cacy of our approach in \u00a75.\n\n2 Background and related work\n\nWe review relevant literature on fair and noise-tolerant machine learning.\n\n2.1 Fair machine learning\n\nAlgorithmic fairness has gained signi\ufb01cant attention recently because of the undesirable social impact\ncaused by bias in machine learning algorithms (Angwin et al., 2016; Buolamwini & Gebru, 2018;\nLahoti et al., 2018). There are two central objectives: designing appropriate application-speci\ufb01c\nfairness criterion, and developing predictors that respect the chosen fairness conditions.\nBroadly, fairness objectives can be categorised into individual- and group-level fairness. Individual-\nlevel fairness (Dwork et al., 2011; Kusner et al., 2017; Kim et al., 2018) requires the treatment of\n\u201csimilar\u201d individuals to be similar. Group-level fairness asks the treatment of the groups divided based\non some sensitive attributes (e.g., gender, race) to be similar. Popular notions of group-level fairness\ninclude demographic parity (Calders et al., 2009) and equality of opportunity (Hardt et al., 2016); see\n\u00a73.2 for formal de\ufb01nitions.\nGroup-level fairness criteria have been the subject of signi\ufb01cant algorithmic design and analysis, and\nare achieved in three possible ways:\n\u2013 pre-processing methods (Zemel et al., 2013; Louizos et al., 2015; Lum & Johndrow, 2016;\nJohndrow & Lum, 2017; Calmon et al., 2017; del Barrio et al., 2018; Adler et al., 2018), which\nusually \ufb01nd a new representation of the data where the bias with respect to the sensitive feature is\nexplicitly removed.\n\n\u2013 methods enforcing fairness during training (Calders et al., 2009; Woodworth et al., 2017; Zafar\net al., 2017b; Agarwal et al., 2018), which usually add a constraint that is a proxy of the fairness\ncriteria or add a regularization term to penalise fairness violation.\n\n2\n\n\f\u2013 post-processing methods (Feldman, 2015; Hardt et al., 2016), which usually apply a thresholding\n\nfunction to make the prediction satisfying the chosen fairness notion across groups.\n\n2.2 Noise-tolerant classi\ufb01cation\nDesigning noise-tolerant classi\ufb01ers is a classic topic of study, concerned with the setting where one\u2019s\ntraining labels are corrupted in some manner. Typically, works in this area postulate a particular\nmodel of label noise, and study the viability of learning under this model. Class-conditional noise\n(CCN) (Angluin & Laird, 1988) is one such effective noise model. Here, samples from each class\nhave their labels \ufb02ipped with some constant (but class-speci\ufb01c) probability. Algorithms that deal with\nCCN corruption have been well studied (Natarajan et al., 2013; Liu & Tao, 2016; Northcutt et al.,\n2017). These methods typically \ufb01rst estimate the noise rates, which are then used for prediction. A\nspecial case of CCN learning is learning from positive and unlabelled data (PU learning) (Elkan &\nNoto, 2008), where in lieu of explicit negative samples, one has a pool of unlabelled data.\nOur interest in this paper will be the mutually contaminated (MC) learning noise model (Scott et al.,\n2013a). This model (described in detail in \u00a73.3) captures both CCN and PU learning as special\ncases (Scott et al., 2013b; Menon et al., 2015), as well as other interesting noise models.\n\n3 Background and notation\n\nWe recall the settings of standard and fairness-aware binary classi\ufb01cation2, and establish notation.\nOur notation is summarized in Table 1.\n\n3.1 Standard binary classi\ufb01cation\nBinary classi\ufb01cation concerns predicting the label or target feature Y 2{ 0, 1} that best corresponds\nto a given instance X 2X . Formally, suppose D is a distribution over (instance, target feature) pairs\nfrom X\u21e5{ 0, 1}. Let f : X! R be a score function, and F\u21e2 RX be a user-de\ufb01ned class of such\nscore functions. Finally, let ` : R\u21e5{ 0, 1}! R+ be a loss function measuring the disagreement\nbetween a given score and binary label. The goal of binary classi\ufb01cation is to minimise\n(1)\n\nLD(f ) := E(X,Y )\u21e0D[`(f (X), Y )].\n\n3.2 Fairness-aware classi\ufb01cation\nIn fairness-aware classi\ufb01cation, the goal of accurately predicting the target feature Y remains. How-\never, there is an additional sensitive feature A 2{ 0, 1} upon which we do not wish to discriminate.\nIntuitively, some user-de\ufb01ned fairness loss should be roughly the same regardless of A.\nFormally, suppose D is a distribution over (instance, sensitive feature, target feature) triplets from\nX\u21e5{ 0, 1}\u21e5{ 0, 1}. The goal of fairness-aware binary classi\ufb01cation is to \ufb01nd3\n\nf\u21e4 := arg min\n\nLD(f ), such that \u21e4D(f ) \uf8ff \u2327\n\nf2F\n\nLD(f ) := E(X,A,Y )\u21e0D[`(f (X), Y )],\n\n(2)\n\nfor user-speci\ufb01ed fairness tolerance \u2327 0, and fairness constraint \u21e4D : F! R+. Such constrained\noptimisation problems can be solved in various ways, e.g., convex relaxations (Donini et al., 2018),\nalternating minimisation (Zafar et al., 2017b; Cotter et al., 2018), or linearisation (Hardt et al., 2016).