Practical Approaches for Fair Learning with Multitype and Multivariate Sensitive Attributes
Authors: Tennison Liu
Alex Chan
Boris van Breugel
Mihaela van der Schaar
Venue: Submitted to ICLR 2023 (2023)
Keywords: Fairness
Metrics
Regularization

Abstract: It is important to guarantee that machine learning algorithms deployed in the real world do not result in unfairness or unintended social consequences. Fair ML has largely focused on the protection of single attributes in the simpler setting where both attributes and target outcomes are binary. However, the practical application in many a real-world problem entails the simultaneous protection of multiple sensitive attributes, which are often not simply binary, but continuous or categorical. To address this more challenging task, we introduce FairCOCCO, a fairness measure built on cross-covariance operators on reproducing kernel Hilbert Spaces. This leads to two practical tools: first, the FairCOCCO Score, a normalized metric that can quantify fairness in settings with single or multiple sensitive attributes of arbitrary type; and second, a subsequent regularization term that can be incorporated into arbitrary learning objectives to obtain fair predictors. These contributions address crucial gaps in the algorithmic fairness literature, and we empirically demonstrate consistent improvements against state-of-the-art techniques in balancing predictive power and fairness on real-world datasets.

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PRACTICAL APPROACHES FOR FAIR LEARNING WITH MULTITYPE AND MULTIVARIATE SENSIIIVE ATTRIBUTES
Anonymous authors Paper under double-blind review
ABSTRACT
It is important to guarantee that machine learning algorithms deployed in the real world do not result in unfairness or unintended social consequences. Fair ML has largely focused on the protection of single attributes in the simpler setting where both attributes and target outcomes are binary. However, the practical application in many a real-world problem entails the simultaneous protection multiple sensitive attributes, which are often not simply binary, but continuous categorical. To address this more challenging task, we introduce Fai rcocco, fairesss measure built cross-covariance operators Hilbert Spaces. This leads to two practical tools: first, the FairCocco Score, normalized metric that can quantify fairness in settings with single or multiple sensitive attributes of arbitrary type; and second, a subsequent regularization term that can be incorporated into arbitrary learning objectives obtain fair predictors. These contributions address crucial gaps in the algorithmii fairesss literature, and we empirically demonstrate consistent improvements against ofate-of-the-art techniques in balancing predictive power and fairness on real-world datasets.
reproducing kernel
INTRODUCTION
There clear need for scalable and practical methods that can be easily incorporated into machine learning (ML) operations, in order to make sure they don't inadvertently disadvantage one group over another. The MI. community has responded with number of methods designed to ensure that predictive models are fair (under variety of definitions that we shall explore later) (Caton & Haas, 2020). Perhaps due to the archetypal fairness example, an investigation into the COMPAS software that found racial discrimination in the assessment of risk of recidivism (Angwin et al., 2016), most of the focus has been on single, binary variables in this case race being treated as an indicator of whether individual was black white. This, combined with discrete target, allows for easy analysis of fairness criteria such as demographic parity and equalized odds (Barocas & Selbst, 2016; Hardt et al., 2016), through the rates of outcomes in the confusion matrix of the subgroups.
The problem is, however, that in many practical applications we may have multiple attributes which we would like to protect, for example both race and sex indeed U.S. federal law protects groups from discrimination based on nine protected classes (EEOC, 2021). Algorithms deployed in the real-world therefore need be capable of protecting multiple attributes both jointly (e.g. 'black woman') ') and individually (e.g. 'black' and 'woman'). This is non-trivial and cannot be simply achieved by introducing separate fairness conditions for each attribute. Such an approach both does not provide joint protection of sensitive attributes and complicates matters by introducing additional hyperparameters that need to be traded-off against each other. Matters are further complicated by the fact that many sensitive attributes (e.g. age) and outcomes (c.g. credit limit) take continuous values, for which calculated rates do not make sense. Existing methods simply discretise these into categorical bins, which leads several issues in practice, it entails thresholding and data sparsity effects while discarding element order information. As we shall see later in Section 4, this approach is unlikely to be optimal in delivering discriminative yet fair predictors.
Contributions and Outline. Consequently, we introduce two practical tools to the community, which hope can be used to more easily incorporate fairness into standard ML pipeline: (1) Fairness

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metric. We introduce the FairCoCCo score, flexible normalized metric that can quantify the level of independence- based fairness in tasks with multitype and multivariate sensitive attributes by employing the cross-covariance operator reproducing kernel Hilbert Spaces (RKHS); and (2) Fairness regulariser. Based on the Fa rcocco score, we construct a fairness regulariser that can be easily added arbitrary learning objectives for fairness-aware learning.
In what follows, we introduce current notions of faimess alongside contemporary methods to ensure fair learning (Section 2), before introducing our contributions and explain how they plug the crucial gaps in the literature (Section 3). With that established, we illustrate the practical advantages of Fairoooco in a series of demonstrations on multiple real-world datasets across a variety of modalities, quantitatively demonstrating consistent improvements over state-of-the-arr techniques (Section 4). We conclude with discussion on future work and societal implications (Section 5).
BACKGROUND
Fairness Notions Let dx, dy, dA be dimensions of measur- able space X' C IRdx, y Rdr and A C IRd,, respectively. We introduce random variable X defined on X' to denote the features; Y and A are similarly defined and denote the target and sensitive attribute(s) that we want to protect (e.g, gender or race). Note that A can be part of X, i.c. with a slight abuse of notation, we can write A C X.
Table I: Popular definitions of fair- ness. Defined in terms of (condi- tional) independence requirements.