\nA number of fairness constraints \u21e4D(\u00b7) have been proposed in the literature. We focus on two\nimportant and speci\ufb01c choices in this paper, inspired by Donini et al. (2018):\n(3)\n(4)\n\n\u21e4DP\n\u21e4EO\n\n(f ) \u00afLD1,\u00b7\n\nD (f ) := \u00afLD0,\u00b7\n(f )\nD (f ) := \u00afLD0,1(f ) \u00afLD1,1(f ) ,\n\n2For simplicity, we consider the setting of binary target and sensitive features. However, our derivation and\n\nmethod can be easily extended to the multi-class setting.\n\n3Here, f is assumed to not be allowed to use A at test time, which is a common legal restriction (Lipton\n\net al., 2018). Of course, A can be used at training time to \ufb01nd an f which satis\ufb01es the constraint.\n\n3\n\n\fTable 1: Glossary of commonly used symbols\n\nSymbol Meaning\nSymbol Meaning\ninstance\nDcorr\nX\nsensitive feature\nf\nA\ntarget feature\n`\nY\ndistribution P(X, A, Y )\nLD\nD\n\u00af`\ndistribution P(X, A, Y |A = a)\nDa,\u00b7\n\u00afLD\ndistribution P(X, A, Y |Y = y)\nD\u00b7,y\ndistribution P(X, A, Y |A = a, Y = y)\u21e4 D\nDa,y\n\ncorrupted distribution D\nscore function f : X! R\naccuracy loss ` : R\u21e5{ 0, 1}! R+\nexpected accuracy loss on D\nfairness loss \u00af` : R\u21e5{ 0, 1}! R+\nexpected fairness loss on D\nfairness constraint\n\nwhere we denote by Da,\u00b7, D\u00b7,y, and Da,y the distributions over X\u21e5{ 0, 1}\u21e5{ 0, 1} given by\nD|A=a, D|Y =y, and D|A=a,Y =y and \u00af` : R\u21e5{ 0, 1}! R+ is the user-de\ufb01ned fairness loss with\ncorresponding \u00afLD(f ) := E(X,A,Y )\u21e0D[\u00af`(f (X), Y )]. Intuitively, these measure the difference in the\naverage of the fairness loss incurred among the instances with and without the sensitive feature.\nConcretely, if \u00af` is taken to be \u00af`(s, y) = 1[sign(s) 6= 1] and the 0-1 loss \u00af`(s, y) = 1[sign(s) 6= y]\nrespectively, then for \u2327 = 0, (3) and (4) correspond to the demographic parity (Dwork et al., 2011)\nand equality of opportunity (Hardt et al., 2016) constraints. Thus, we denote these two relaxed\nfairness measures disparity of demographic parity (DDP) and disparity of equality of opportunity\n(DEO). These quantities are also known as the mean difference score in Calders & Verwer (2010).4\n\n3.3 Mutually contaminated learning\nIn the framework of learning from mutually contaminated distributions (MC learning) (Scott et al.,\n2013b), instead of observing samples from the \u201ctrue\u201d (or \u201cclean\u201d) joint distribution D, one ob-\nserves samples from a corrupted distribution Dcorr. The corruption is such that the observed\nclass-conditional distributions are mixtures of their true counterparts. More precisely, let Dy denote\nthe conditional distribution for label y. Then, one assumes that\n\nD1,corr = (1 \u21b5) \u00b7 D1 + \u21b5 \u00b7 D0\nD0,corr = \u00b7 D1 + (1 ) \u00b7 D0,\n\n(5)\n\nwhere \u21b5, 2 (0, 1) are (typically unknown) noise parameters with \u21b5+< 1.5 Further, the corrupted\nbase rate \u21e1corr := P[Ycorr = 1] may be arbitrary. The MC learning framework subsumes CCN and\nPU learning (Scott et al., 2013b; Menon et al., 2015), which are prominent noise models that have\nseen sustained study in recent years (Jain et al., 2017; Kiryo et al., 2017; van Rooyen & Williamson,\n2018; Katz-Samuels et al., 2019; Charoenphakdee et al., 2019).\n\n4 Fairness under sensitive attribute noise\n\nThe standard fairness-aware learning problem assumes we have access to the true sensitive attribute,\nso that we can both measure and control our classi\ufb01er\u2019s unfairness as measured by, e.g., Equation 3.\nNow suppose that rather than being given the sensitive attribute, we get a noisy version of it. We will\nshow that the fairness constraint on the clean distribution is equivalent to a scaled constraint on the\nnoisy distribution. This gives a simple reduction from fair machine learning in the presence of noise\nto the regular fair machine learning, which can be done in a variety of ways as discussed in \u00a72.1.\n\n4.1 Sensitive attribute noise model\nAs previously discussed, we use MC learning as our noise model, as this captures both CCN and PU\nlearning as special cases; hence, we automatically obtain results for both these interesting settings.\n4Keeping \u00af` generic allows us to capture a range of group fairness de\ufb01nitions, not just demographic parity and\nequality of opportunity; e.g., disparate mistreatment (Zafar et al., 2017b) corresponds to using the 0-1 loss and\nD , and equalized odds can be captured by simply adding another constraint for Y = 0 along with \u21e4EO\nD .\n\u21e4DP\n\n5The constraint imposes no loss of generality: when \u21b5 + > 1, we can simply \ufb02ip the two labels and apply\nour theorem. When \u21b5 + = 1, all information about the sensitive attribute is lost. This pathological case is\nequivalent to not measuring the sensitive attribute at all.\n\n4\n\n\fOur speci\ufb01c formulation of MC learning noise on the sensitive feature is as follows. Recall from\n\u00a73.2 that D is a distribution over X \u21e5{ 0, 1}\u21e5{ 0, 1}. Following (5), for unknown noise parameters\n\u21b5, 2 (0, 1) with \u21b5 + < 1, we assume that the corrupted class-conditional distributions are:\n\nD1,\u00b7,corr = (1 \u21b5) \u00b7 D1,\u00b7 + \u21b5 \u00b7 D0,\u00b7\nD0,\u00b7,corr = \u00b7 D1,\u00b7 + (1 ) \u00b7 D0,\u00b7,\n\nand that the corrupted base rate is \u21e1a,corr (we write the original base rate, P(X,A,Y )\u21e0D[A = 1] as \u21e1a).