Definition
Requirement
FTU
A LL (Y X \ A) A Il Ý
We are mainly concerned with quantifying group fairness, which requires that protected groups (e.g. black applicants) be treated similarly to advantaged groups (e.g. white applicants) (Caton & Haas, 2020). In Table 1, we highlight four popular definitions and how each quantifies a different aspect of fairness. Fairness through unawareness (FTU) (Grgic- -llaa et al., 2016) prohibits the algorithm from using sensitive attributes explicitly in making predictions. While straightforward to implement, this method ignores the indirect dis- criminatory effect of proxy covariates that are correlated with A, e.g. "redlining" (Avery al., 2009). Demographic parity (DP) (Barocas & Selbst, 2016; Zafar et al., 2017) accounts for indirect discrimination, by requiring statistical independence between predictions and attributes Y IL A. Evidently, this strict notion sacrifices predictive utility by ignoring all correlations between Y and A, thereby precluding the optimal predictor. Dwork et al. (2012), most notably, argues that this approach permits laziness, which can hurt fairness in the long run. To address some of these concerns, Hardt et al. (2016) introduced equalized odds (EO), requiring that predictions Y and attributes are independent given the true outcome Y, i.e. Y JL. A Y. This approach recognizes that sensitive attributes have predictive value, but only allows to influence Y to the extent allowed for by the true outcome Y. For binary predictions and sensitive attributes, metric known difference in equal opportunity (DEO) highlights the different predictions made based on different group memberships:
DP
EO
(A ll L)LY
CAL
(A Il IALYY) Y)
DEO IP( A 1,Y P( = 0, = =1]
Additional notions of fairness include calibration (CAL.) (Kleinberg et al., 2016), which ensures that predictions are calibrated between subgroups, i.e. TL A Y. For comprehensive review of fairness notions, we defer to $3 in Caton & Haas (2020). In the remaining sections, we illustrate our proposed methods using the framework of EO. but this without loss generality, as our method is compatible with any dependency-base based fairness measure. It is important to note that there is no universal measure of fairness, and the correct notion depends on ethical, legal and technical contexts.
2.1 RELATED WORKS
Technical approaches to algorithmic fairness can be categorized into three main types: prior to modelling (pre-processing), during modelling (in-processing) or after modelling (post-processing) (del Barrio et al., 2020). The work herein falls into the category of in-processing techniques, which achieve fairness by incorporating either constraints regularisers. Table 1 makes explicit the connection between fairness notions and (conditional) dependence. At the core of many algorithmic fairnsss techniques is how fairness is estimated and constrained. Much of the literature focuses
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Table 2: Overview of related work for fairness-aware learning. Comparison made on method of fairness estimation, underlying model class and the following desiderata: (1) supports contin-
uous outcomes; (2) continuous attributes; (3) protects multiple attributes; (4) compatible with all
dependency-basee based notions of fairness (as in Table 1).
Method
Fairness Estimation
Predictive Model
(1)
(2)
(3)
(4)
7emel al. (2013)
Mutual information
Linear
X
V
Zafar et al. (2017)
Conditional covariance
Linear/Kemel
X
X
X
X
Donini et al. (2018)
Linear loss
Linear/Kemel
X
X
X
Cho et al. (2020)
Linear loss
Any
X
V
Mary et al. (2019)
Rényi correlation
Any
V
V
Steinberg al. (2020h)
Mutual information
Any
X
X
V
Pérez-Suay al. (2017)
Kemel measure
Linear/Kemel
X
FairCocco
Kernel measure
Any
v
v
v
settings with a single, binary label and attribute (Kamishima et al., 2012; Goel et al., 2018; Jiang et al., 2020; Donini et al., 2018), where fairness quantification is straightforward by comparing rates of outcomes between subgroups. However, settings involving continuous variables are significantly more challenging (Bergsma, 2004). Recent efforts in fair regression (where only outcomes continuous) (Agarwal et al., 2019; Chzhen et al., 2020) discretise continuous variables, but such approaches introduce unwanted threshold effects, discards order information and requires sufficient sample coverage in each bin.
Protecting continuous attributes. To pursue a fully continuous treatment, recent methods have made parametric or other assumptions to simplify conditional dependence criteria. Calders et al. (2013), Johnson al. (b016b) and Bechavod & Ligett (2017) reduce the task of dependence minimization to minimizing the distances between moments of distributions. Donini et al. (2018) generalizes this to minimizing the distance between first moments of functions. Woodworth al. (2017) and Zafar et al. (2017) similarly employ second moment relaxation to regularize only conditional covariance, corresponding removing linear correlations only. Kamishima et al. (2012) introduced first moment relaxation of mutual information (MI). However, such approaches are limiting as weak fairness measures that cannot fully capture important fairness effects and potentially lead to harm the distribution assumptions miss speciieee (Daudin, 1980).
Ideally, we hope for a strong measure that can accurately identify the level of fairness. Key approaches include kernel methods, MI, maximal correlation. Cho et al. (2020) employ kernel density estimation (KDE) to compute MI to enforce fairness. Lowy et al. (2021) and Mary et al. (2019) developed regularization using maximal correlation, but similarly rely KDE, which does not scale to higher dimensions. Steinberg et al. (2020a) and Steinberg et al. (2020b) adopts a MI-based measure using density ratio estimation, but requires the training of an inner-loop probabilistic classifier.
Protecting multiple attributes. Few existing methods support protection of multiple attributes, even though this is common and necessary requirement in practice. Kearns al. (2018) highlighted fairness gerrymandering. in which a predictor appears to be fair on each individual attribute (c.g. black) but badly violates fairness when considering multiple sensitive attributes (e.g. black woman). Put formally, the prediction should be jointly independent (i.e. fair) to multiple sensitive attributes while also being independent to each individual attribute. Fortunately, this is already implied due to the decomposition property:
Y IL (A1 Ad.....,AA))I =
Ý IL LLLAY fe. .d.
However, we cannot naively extend existing methods protect multiple attributes by introducing separate conditions on each attribute. This is evident, as the inverse proposition of (1) does not hold in general. In other words, while this naive approach can ensure individual protection, does not guarantee protection of all attributes simultancously. A related stream of research investigates inter- sectional fairness (Kearns al., 2018; Foulds et al., 2020), which models combinatorial intersection of various subgroups. However, this only considers discrete attributes and outcomes, and one notion of fairess (DP). Table provides an overview and comparison of related works.