\nThat is, the distribution over (instance, label) pairs for the group with A = 1, i.e. P(X, Y | A = 1),\nis assumed to be mixed with the distribution for the group with A = 0, and vice-versa.\nNow, when interested in the EO constraint, it can be simpler to assume that the noise instead satis\ufb01es\n\n(6)\n\n(7)\n\nD1,1,corr = (1 \u21b50) \u00b7 D1,1 + \u21b50 \u00b7 D0,1\nD0,1,corr = 0 \u00b7 D1,1 + (1 0) \u00b7 D0,1,\n\nfor noise parameters \u21b50, 0 2 (0, 1). As shown by the following, this is not a different assumption.\nLemma 1. Suppose there is noise in the sensitive attribute only, as given in Equation (6). Then, there\nexists constants \u21b50, 0 such that Equation (7) holds.\n\nAlthough the lemma gives a way to calculate \u21b50, 0 from \u21b5, , in practice it may be useful to consider\n(7) independently. Indeed, when one is interested in the EO constraints we will show below that only\nknowledge of \u21b50, 0 is required. It is often much easier to estimate \u21b50, 0 directly (which can be done\nin the same way as estimating \u21b5, simply by considering D\u00b7,1,corr rather than Dcorr).\n4.2 Fairness constraints under MC learning\nWe now show that the previously introduced fairness constraints for demographic parity and equality\nof opportunity are automatically robust to MC learning noise in A.\nTheorem 2. Assume that we have noise as per Equation (6). Then, for any \u2327> 0 and f : X ! R,\n\n\u21e4DP\nD (f ) \uf8ff \u2327 () \u21e4DP\n\u21e4EO\nD\u00b7,1(f ) \uf8ff \u2327 () \u21e4EO\n\nDcorr(f ) \uf8ff \u2327 \u00b7 (1 \u21b5 )\nDcorr,\u00b7,1(f ) \uf8ff \u2327 \u00b7 (1 \u21b50 0),\n\nwhere \u21b50 and 0 are as per Equation (7) and Lemma 1.\n\nThe above can be seen as a consequence of the immunity of the balanced error (Chan & Stolfo, 1998;\nBrodersen et al., 2010; Menon et al., 2013) to corruption under the MC model. Speci\ufb01cally, consider\na distribution D over an input space Z and label space W = {0, 1}. De\ufb01ne\n\nBD := EZ|W =0[h0(Z)] + EZ|W =1[h1(Z)]\n\nBDcorr = (1 \u21b5 ) \u00b7 BD,\n\nfor functions h0, h1 : Z ! R. Then, if for every z 2 R h0(z) + h1(z) = 0, we have (van Rooyen,\n2015, Theorem 4.16), (Blum & Mitchell, 1998; Zhang & Lee, 2008; Menon et al., 2015)\n(8)\nwhere Dcorr refers to a corrupted version of D under MC learning with noise parameters \u21b5, . That\nis, the effect of MC noise on BD is simply to perform a scaling. Observe that BD = \u00afLD(f ) if we set\nZ to X \u21e5 Y , W to the sensitive feature A, and h0((x, y)) = +\u00af`(y, f (x)), h1((x, y)) = \u00af`(y, f (x)).\nThus, (8) implies \u00afLD(f ) = (1 \u21b5 ) \u00b7 \u00afLDcorr(f ), and thus Theorem 2.\n4.3 Algorithmic implications\nTheorem 2 has an important algorithmic implication. Suppose we pick a fairness constraint \u21e4D and\nseek to solve Equation 2 for a given tolerance \u2327 0. Then, given samples from Dcorr, it suf\ufb01ces to\nsimply change the tolerance to \u23270 = \u2327 \u00b7 (1 \u21b5 ).\nUnsurprisingly, \u23270 depends on the noise parameters \u21b5, . In practice, these will be unknown; however,\nthere have been several algorithms proposed to estimate these from noisy data alone (Scott et al.,\n2013b; Menon et al., 2015; Liu & Tao, 2016; Ramaswamy et al., 2016; Northcutt et al., 2017). Thus,\nwe may use these to construct estimates of \u21b5, , and plug these in to construct an estimate of \u23270.\n\n5\n\n\fIn sum, we may tackle fair classi\ufb01cation in the presence of noisy A by suitably combining any\nexisting fair classi\ufb01cation method (that takes in a parameter \u2327 that is proportional to mean-difference\nscore of some fairness measures), and any existing noise estimation procedure. This is summarised in\nAlgorithm 1. Here, FairAlg is any existing fairness-aware classi\ufb01cation method that solves Equation 2,\nand NoiseEst is any noise estimation method that estimates \u21b5, .\n\nconstraint \u21e4(\u00b7), fair classi\ufb01cation algorithm FairAlg, noise estimation algorithm NoiseEst\n\nAlgorithm 1 Reduction-based algorithm for fair classi\ufb01cation given noisy A.\nInput: Training set S = {(xi, yi, ai)}n\nOutput: Fair classi\ufb01er f\u21e4 2F\n1: \u02c6\u21b5, \u02c6 NoiseEst(S)\n2: \u23270 (1 \u02c6\u21b5 \u02c6) \u00b7 \u2327\n3: return FairAlg(S,F, \u21e4,\u2327 0)\n\ni=1, scorer class F, fairness tolerance \u2327 0, fairness\n\n4.4 Noise rate and sample complexity\nSo far, we have shown that at a distribution level, fairness with respect to the noisy sensitive attribute is\nequivalent to fairness with respect to the real sensitive attribute. However, from a sample complexity\nperspective, a higher noise rate will require a larger number of samples for the empirical fairness to\ngeneralize well, i.e., guarantee fairness at a distribution level. A concurrent work by Mozannar et al.\n(2019) derives precise sample complexity bounds and makes this relationship explicit.\n\n1\n\nexp (\u270f)+1 to the sensitive attribute.\n\n4.5 Connection to privacy and fairness\nWhile Algorithm 1 gives a way of achieving fair classi\ufb01cation on an already noisy dataset such as\nthe use case described in example (2) of \u00a71, it can also be used to simultaneously achieve fairness\nand privacy. As described in example (1) of \u00a71, the very nature of the sensitive attribute makes it\nlikely that even if noiseless sensitive attributes are available, one might want to add noise to guarantee\nsome form of privacy. Note that simply removing the feature does not suf\ufb01ce, because it would\nmake dif\ufb01cult the task of developing fairness-aware classi\ufb01ers for the dataset (Gupta et al., 2018).