Weak strong by (Daudin, 1980), refer strength characterizations dependence.
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3 EVALUATING AND LEARNING FAIRNESS
In this section, we introduce FairCocco, a strong fairness measure from which we develop a metric and regulariser for fair learning. It applies kemel measures to quantify and control the level of dependence between algorithm predictions and protected attributes, such that the fairness requirements in Table 1 hold.
3.1 KERNEL MEASURE OF FAIRNESS
Setup. Let Hy denote the RKHS on y, with positive definite kernel ky. KA and H.d are defined similarly. 2 Formally, the problem of interest is quantifying the conditional fairness between and A given Y finite samples.
We propose measure based on the conditional cross-covariance operator Reproducing Kernel Hilbert Space (RKHS). A RKHS Hy is a Hilbert space of functiion, which each point evaluation fv), for any E y/ and f E Hy. is a bounded linear functional. Distributions of variables can be embedded into the RKHS through kernels, where inference of higher order moments and dependence between distributions can be performed (Bach & Jordan, 2002; Gretton et al., 2005).
Unconditional fairness. We start by describing how operators in the RKHS can be used to evaluate fairness in the unconditional case (DP), by quantifying reliance of model predictions Y on sensitive attributes A. The cross-covariance operator (CCO) Ev A HA - Hy is the unique, bounded operator that satisfies the relation:
(g. Ev Afy = Ey ((())(()))))) - Ey (EY))
for f E Hy and g E H.. Intuitively, the Σ operator extends the covariance matrix defined on Euclidean spaces to represent higher (possibly infinite) order covariance between and A through kernel mappings fX) and g Additionally, we can obtain normalized operator, i.c. the normalized crossccovaiiance operator (NOCCO) Vy (Baker, 1973):
-ー E Vy E b
where Eg AAA are defined similarly (2). This normalization analogous to the relationship between covariance and correlation, and disentangles the influence of marginals while retaining the same dependence information. Intuitively, we have obtained strong measure of correlation between sensitive attributes and faimess by leveraging the RKHS to represent higher-order moments.
Conditional fairness. For many notions fairness (i.e. BO and CAL), we also require a measure of conditional fairness. We will frame the discussion around EO, where the prediction should be independent of the sensitive attribute given the true outcome Ý IL Y. It is straightforward to adapt this for CAL by swapping variables around. We can derive a normalized, conditional cross-covariance operator, by manipulating (3), i.e. VYAIY (COCCO):
VY Vy Vyy VY
(4)
where Vy Vy are defined similarly to (3). In line with the intuition established previously, this operator measures higher-order partial correlation through function transformations ((A) V H.
and g(\) ( ((Y) V g, h E Hy. We round up this discussion by characterizing the relation between the y A|y operator and conditional fairness.
Lemma 3.1 (COCCO and Conditional Fairness (Fukumizu et al., 2007))
Denote & (A,Y), and the product of kernels k A kaky, and further assuming 1 characteristic kernel. Then:
Vy =0 - IL A
(5)
Note that À denotes the extended variable set. For ease of notation, we write VY in place of Vy Ajy from this point onward. (3) and (4) gives us a way to measure unconditional and conditional fairness, respectively, and lower values will indicate higher levels of fairness. Additionally, we note
that (3) can be viewed as a special case of (4), where y = 0, i... Vy = +++ Y 11 A.
We make mild ussumptions on the invulved RKIISs, assuming they are sepurable and square integrable (Grettion et al., 2005); and employ characteristic kernels, egg. Gaussian and Laplacian kemels.
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3.2 METRIC: FAIRCOCCO SCORE
Having described a kernel-based measure of fairness, we propose faimess metric that is applicable conditional and unconditional faimesss well settings with multiple sensitive attributes of arbitrary (continuous or discrete) type. Many metrics (e.g. DEO (Hardt et al., 2016) to evaluate EO, and DI (Feldman al., 2015) to evaluate DP) have been proposed for binary fairness settings. However, their utility is limited to classification tasks with single binary sensitive attributes. This insufficient in real-world conditions, where there often exists many sensitive attributes that can be discrete or continuous. To address these challenges, we propose PcicCo Score that can evaluate fairness of several attributes of mixed type and for both continuous and discrete outcomes.
We start by summarizing the information contained in VY into single statistic using the squared Hilbert-Schmidt (HS) norm (Bach Jordan, 2002):
11V A Il
This scalar value can be estimated from samples analytically, and we provide the complete closed- form expression Appendix A. By Lemma 3.1, we know that AlVyallis = 0 + Y TL A. Thus, values closer zero indicate higher levels of conditional fairness. However, while (6) is non-negative, can be arbitrarily large. This makes it hard to interpret and compare across different tasks. To address this, we propose the normalized metric F FairCOcco score: Definition 3.2 (FairCOCCO Score)
IVV Iliss V ||s||| VAA
Faircocco Score (unconditional)
(7)
HS
|1 llus
FairCocco Score (conditional)
(8)
V || AarsslVaa
which takes values in [0, 1]. where value closer to indicates higher levels of fairness, and vice versa.
This normalization scheme is derived from the Cauchy-Schwarz inequality and can be understood as taking into account the (conditional) variance within each variable (c.f. relationship between covariance and correlation). In Appendix A, we derive the metric and its conditional counterpart and additionally demonstrate how the measure (6) can be used to perform (conditional) independence testing for additional transparency and interpretabilityy
The Fairoocco Score can be used to measure any of the independence based notions of fairness. In particular, to make the connection with the Table clear, the terms for the different notions can be expressed as:
IEO IIV
ICAL = Il
Ine ||V lius
3.3 LEARNING: FAIRCOCCO LEARNING
Now that have established the FairCOCCO Score that can be used to detect (un-)fairness, we move on to how it can be employed in order to obtain fair predictors. We focus on a standard supervised learning setup with the task learn the map X' x A + V, with given fairness condition F that should be satisfied. Given a batch of N training triplets {(X,, Yi, A;) a learning function fo(-) with learnable parameters 0 O, training loss L, and 1F denoting one of the fairness statistics from (9) that takes both the batch and learning function and returns the corresponding score, we arrive a constrained optimization problem:
N
min
Σ C(fo(z;), v.) subject to Ip( fo) = OEE N
(10)
Practically speaking, this can be relaxed via Lagrangian in order to obtain an unconstrained optimization problem that can be solved significantly more easily:
N
min BEO N
Σ
c(fo(zs).u)) (D,
(11)
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The summary statistic (6) and therefore (p) is differentiable and as shown here can be employed as a regulariser in any gradient-based method, with À > 0 hyperparameter that determines the fairness-peffomaane performance trade-off: higher À guarantees higher faireess, but this typically leads to lower predictive performance. Consequently, this measure can be used to quantify and enforce fairness notions by controlling dependence between Y, A and A. We term this regularization scheme FairCOCCo Learning.
3.4 IN SUMMARY
The proposed kernel faimess measure provides a non- non-parametric, and strong characterization of fairness. The mappings allow both multivariate continuous and discrete variables to be embedded into the RKHS, from which we infer higher-order dependencies, and thus fairness effects. This enables the evaluation of multivariate, multitype fairness problems as commonly encountered in the real world. Additionally, the proposed metric and regularization methods are compatible with all dependency- based notions of faimess (as in Table 1), giving practitioners more flexibility in choosing the appropriate definitions for their scenario.
In our experiments, we use Gaussian kernel: k(X;, = exp (- IX,-X,111 Vi,j E N where
the bandwidth parameter, is selected with the median heuristic, o median [ -zyl Vi デ E N} (Schölkopf et al., 2002). As the calculation of (9) comprises a matrix inversion operation, the computational complexity scales with the number of samples (N We improve the scaling with training samples in two ways, (1) by employing a low-rank Cholesky decomposition of the Gram matrix (of rank r), resulting in ((22N) complexity (Harbrecht et al., 2012) and (2) by estimating regulariser on mini-batches, We empirically investigate the effect of these relaxations on fairness estimation in Appendix C.2 and demonstrate that they lead to strong results in real-world experiments.
4 EXPERIMENTAL DEMONSTRATION
We now turn our attention how our proposed methods works in practice. We perform experiments within the EO framework, since it is usually considered the most challenging, and it covers the middle middle-ground between the strict DP and lenient FTU definitions. However, we re-iterate that our method is frumework-agnostic and attach further results under alternative definitions in Appendix C.I. There are a number of areas that require empirical demonstration, and so we proceed as follows:
1. First, in Section 4.1, employ standard real-world benchmarks to compare against existing methods on single binary attributes and outcomes, resulting in competitive (and usually superior) predictive performance Then, in Section 4.2, we apply Fai i roccO to real data with multiple attributes and continuous outcomes. This is an area that to the best of our knowledge other method naturally extends to, and one that FaircooCk now sets a strong benchmark for future work.
these tasks while consistently producing the best DEO score.
3. Finally, in Section 4.3, consider the more complicated setting of fair learning in image data and time series. Here, we demonstrate how the problems of sepsis treatment and facial recognition are important applications of our method.
In the interest of limited space, we attach additional results in Appendix C. Specifically, we include experiments on: Different notions of fairness: evaluating accuracy-fainee fairness trade-off on dif- ferent definitions of fairesss (specifically DP and CAL); 5. Statistical testing: demonstrating the Paircocco Score as a test statistic for stronger faimnsss transparency: Sensitivity analysis: to better evaluate the performance of our method on varying numbers of sensitive attributes.
Benchmarks. We compare against state-of-the-ar fairness methods, including classic baselines (Zafar et al., 2017; Hardt et al., 2016; Donini et al., 2018) and more recent methods that adopt stronger fairness quantification: FACL (Mary et al., 2019) and FARMI (Steinberg et al., 2020b), which leverages MCC and MI, respectively.
Following the experiment design in recent works (Hardt et al., 2016; Donini et al., 201 2018), we employ 9 real-world datasets from the UCI machine learning repository (Dua & Graff, 201 70.7). Specifically, we consider 4 datasets contain single sensitive attributes and binary outcomes and datasets with multiple sensitive attributes and outcome of arbitrary type. We employ datasets with
Datasets.

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Table 3: Performance in binary setting. Accuracy (ACC) and DEO benchmark datasets. NN an unregularised neural network, on top of which the regularizers from competitor methods and Fa FairCocco applied to. Best results emboldened.
COMPAS
German
Drug
Adult
Method
ACC
DEO
ACC
DEO
ACC
DEO
ACC
DEO
Zafar et al. (2017)
0.69 0.02
0.10 +: 0.06
0.62 +: 0.09
0.13
0.69
0.03
0.02 1: 0.07
0.78
0.05
Hardt (2016)
0.01
0.11
Donini et al. (2018)
0.73 = 0.01
0.05 + 0.03
0.73 + 0.04
0.05 0.03
0.80 + 0.03
0.07 + 0.05
0.81
0.01
NY
0.90 0.02
0.06 + 0.00
0.74 0.07
0.31
0.80 + 0.08
0.06 0.12
0.84
0.19
Mary (2019)
0.02
0.04 0.01
Steinberg et al. (2020b)
0.85 0.01
0.03 + 0.01
0.71 = 0.10
0.09
0.79 0.05
0.01 0.02
0.80
0.10
FairCOCCo
0.89 0.01
0.00 1: 0.01
0.74 1: 0.03
0.02 0.09
0.80 0.06
0.02 4: 0.01
0.83
0.04
different number of samples (ranging from 649 to 299285) and different feature counts (ranging from 10 128) gain better understanding of our method's performance profile.