\nFormally, we can give the following privacy guarantee by adding CCN noise to the sensitive attribute.\nLemma 3. To achieve (\u270f, = 0) differential privacy on the sensitive attribute we can add CCN noise\nwith \u21e2+ = \u21e2 = \u21e2 \nThus, if a desired level of differential privacy is required before releasing a dataset, one could simply\nadd the required amount of CCN noise to the sensitive attributes, publish this modi\ufb01ed dataset as\nwell as the noise level, and researchers could use Algorithm 1 (without even needing to estimate the\nnoise rate) to do fair classi\ufb01cation as usual.\nThere is previous work that tries to preserve privacy of individuals\u2019 sensitive attributes while learning a\nfair classi\ufb01er. Kilbertus et al. (2018) employs the cryptographic tool of secure multiparty computation\n(MPC) to try to achieve this goal. However, as noted by Jagielski et al. (2018), the individual\ninformation that MPC tries to protect can still be inferred from the learned model. Further, the method\nof Kilbertus et al. (2018) is limited to using demographic parity as the fairness criteria.\nA more recent work of Jagielski et al. (2018) explored preserving differential privacy (Dwork, 2006)\nwhile maintaining fairness constraints. The authors proposed two methods: one adds Laplace noise\nto training data and apply the post-processing method in Hardt et al. (2016), while the other modi\ufb01es\nthe method in Agarwal et al. (2018) using the exponential mechanism as well as Laplace noise. Our\nwork differs from them in three major ways:\n\n(1) our work allows for fair classi\ufb01cation to be done using any in-process fairness-aware\nclassi\ufb01er that allows user to specify desired fairness level. On the other hand, the \ufb01rst\nmethod of Jagielski et al. (2018) requires the use of a post-processing algorithm (which\ngenerally have worse trade-offs than in-processing algorithms Agarwal et al. (2018)), while\nthe second method requires the use of a single particular classi\ufb01er.\n\n(2) our focus is on designing fair-classi\ufb01ers with noise-corrupted sensitive attributes; by contrast,\nthe main concern in Jagielski et al. (2018) is achieving differential privacyand thus they do\nnot discuss how to deal with noise that is already present in the dataset.\n\n6\n\n\f(3) our method is shown to work for a large class of different fairness de\ufb01nitions.\n\nFinally, a concurrent work of Mozannar et al. (2019) builds upon our method for the problem of\npreserving privacy of the sensitive attribute. The authors use a randomized response procedure on\nthe sensitive attribute values, followed by a two-step procedure to train a fair classi\ufb01er using the\nprocessed data. Theoretically, their method improves upon the sample complexity of our method and\nextends our privacy result to the case of non-binary groups. However, their method solely focuses on\npreserving privacy rather than the general problem of sensitive attribute noise.\n\n5 Experiments\n\nWe demonstrate that it is viable to learn fair classi\ufb01ers given noisy sensitive features.6 As our\nunderlying fairness-aware classi\ufb01er, we use a modi\ufb01ed version of the classi\ufb01er implemented in\nAgarwal et al. (2018) with the DDP and DEO constraints which, as discussed in \u00a73.2, are special\ncases of our more general constraints (3) and (4). The classi\ufb01er\u2019s original constraints can also be\nshown to be noise-invariant but in a slightly different way (see Appendix C for a discussion). An\nadvantage of this classi\ufb01er is that it is shown to reach levels of fairness violation that are very close to\nthe desired level (\u2327), i.e., for small enough values of \u2327 it will reach the constraint boundary.\nWhile we had to choose a particular classi\ufb01er, our method can be used before using any downstream\nfair classi\ufb01er as long as it can take in a parameter \u2327 that controls the strictness of the fairness constraint\nand that its constraints are special cases of our very general constraints (3) and (4).\n\n5.1 Noise setting\nOur case studies focus on two common special cases of MC learning: CCN and PU learning. Under\nCCN noise the sensitive feature\u2019s value is randomly \ufb02ipped with probability \u21e2+ if its value was 1, or\nwith probability \u21e2 if its value was 0. As shown in Menon et al. (2015, Appendix C), CCN noise is a\nspecial case of MC learning. For PU learning we consider the censoring setting (Elkan & Noto, 2008)\nwhich is a special case of CCN learning where one of \u21e2+ and \u21e2 is 0. While our results also apply to\nthe case-controlled setting of PU learning (Ward et al., 2009), the former setting is more natural in\nour context. Note that from \u21e2+ and \u21e2 one can obtain \u21b5 and as described in Menon et al. (2015).