Additionally, we also employ time-series dataset on sepsis treatment from the MIMIC-III ICU database (Johnson al., 201 and an image dataset CelebA (Liu et al., 2015) for face attribute recognition. We provide additional information about benchmarks, datasets, model design, hyper- parameters, and evaluation methods Appendix B. For results, we report mean + std over 10 runs.
4.1 BINARY ATTRIBUTES AND OUTCOMES
While the focus of this work is on introducing practical methods for fairness in multitype, multivariate settings, we want to first prove that Fa rCocCo is also com- petitive with state-of-the-arr methods on problems with binary sensitive attributes and outcomes. We reproduce benchmarks based on UCT's Drugs, German, Adult and COMPAS datasets. We compare against representative methods in literature as well as standard (unfair) neural network (NN). For strong fairness methods, specifically including our method, FACL and FARMI, we employ the same NN the underlying predictive model en- sure comparability.' We report our results in Table 3. Paircocco achieves higher levels of fairness (lower DEO) while maintaining strong predictive accuracy on all datasets except Drug. We note that (Mary et al., 2019) is specifically tailored for settings with binary sensitive attribute and outcome, but our method is more gener- ally applicable to settings with multitype, multivariate sensitive attributes.
Fair-COCCO
0.225
FACL
0.200
FARMI
Fair-Cocco
0.175
0.150
0.125
0.100
0.075
0.0
0.2
0.4
0.6
0.8
1.0
MSE
Fair-COCCO
FACL
0.125
FARMI
Fair-COcco
0.120
4.2 CONTINUOUS ATTRIBUTES AND OUTCOMES
0.115
Next, we illustrate the main contributions of our work, by demonstrating PairCoccO protect fairness in settings involving multiple sensitive attributes and out- comes of arbitrary type. We employ Crimes and Commu- nities (C&C), Credit Card, KDD-Census, Law School, and Students datasets from the UCI repository. We start by looking at protection of single continuous attributes, before examining the joint protection of multiple sensi- tive attributes.
0.110
0.105
0.3
0.4
0.5 MSE
0.6
0.7
Figure 1: Fairness accuracy trade-off. Crimes and Communities (top) and Stu- dents (bottom). Optimum desiderata the origin, where both MSE and unfaimesss are minimized.
Single continuous attribute. We compare our method
against our closest competitors FACL and FARMI.
re-man available implementation own pipeline, reporting
re-runs original reported scores.
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Table 4: Protection of multiple attributes. Investigation on joint fairness effects and fairness protection with respect to individual sensitive attributes on array of benchmarks. Lowest MSE/ACC, Faircocco and DEO scores are emboldened.
Crimes and Communities racePetBlack
Joint
racePetAsiin
racePetHisp
Method
MSE
COCCO
COCCO
DEO
COCCO
DEO
COCCO
DEO
NN
0.22 = 0.01
0.27 0.01
0.24 = 0.02
0.25 + 0.03
0.10 0.00
0.14=0022
0.16 = 0.01
0.08 :: 0.04
FACL
0.66 = 0.01
0.15 0.02
0.10 1 0,02
0,10 2: 0.08
0.0000000
0.13 0.06
0,09 = 0.02
0.09 0.03
PARMI
0.65 = 0.01
0.20 2: 0.02
0.14 = 0.02
0.10 + 0.06
0.11 + 0.02
0.11 = 0.05
0.13 = 0.02
0.08 0.02
Faircocco
0.63 0.01
0.11 0.01
0.08
0.01
0.07 1 0.05
0.07 1 0.01
0.11 0.05
0.07
0.02
0.05 0.03
KDD-Census
Joint
age
S6X
race
Method
ACC
COCCO
COCCO
DEO
COCCO
DEO
COCCO
DEO
0.95
0.04
0.17
0.06
0.00
0.00
0.07 0.01
0.02
TACI
0.93 0.01
0.10 0.02
0.10 + 0.03
0.12 0.03
0.04 + 0.01
0.03 0.00
0.08 = 0.02
0.09 0.02
FARMT
0.88 + 0:03
0.15 0.05
0.13 0.05
0.18 0.05
0.05 0.00
0.04 0.01
0.07 0.01
0.05 0.02
Pair00000
0.94 0.02
0.02 0.00
0.02 = 0.00
0.01 + 0.00
0.00
: 0.01
0.02 = 0.01
0.00 = 0.01
0.02 0.00
Credit Card
Juint
education
marriuge
Method
ACC
COCCO
COCCO
DRO
COCCO
COCCO
DEO
AN
0.82 = 0.02
0.13 0.01
0.06 0.01
0.08 :: 0.00
0.04 1: 0.01
0.02 = 0.00
0.04 =: 0.02
0.03 0.02
FACL
0.80
0.00
0:04
0.03 0.00
0.01
0.02
0.05 0.01
0.01
FARMI
0.81 = 0:012
0.05 1: 0.00
0.02 4 0.00
0.02 4: 0.00
0,03 1 0.00
0.02 = 0.00
0.04 = 0.01
0.03 0.00
FairCoCCo
0.01 0.00
0.00 0.00
1:
0.00 0.000
0.02
0.01 0.00
Low School
Joint
male
race
Method
ACC
COCCO
COCCO
DEO
COCCO
DEO
NV
0.89 0.03
8.07::0.04
0:01 0.00
0.02 0.00
0.11 + 0.01
0.12 +:
FACT.