\n\n5.2 Benchmarks\nFor each case study, we evaluate our method (termed cor scale); recall this scales the input parameter\n\u2327 using Theorem 2 and the values of \u21e2+ and \u21e2, and then uses the fair classi\ufb01er to perform\nclassi\ufb01cation. We compare our method with three different baselines. The \ufb01rst two trivial baselines\nare applying the fair classi\ufb01er directly on the non-corrupted data (termed nocor) and on the corrupted\ndata (termed cor). While the \ufb01rst baseline is clearly the ideal, it won\u2019t be possible when only the\ncorrupted data is available. The second baseline should show that there is indeed an empirical need to\ndeal with the noise in some way and that it cannot simply be ignored.\nThe third, non-trivial, baseline (termed denoise) is to \ufb01rst denoise A and then apply the fair classi\ufb01er\non the denoised distribution. This denoising is done by applying the RankPrune method in Northcutt\net al. (2017). Note that we provide RankPrune with the same known values of \u21e2+ and \u21e2 that we\nuse to apply our scaling so this is a fair comparison to our method. Compared to denoise, we do not\nexplicitly infer individual sensitive feature values; thus, our method does not compromise privacy.\nFor both case studies, we study the relationship between the input parameter \u2327 and the testing error\nand fairness violation. For simplicity, we only consider the DP constraint.\n\n5.3 Case study: privacy preservation\nIn this case study, we look at COMPAS, a dataset from Propublica (Angwin et al., 2016) that is widely\nused in the study of fair algorithms. Given various features about convicted individuals, the task\nis to predict recidivism and the sensitive attribute is race. The data comprises 7918 examples and\n10 features. In our experiment, we assume that to preserve differential privacy, CCN noise with\n\u21e2+ = \u21e2 = 0.15 is added to the sensitive attribute. As per Lemma 3, this guarantees (\u270f, = 0)\n\n6Source code is available at https://github.com/AIasd/noise_fairlearn.\n\n7\n\n\f(a) COMPAS dataset (privacy case study).\n\n(b) law school dataset (PU learning case study).\n\nFigure 1: Relationship between input fairness tolerance \u2327 versus DP fairness violation (left panels), and versus\nerror (right panels). Our method (cor scale) achieves approximately the ideal fairness violation (indicated\nby the gray dashed line in the left panels), with only a mild degradation in accuracy compared to training on\nthe uncorrupted data (indicated by the nocor method). Baselines that perform no noise-correction (cor) and\nexplicitly denoise the data (denoise) offer suboptimal tradeoffs by comparison; for example, the former achieves\nslightly lower error rates, but does so at the expense of greater fairness violation.\n\ndifferential privacy with \u270f = 1.73. We assume that the noise level \u21e2 is released with the dataset (and\nis thus known). We performed fair classi\ufb01cation on this noisy data using our method and compare the\nresults to the three benchmarks described above.\nFigure 1a shows the average result over three runs each with a random 80-20 training-testing split.\n(Note that fairness violations and errors are calculated with respect to the true uncorrupted features.)\nWe draw two key insights from this graph:\n\n(i) in terms of fairness violation, our method (cor scale) approximately achieves the desired\nfairness tolerance (shown by the gray dashed line). This is both expected and ideal, and it\nmatches what happens when there is no noise (nocor). By contrast, the na\u00efve method cor\nstrongly violates the fairness constraint.\n\n(ii) in terms of accuracy, our method only suffers mildly compared with the ideal noiseless method\n(nocor); some degradation is expected as noise will lead to some loss of information. By\ncontrast, denoise sacri\ufb01ces much more predictive accuracy than our method.\n\nIn light of both the above, our method is seen to achieve the best overall tradeoff between fairness\nand accuracy. Experimental results with EO constraints, and other commonly studied datasets in\nthe fairness literature (adult, german), show similar trends as in Figure 1a, and are included in\nAppendix D for completeness.\n\n5.4 Case study: PU learning\n\nIn this case study, we consider the dataset law school, which is a subset of the original dataset from\nLSAC (Wightman, 1998). In this dataset, one is provided with information about various individuals\n(grades, part time/full time status, age, etc.) and must determine whether or not the individual passed\nthe bar exam. The sensitive feature is race; we only consider black and white. After prepossessing\n\n8\n\n\fFigure 2: Relationship between the estimated noise level \u02c6\u21e2 and fairness violation/error on the law school\ndataset using DP constraint (testing curves), with \u02c6\u21e2+ = 0 and \u2327 = 0.2. Our method (cor scale) is not overly\nsensitive to imperfect estimates of the noise rate, evidenced by its fairness violation and accuracy closely tracking\nthat of training on the uncorrupted data (nocor) as \u02c6\u21e2 is varied. That is, red curve in the left plot closely tracks\nthe yellow reference curve. By contrast, the baseline that explicitly denoises the data (denoise) deviates strongly\nfrom nocor, and is sensitive to small changes in \u02c6\u21e2. This illustrates that our method performs well even when\nnoise rates must be estimated.\n\nthe data by removing instances that had missing values and those belonging to other ethnicity groups\n(neither black nor white) we were left with 3738 examples each with 11 features.