0.85 + 0.02
0.04 0.02
0.00 = 0.01
0.01 +-0.0
0.04 + 0.02
0.06 + 0.03
FARM1
0,86 : 0.02
0.01
0.01 0.00
0,01 0.00
0.03 1 0.01
0.01 0.02
FairCocco
0.89 :: 0.01
0.02 0.00
0.00 = 0.00
0.00 4: 0.00
0.01 4: 0.01
0.04 :: 0.00
Studenty
Joint
sex
Method
MSR
COCCO
COCCO
DEO
COCCO
DEO
NV
0.25 + 0.05
0.16 0.03
0.12 0.02
0.06 + 0.03
0.09+ 0.03
0.97 + 0.02
FACL
0.324: 0.03
0,142 0.02
0.12 0.02
0.07-t: 0,03
0.07 0.02
0,04: 0.05
ZARMI
0.3611
::
FaircoCCC
0.29 1: 0.05
0.14: 0.02
0.10 1: 0.03
0.06 4: 0.03
0.07.1 0.01
0.03 =: 0.03
While FACL does not support multiple attributes, it is applicable to protect a single continuous variable. FARMI is only compatible with discrete sensitive attributes; we thus binarise the sensitive attributes at the median during training. We take the datasets C&C and Students, and use protected at- tributes ra ceecctttBBeccc ck and age respectively. We plot the performance versus fairness by varying the fairness penalty in Figure 1. Notably, F FairCocco obtains a better trade-off between fairness and MSE than both methods (optimum desiderata the origin).
Multiple (arbitrary type) attributes. Going one step further, we want to evaluate the concurrent protection of multiple sensitive attributes. We note that while this is natural for FairCocco, to the best of our knowledge, there are no existing methods that can jointly protect multiple sensitive attributes of arbitrary type. To enable adequate comparison, we adapt FACL and FARMT by including a separate regularization term for each attribute. In contrast, the Fa rCOCCO regularization is applied directly and jointly on all sensitive attributes. Previously, we showed that the protection of individual fairness effects does not guarantee protection of joint fairness. To that end, we are interested in analyzing both joint fairness effects and protection w.r.t. individual attributes. In Table 4, we evaluate the joint fairness (Joint) and fairness on individual attributes (e.g. raceP etBack, racePetAsian, racePetHisp on C&C). To evaluate individual fairness, we also calculate the DEO by binarising the attributes at the median during evaluation.
We first note that Fai rcoco and DEO scores are highly correlated in their respective estimation of unfairness. However, the key result we wish to highlight is that not only does Fa iroocoo successfully minimize joint fairness effects, also consistently minimizes the levels of unfairness for each sensitive attribute. The same cannot be said for F FARMI and FACL, where the joint fairness outcomes are inadequate as the protection granted to individual attributes are traded-off to the detriment of other attributes. To better investigate the sensitivity of our method to the number of sensitive attributes, the performance faimess trade-off by varying the number of protected attributes in A Appendix C.4.
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Table 5: Facial attribute recognition. Accuracy (ACC) and DEO on three separate classification tasks attractive, smile, and wavy hair. Best results are emboldened.
altractive
smile
wavy hair
Method
ACC
DEO
ACC
DEO
ACC
DEO
NN
0.82 = 0.02
0.43 + 0.00
0.98 + 0.03
0.05 + 0.01
0.81 + 0.02
0.18 0.02
FACL
0.78 + 0.02
0.11 + 0.02
0.95 +: 0.03
0.02 +: 0.00
0.78 + 0.02
0.10 0.01
FARMT
0.79 + 0.03
0.07 0.01
0.96 0.02
0.01 + 0.00
0.74 + 0.01
0.04 0.00
FairCocc0
0.80 + 0.03
0.03 : 0.00
0.96 1 0.04
0.01 1 0.00
0.80 + 0.02
0.02 = 0.00
4.3 BEYOND TABULAR DATA
CelebA facial attributes recognition. In this section, highlight that Fa f rCOCCO can be applied beyond tabular data by experimenting on the CelebA dataset (Liu et al., 2015). The CelebA dataset contains images of celebrity faces, where each face is associated with binary sensitive attributes, including gender. We follow the experimental design in (Chuang & Mroueh, 2021) and form binary classification tasks using attributes attractive, smile, and wavy hair, and treat gender as the sensitive attribute. We fine-tune ResNet-18 (He et al., 2016) with two additional hidden layers to perform the classification task. We report the results in Table 5, noting similar improvements in fairness with little decrease in accuracy, especially the classifying attractive and wavy hair.
Sepsis treatment. Finally, we emphasize that FairCocCO is not limited to the standard su- pervised learning setup and demonstrate how our approach can be applied for learning fairer policies in time series setting. We employ the
Table 6:
Sepsis treamtent. Accuracy (ACC), DEO and Fai Pcicooooo score learning fair sepsis treatment policies; the best results are emboldened.
Method
ACC
DEO
COCCO
MIMIC-III ICU database (Johnson et al., 201 6a), containing data routinely collected from adult patients in the United States. We analyze the de- cisions made by clinicians to treat sepsis, using a patient cohort fulfilling the Sepsis-3 criteria, delineated by Komorowski et al. (2018). For each patient, we have relevant physiological parameters recorded at 4 hour resolution, and static demographic context. The task is to predict the clinical intervention to treat sepsis by learning from clinician's 's actions. For this, we have access to binary variable corresponding to clinical interventions targeting sepsis. Ground-truth treatment outcomes computed from SOFA scores (measuring sequential organ failure) and lactate levels (correlated with severity of sepsis) in the subsequent time step. and we consider gender as the sensitive attribute. For the complete problem setup, refer to the Appendix B.2. Table 6 indicates that Fa i rcocco successfully reduced any bias contained in expert demonstrations and achieved the best predictive and fair performance when compared to FACL and FARMI.
NN
0.82 +: 0.03
0.05 + 0.03
0.13 +: 0.02
FACL
0.81 : 0.04
0.02 :: 0.01
0.08 1 0.02
FARMT
0.78
0.03 0.01
0.10 0.01
FairCocco
0.81 0.02 0.00 0.01
0.01
DISCUSSION
In this work, we proposed FairCocco, kernel-based fairness measure that strongly quantifies the level of unfairness in the presence of multiple sensitive attributes of mixed type. Specifically, introduced a normalized fairness metric (Fai rcoco Score), applicable to different problem settings and dependency-basee -bsed fairness notions, and faimess regularization scheme. Through our experiments, we empirically demonstrated superior mhiresss-predictis trade-off and protection of multiple and individual fairness outcomes.
Limitations and future works. The main limitation of our work is computational complexity the matrix operations, required to kernalise the data and embed it in the RKHS, has complexity O( We propose two directions to alleviate this (i.e. low-rank approximation, mini-batch evaluations), which empirically do not noticeably impact performance. Future works should consider speeding up kernel operations using methods proposed in (Zhang et al., 2012). Additionally, while our regularizer can be applied to models trained using seadienttbased methods, future works should extend our approach to be compatible with powerful decision- tree based algorithms.

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ETHICS AND REPRODUCIBILITY STATEMENT
Ethics statement. We caution against using our proposed methods as a certificate fairness. As Corbett-Davies et al. (2017) rightfully emphasize, fairness measures do not rule unfair practices. Additionally, future works should focus on interpretable fairness quantification that sheds insight on root causes of unfairness, allowing them to be eliminated through procedural changes rather than solely in prediction tasks. Lastly, we encourage more lively discourse on philosophical implications of ML methods on justice and fairness (Kuppler al., 2021) that is critical to Fair ML deployment.
Reproducibility statement. We detailed exact implementation details, including dataset prepro- cessing, implementation of benchmark methods, architecture design, hyperparameter tuning, and evaluation methods in Section 3, Section 4 and Appendix B. We will release code upon acceptance of the paper for the camera-ready version.
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A MORE ON FAIRCOCCO
A.1 CLOSED FORM EXPRESSION
We introduced covariance operators on RKHSs, which can be used quantify unconditional (Vg.) and conditional fairness (VEAlv) FairCocco based on the HiilbertSchhmin (HS) norm the AΦ;)
covariance operators. An operator A H/1 + H2 is called HS if, for complete orthonormal systems { of M1 and {Φ)} } of H2, the sum Σx, AØ; Hs is finite (Reed Simon, 1980). Thus, for
an HS operator A, the HS norm, ||A||s is defined as || Hs Ei, (1l A Hs- Provided that Vy A and V are HS operators, F aircocco scores can be expressed as:
V A y llis (conditional fairness measure)
V A (unconditional fairness measure)
The umlaut on A represent extended variable sets, i.c. Å= (A, Y). Here, we briefly flesh out the closed-form expression of the empirical estimators, while more details can be found at (Fukumizu al., 2007; Gretton al., 2005). Let Gy be the centered Gram matrices, such that:
GY, (ky(, YJ) - my (N) Myy(:
(N) my Hl
We choose Gaussian RBF kernel, k(Y,, Yj) = exp |11 median heuristic introduced by Schölkopf et al. (2002), i.e. a = median ( Yjl. v &i E to select bandwidth a. Additionally, my (N) 1/N Σ ky(: Y,) is the empirical mean. G Gy are defined similarly. Based on this, proxy Gram matrices Ry can be defined as follows:
V Vi E N, and employ the
2n
Ry = Gy (Gy + cNTN)
where -e-eellh is regularization constant, used in the same way as Bach & Jordan (2002), IN is an identity matrix and Ry. R defined similarly. The empirical estimator of ||V X 1l can then be computed:
i= 1=1iva 1l = -Tr(Ry Rá 2Ry Ry
(12)
(13)
The unconditional fairness score can similarly be estimated empirically as follows (note that uncondi- tional dependence does not entail using extended variables):
|1 llis = Tr(Rq- RAl
(14)
(15)
Choice of Kernels. While, in general, kernel dependence measures depend not only on variable distributions, but also the choice of kernel, Fukumizu et al. (2007) showed that, in the limit of infinite data and assumptions on richness of the RKHS, the estimates converges to kennee-independent value. We employ Gaussian RBF (characteristic kernel) in our experiments.
On the computational complexity. For our experiments, we use a Gaussian RBF kernel: -|X,- A Vij E N where is the tuneable bandwidth parameter. We employ the median heuristic introduced by Schölkopf et al. (2002), i.c. median æz1, V Vi j E N} to select bandwidth.
A(X,, X,) exp
20
As the calculation of (9) comprises a matrix inversion operation, the computational complexity scales with the number of samples O(N3). We improve the scaling with training samples two ways, (I) by employing low-rank Cholesky decomposition of the Gram matrix (of rank r), resulting in Orr N) complexity (Harbrecht et al., 2012) and (2) by estimating regulariser on empirically demonstrate that these lead to strong results in real-world experiments.
We
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A.2 FAIRCOCCO SCORE
Here, we derive FairCocco score from the underlying measure using the Cauchy-Schwarz In- equality, The Fai rcocco score for conditional fairness and unconditional fairness can be written as:
y liss
FairCocco Score (unconditional)
V[Vy | [Hs||VAlll VAAllH
y A 1l
Fai rCocCo Score (conditional)
|Rg - Rg Ry |lslll - R2 Ryllas lls
We start by looking unconditional version of Faircocco, we know from (14) and the Cauchy- Schwarz inequality for the inner-product ( that:
V 1l
[Tr[Ry RA]] 1 RAY
Ry HS RAIlHS= V Ryl RA
||Vy |ls | VAAlIS
By the inequality, Fair Score (unconditional) E 1-1, Additionally, as the score is also non-negative, it takes value E [0, 1] where indicates perfect fairness (as indicated by Lemma 3.1). By contrast, the score takes value iff gram matrices, Ry and RA, are linearly dependent (i.e. perfectly unfair). The derivation and interpretation can similarly be shown for the conditional case:
III Y y Fisl [Tr(RR RA RRR Ry + Ry Ry RARyl
[Tr((Rp - (y) RARy K(Rg RR RABy))
$ || - Ry sS RA - RA ||Hs
Here, Ry- Ry is related the conditional covariance operator, i... Vy | which captures the conditional covariance Y given Y. See (Fukumizu al., 2007; 2009; Baker, 1973) and others for more.