\nWhile the data ostensibly provides the true values of the sensitive attribute, one may imagine having\naccess to only PU information. Indeed, when the data is collected one could imagine that individuals\nfrom the minority group would have a much greater incentive to conceal their group membership due\nto fear of discrimination. Thus, any individual identi\ufb01ed as belonging to the majority group could\nbe assumed to have been correctly identi\ufb01ed (and would be part of the positive instances). On the\nother hand, no de\ufb01nitive conclusions could be drawn about individuals identi\ufb01ed as belonging to the\nminority group (these would therefore be part of the unlabelled instances).\nTo model a PU learning scenario, we added CCN noise to the dataset with \u21e2+ = 0 and \u21e2 = 0.2. We\ninitially assume that the noise rate is known. Figure 1b shows the average result over three runs under\nthis setting each with a random 80-20 training-testing split. We draw the same conclusion as before:\nour method achieves the highest accuracy while respecting the speci\ufb01ed fairness constraint.\nUnlike in the privacy case, the noise rate in the PU learning scenario is usually unknown in practice,\nand must be estimated. Such estimates will inevitably be approximate. We thus evaluate the impact of\nthe error of the noise rate estimate on all methods. In Figure 2, we consider a PU scenario where we\nonly have access to an estimate \u02c6\u21e2 of the negative noise rate, whose true value is \u21e2 = 0.2. Figure 2\nshows the impact of different values of \u02c6\u21e2 on the fairness violation and error. We see that that as\nlong as this estimate is reasonably accurate, our method performs the best in terms of being closest to\nthe case of running the fair algorithm on uncorrupted data.\nIn sum, these results are consistent with our derivation and show that our method cor scale can\nachieve the desired degree of fairness while minimising loss of accuracy. Appendix E includes results\nfor different settings of \u2327, noise level, and on other datasets showing similar trends.\n\n6 Conclusion and future work\n\nIn this paper, we showed both theoretically and empirically that even under the very general MC\nlearning noise model (Scott et al., 2013a) on the sensitive feature, fairness can still be preserved by\nscaling the input unfairness tolerance parameter \u2327. In future work, it would be interesting to consider\nthe case of categorical sensitive attributes (as applicable, e.g., for race), and the more challenging case\nof instance-dependent noise (Awasthi et al., 2015). We remark also that in independent work, Awasthi\net al. (2019) studied the effect of sensitive attribute noise on the post-processing method of Hardt\net al. (2016). In particular, they identi\ufb01ed conditions when such post-processing can still yield\nan approximately fair classi\ufb01er. Our approach has an advantage of being applicable to a generic\nin-processing fair classi\ufb01er; however, their approach also handles the case where the sensitive feature\nis used as an input to the classi\ufb01er. Exploration of the synthesis of the two approaches is another\npromising direction for future work.\n\n9\n\n\fReferences\nBank marketing dataset (a sample taken from UCI). https://www.kaggle.com/rouseguy/\n\nbankbalanced/kernels.\n\nAdler, P., Falk, C., Friedler, S. A., Nix, T., Rybeck, G., Scheidegger, C., Smith, B., and Venkatasub-\nramanian, S. Auditing black-box models for indirect in\ufb02uence. Knowl. Inf. Syst., 54(1):95\u2013122,\nJanuary 2018. ISSN 0219-1377.\n\nAgarwal, A., Beygelzimer, A., Dud\u00edk, M., Langford, J., and Wallach, H. A reductions approach\nto fair classi\ufb01cation. In Dy, J. G. and Krause, A. (eds.), Proceedings of the 35th International\nConference on Machine Learning, volume 80, pp. 60\u201369, Stanford, CA, 2018. JMLR.\n\nAngluin, D. and Laird, P. Learning from noisy examples. Machine Learning, 2(4):343\u2013370, Apr\n\n1988. ISSN 1573-0565.\n\nAngwin, J., Larson, J., Mattu, S., and Kirchner, L. Machine bias. 2016.\n\nAwasthi, P., Balcan, M.-F., Haghtalab, N., and Urner, R. Ef\ufb01cient learning of linear separators under\n\nbounded noise. In Conference on Learning Theory (COLT), volume 40, pp. 167\u2013190, 2015.\n\nAwasthi, P., Kleindessner, M., and Morgenstern, J. Equalized odds postprocessing under imperfect\n\ngroup information, 2019.\n\nBlum, A. and Mitchell, T. Combining labeled and unlabeled data with co-training. In Conference on\n\nLearning Theory (COLT), pp. 92\u2013100, 1998.\n\nBrodersen, K. H., Ong, C. S., Stephan, K. E., and Buhmann, J. M. The balanced accuracy and\nits posterior distribution. In Proceedings of the 2010 20th International Conference on Pattern\nRecognition, ICPR \u201910, pp. 3121\u20133124, Washington, DC, USA, 2010. IEEE Computer Society.\nISBN 978-0-7695-4109-9.\n\nBuolamwini, J. and Gebru, T. Gender shades: Intersectional accuracy disparities in commercial\nIn Proceedings of the 1st Conference on Fairness, Accountability and\n\ngender classi\ufb01cation.\nTransparency, pp. 77\u201391, 2018.\n\nCalders, T. and Verwer, S. Three naive bayes approaches for discrimination-free classi\ufb01cation. Data\n\nMining and Knowledge Discovery, 21(2):277\u2013292, Sep 2010. ISSN 1573-756X.\n\nCalders, T., Kamiran, F., and Pechenizkiy, M. Building classi\ufb01ers with independency constraints. In\n\n2009 IEEE International Conference on Data Mining Workshops, pp. 13\u201318, Dec 2009.\n\nCalmon, F., Wei, D., Vinzamuri, B., Natesan Ramamurthy, K., and Varshney, K. R. Optimized\npre-processing for discrimination prevention. In Advances in Neural Information Processing\nSystems 30, pp. 3992\u20134001, 2017.\n\nChan, P. K. and Stolfo, S. J. Toward scalable learning with non-uniform class and cost distributions:\nA case study in credit card fraud detection. In Proceedings of the Fourth International Conference\non Knowledge Discovery and Data Mining, KDD\u201998, pp. 164\u2013168, 1998.\n\nCharoenphakdee, N., Lee, J., and Sugiyama, M. On symmetric losses for learning from corrupted\n\nlabels. In ICML, 2019.\n\nCotter, A., Gupta, M. R., Jiang, H., Srebro, N., Sridharan, K., Wang, S., Woodworth, B. E., and You,\nS. Training well-generalizing classi\ufb01ers for fairness metrics and other data-dependent constraints.\nCoRR, abs/1807.00028, 2018. URL http://arxiv.org/abs/1807.00028.\n\ndel Barrio, E., Gamboa, F., Gordaliza, P., and Loubes, J.-M. Obtaining fairness using optimal\n\ntransport theory. arXiv e-prints, art. arXiv:1806.03195, June 2018.\n\nDenis, F. PAC Learning from Positive Statistical queries, 1998.\nDheeru, D. and Karra Taniskidou, E. UCI machine learning repository, 2017. URL http://\n\narchive.ics.uci.edu/ml.\n\n10\n\n\fDonini, M., Oneto, L., Ben-David, S., Shawe-Taylor, J. S., and Pontil, M. Empirical risk minimization\nunder fairness constraints. In Advances in Neural Information Processing Systems 31, pp. 2796\u2013\n2806, 2018.\n\nDwork, C. Differential privacy. In Bugliesi, M., Preneel, B., Sassone, V., and Wegener, I. (eds.),\nAutomata, Languages and Programming, pp. 1\u201312, Berlin, Heidelberg, 2006. Springer Berlin\nHeidelberg. ISBN 978-3-540-35908-1.\n\nDwork, C., Hardt, M., Pitassi, T., Reingold, O., and Zemel, R. S. Fairness through awareness. CoRR,\n\nabs/1104.3913, 2011.\n\nDwork, C., Immorlica, N., Kalai, A. T., and Leiserson, M. Decoupled classi\ufb01ers for group-fair and\nef\ufb01cient machine learning. In Proceedings of the 1st Conference on Fairness, Accountability and\nTransparency, pp. 119\u2013133, 2018.\n\nElkan, C. and Noto, K. Learning classi\ufb01ers from only positive and unlabeled data. In Proceedings of\nthe 14th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp.\n213\u2013220, 2008.\n\nFeldman, M. Computational fairness: Preventing machine-learned discrimination. 2015.\n\nGupta, M. R., Cotter, A., Fard, M. M., and Wang, S. Proxy fairness. CoRR, abs/1806.11212, 2018.\n\nURL http://arxiv.org/abs/1806.11212.\n\nHardt, M., Price, E., and Srebro, N. Equality of opportunity in supervised learning. In Proceedings\nof the 30th International Conference on Neural Information Processing Systems, NIPS\u201916, pp.\n3323\u20133331, USA, 2016. ISBN 978-1-5108-3881-9.\n\nHashimoto, T. B., Srivastava, M., Namkoong, H., and Liang, P. Fairness without demographics in\n\nrepeated loss minimization. arXiv preprint arXiv:1806.08010, 2018.\n\nHeidari, H., Loi, M., Gummadi, K. P., and Krause, A. A moral framework for understanding of fair ml\nthrough economic models of equality of opportunity. ACM Conference on Fairness, Accountability,\nand Transparency (ACM FAT*), January 2019.\n\nJagielski, M., Kearns, M., Mao, J., Oprea, A., Roth, A., Shari\ufb01-Malvajerdi, S., and Ullman, J.\n\nDifferentially private fair learning. 2018. URL https://arxiv.org/pdf/1812.02696.pdf.\nJain, S., White, M., and Radivojac, P. Recovering true classi\ufb01er performance in positive-unlabeled\n\nlearning. In AAAI, 2017.\n\nJohndrow, J. E. and Lum, K. An algorithm for removing sensitive information: application to\n\nrace-independent recidivism prediction. arXiv e-prints, art. arXiv:1703.04957, March 2017.\n\nKatz-Samuels, J., Blanchard, G., and Scott, C. Decontamination of mutual contamination models.\n\nJMLR, 20(41):1\u201357, 2019.\n\nKilbertus, N., Gascon, A., Kusner, M., Veale, M., Gummadi, K., and Weller, A. Blind justice:\nFairness with encrypted sensitive attributes. In Dy, J. and Krause, A. (eds.), Proceedings of the 35th\nInternational Conference on Machine Learning, volume 80 of Proceedings of Machine Learning\nResearch, pp. 2630\u20132639, Stockholmsm\u00e4ssan, Stockholm Sweden, 10\u201315 Jul 2018. PMLR. URL\nhttp://proceedings.mlr.press/v80/kilbertus18a.html.\n\nKim, M. P., Reingold, O., and Rothblum, G. N. Fairness through computationally-bounded awareness.\n\nCoRR, abs/1803.03239, 2018. URL http://arxiv.org/abs/1803.03239.\n\nKiryo, R., Niu, G., du Plessis, M., and Sugiyama, M. Positive-unlabeled learning with non-negative\n\nrisk estimator. In NIPS. 2017.\n\nKusner, M. J., Loftus, J., Russell, C., and Silva, R. Counterfactual fairness. In Advances in Neural\n\nInformation Processing Systems 30, pp. 4066\u20134076, 2017.\n\nLahoti, P., Weikum, G., and P. Gummadi, K. ifair: Learning individually fair data representations for\n\nalgorithmic decision making. 06 2018.\n\n11\n\n\fLipton, Z., McAuley, J., and Chouldechova, A. Does mitigating ml\u2019s impact disparity require\ntreatment disparity? In Advances in Neural Information Processing Systems, pp. 8136\u20138146, 2018.\nLiu, T. and Tao, D. Classi\ufb01cation with noisy labels by importance reweighting. IEEE Transactions\n\non Pattern Analysis and Machine Intelligence, 38:447\u2013461, 2016.\n\nLouizos, C., Swersky, K., Li, Y., Welling, M., and Zemel, R. The variational fair auto encoder. 11\n\n2015.\n\nLum, K. and Johndrow, J. E. A statistical framework for fair predictive algorithms. CoRR,\n\nabs/1610.08077, 2016.\n\nMenon, A. K., Narasimhan, H., Agarwal, S., and Chawla, S. On the statistical consistency of\nalgorithms for binary classi\ufb01cation under class imbalance. In Proceedings of the 30th International\nConference on Machine Learning, 2013.\n\nMenon, A. K., van Rooyen, B., Ong, C. S., and Williamson, R. C. Learning from corrupted binary\nlabels via class-probability estimation. In Proceedings of the 32nd International Conference on\nMachine Learning, 2015.\n\nMohri, M., Sivek, G., and Suresh, A. T. Agnostic federated learning. In International Conference on\n\nMachine Learning, 2019.\n\nMozannar, H., Ohannessian, M. I., and Srebro, N. Fair learning with private data. In NeurIPS 2019\nworkshop on machine learning with guarantees, 2019. URL https://drive.google.com/\nfile/d/1dqEexwzlSLK5XPckM00Q7Lggfh5o6saB/view.\n\nNatarajan, N., Tewari, A., Dhillon, I. S., and Ravikumar, P. Learning with noisy labels. In Neural\n\nInformation Processing Systems (NIPS), dec 2013.\n\nNorthcutt, C. G., Wu, T., and Chuang, I. L. Learning with con\ufb01dent examples: Rank pruning\nfor robust classi\ufb01cation with noisy labels. In Proceedings of the Thirty-Third Conference on\nUncertainty in Arti\ufb01cial Intelligence, UAI\u201917. AUAI Press, 2017.\n\nPedreshi, D., Ruggieri, S., and Turini, F. Discrimination-aware data mining. In Proceedings of\nthe 14th ACM SIGKDD international conference on Knowledge discovery and data mining, pp.\n560\u2013568. ACM, 2008.\n\nRamaswamy, H. G., Scott, C., and Tewari, A. Mixture proportion estimation via kernel embedding of\ndistributions. In Proceedings of the 33rd International Conference on International Conference on\nMachine Learning - Volume 48, ICML\u201916, pp. 2052\u20132060. JMLR.org, 2016.\n\nScott, C., Blanchard, G., , and Handy, G. Classi\ufb01cation with asymmetric label noise: Consistency\nand maximal denoising. In Conference on Learning Theory (COLT), volume 30, pp. 489\u2013511,\n2013a.\n\nScott, C., Blanchard, G., and Handy, G. Classi\ufb01cation with asymmetric label noise: Consistency and\nmaximal denoising. In Shalev-Shwartz, S. and Steinwart, I. (eds.), Proceedings of the 26th Annual\nConference on Learning Theory, volume 30 of Proceedings of Machine Learning Research, pp.\n489\u2013511, Princeton, NJ, USA, 12\u201314 Jun 2013b. PMLR.\n\nSpeicher, T., Heidari, H., Grgic-Hlaca, N., Gummadi, K. P., Singla, A., Weller, A., and Zafar, M. B. A\nuni\ufb01ed approach to quantifying algorithmic unfairness: Measuring individual & group unfairness\nvia inequality indices. In Proceedings of the 24th ACM SIGKDD International Conference on\nKnowledge Discovery & Data Mining, KDD \u201918, pp. 2239\u20132248, 2018. ISBN 978-1-4503-5552-0.\nvan Rooyen, B. Machine Learning via Transitions. PhD thesis, The Australian National University,\n\n2015.\n\nvan Rooyen, B. and Williamson, R. C. A theory of learning with corrupted labels. JMLR, 18(228):\n\n1\u201350, 2018.\n\nWard, G., Hastie, T., Barry, S., Elith, J., and Leathwick, J. R. Presence-Only Data and the {EM}\n\nAlgorithm. Biometrics, 65(2):554\u2013563, 2009.\n\n12\n\n\fWightman, L. national longitudinal bar passage study, 1998.\nWilliamson, R. C. and Menon, A. K. Fairness risk measures. In Proceedings of the 36th International\nConference on Machine Learning, ICML 2019, 9-15 June 2019, Long Beach, California, USA, pp.\n6786\u20136797, 2019.\n\nWoodworth, B., Gunasekar, S., Ohannessian, M. I., and Srebro, N. Learning non-discriminatory\npredictors. In Proceedings of the 2017 Conference on Learning Theory, volume 65, pp. 1920\u20131953,\nAmsterdam, Netherlands, 07\u201310 Jul 2017. PMLR.\n\nZafar, M. B., Valera, I., Gomez Rodriguez, M., Gummadi, K., and Weller, A. From parity to\npreference-based notions of fairness in classi\ufb01cation. In Advances in Neural Information Processing\nSystems 30, pp. 229\u2013239, 2017a.\n\nZafar, M. B., Valera, I., Gomez Rodriguez, M., and Gummadi, K. P. Fairness beyond disparate treat-\nment & disparate impact: Learning classi\ufb01cation without disparate mistreatment. In Proceedings\nof the 26th International Conference on World Wide Web, pp. 1171\u20131180, 2017b.\n\nZemel, R., Wu, Y., Swersky, K., Pitassi, T., and Dwork, C. Learning fair representations.\nProceedings of the 30th International Conference on Machine Learning, pp. 325\u2013333, 2013.\n\nIn\n\nZhang, D. and Lee, W. S. Learning classi\ufb01ers without negative examples: A reduction approach.\nIn 2008 Third International Conference on Digital Information Management, pp. 638\u2013643, Nov\n2008.\n\n13\n\n\f", "award": [], "sourceid": 147, "authors": [{"given_name": "Alex", "family_name": "Lamy", "institution": "Columbia University"}, {"given_name": "Ziyuan", "family_name": "Zhong", "institution": "Columbia University"}, {"given_name": "Aditya", "family_name": "Menon", "institution": "Google"}, {"given_name": "Nakul", "family_name": "Verma", "institution": "Columbia University"}]}