B EXPERIMENTAL DETAILS
B.1 SUPERVISED LEARNING TASKS
B.1.1 MODEL DETAILS
For all experiments, we train a two-layer neural network with ReL.U-cctivated nodes. The number of nodes chosen between 40~100 depending on the complexity of the data. The network is trained with Cross Entropy or MSE Loss and is optimized using Adam (Kingma & Ba, 2014). The hyperparameters include batch size E [64, 128, 256), learning rate E (1e-2, 1c-3, le-4), and fairness penalty E (0.0, 0.5, 1.0, 2.0, 5.0) and are chosen through cross-validation. For datasets without a defined test set, the data split 60-20-20 into train, validation and test set and results are averaged over 10 runs. Experiments are run either CPU or NVIDIA Tesla K40C GPU, taking around an hour.
1.1.2 DATASETS
Adult (Kohavi, 1996). The task on the Adult dataset is to classify whether an individual's income exceeded S50K/year based on census data. There are 48842 training instances and 14 attributes, 4 of which are sensitive attributes (age, race, sex, inttivieooutt chosen to be sex, which can be either female male.
Here, the sensitive attribute
Drug Consumption (Drugs) (Mirkes, 2015). The classification problem is whether an individual consumed drugs based on personality traits. The dataset contains 1885 respondents and 12 personality measurements. Respondents are questioned on drug use on 18 drugs, including a fictitious drug Semeron to identify over-claimers. Here, focus Heroin n use, drop the respondents who
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Table 7: Deseription of datasets. 'B' suffix indicates binary variables, 'D' indicates discrete
variables (i.e. >2 classes) '-C" indicates continuous variables.
Dataset
Examples
Features
Sensitive (4)
Outcome (Y)
Adult
45222
Gender-B
Income-B
Single sensitive
Drugs
1885
11
Ethnicity-B B
Drug use-B
German
1700
20
Foreign-B
Income-B B
attributes
COMPAS
6172
10
Ethnicity -B
Recidivism-B
C&C Students
1994
128
Ethnicity-C (x4)
Crime rate- C
Multiple
649
33
Age-C, Gender-B
Performance
sensitive
KDD-Census
299285
40
Sex-B, Race-B, Age-C
Income-
attributes
Credit Card
30000
24
Sex-B, Marriage-D, Education-D
Default-B
Law School
20798
12
Male-B, Race-D
Pass-B
claimed to use Semeron and transform the categorical response into binary outcome: Used" versus "Used", The binary sensitive attribute is Ethnicity.
"Never
South German Credit (German) (Hoffman, 1994). The German dataset contains 1000 instances with 20 predictor variables debtor's financial history and demographic information, which are used to predict binary credit risk (i.e. complied with credit contract not). The sensitive attribute is a binary variable indicating whether the debtor of foreign nationality.
COMPAS (Angwin et al., 2016). COMPAS is commercial software commonly used by judges and parole officers for scoring a criminal defendant's ' likelihood of recidivism. The dataset contains 6172 instances with 10 features. The outcome binary variable corresponding whether violent recidivism occurred ss__iolentt_eecee d) and the sensitive attribute is race, which is binarised into "Caucasian" and "oon-Cacassian" defendants.
Communities and Crime (C&C) (Redmond, 2009). C&C contains socio-economic data from the 1990 US Census, law enforcement data from the 1990 US LEMAS survey and crime data from 1995 FBI UCR. It contains 1994 instances of communities with 128 attributes. The outcome of the regression problem is crime rate within each community ViolentC PerreeeeePPorrre which is continuous value. There are three sensitive attributes, corresponding to ethnic proportions in the communit-raccer
ack, racePct whi te, racePotAsion
Student Performance (Students) (Cortez, 2014). The Students dataset predicts academic perfor- mance in the last year of high school. There are 649 instances with 33 attributes, including past academic information and student demographics. The response variable is continuous variable corresponding to final grade and the sensitive attributes are age (continuous value from 15-22) and sex female, 'M' male).
B.2 TIME SERIES TASK
The data used to develop and evaluate our experiment on fair imitation learning is extracted from the MIMIC-III ICU database (Johnson et al., 2016a), based on the Sepsis-3 cohort defined by Komorowski et al. (2018).
Discrimination in Healthcare. Sepsis is one of the leading causes of mortality in intensive care units (Singer et al., 2016), and while efforts have been made to provide clinical guidelines for treatment, physicians at the bedside largely rely on experience, giving rise to possible variations in fair treatments. Prejudice in healthcare has been reported in many instances- example, healthcare professionals are more likely to downplay women's health concerns (Rogers & Ballantyne, 2008) and racial biases affect pain assessment and treatment prescribed (Hoffman et al., 2016). Thus, critical, when learning to imitate expert policy, that no underlying prejudices are leaked into the learned policy.
Problem Setup. We have access to a set of expert trajectories D = .r..... TN }. where each trajectory is sequence of state-action pairs {($1, 01), (Sr, ar)). The time-vane varying state space is modelled with Markov Decision Process (MDP), i.e. at every time step t, the agent observes current state s, and takes action dt-
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Data. We obtain data from MIMIC-III and use the pre-processing scripts provided by Komorowski et al. (2018) to extract patients satisfying the Sepsis-3 criteria. For each patient, we have relevant physiological parameters, including demographics, lab values, vital signs and intakc/output events. Data are aggregated into hour windows.
State Space. The pre-processing yields 45 feature vectors for each patient at each time step, which are summarized Table 8. We consider gender as the sensitive attribute.
Table 8: MIMIC-III Features. Description of patient features recorded at four hour intervals.
Feature Type
Features
Demographic
Gender, Age. Weight (kg),
Re-admission. Glasgow Coma Scale (GCS). Sequential Organ Failure Assessment (SOFA).
Static
Systematic Inflammatory Response Syndrome (SIRS), Shock Index,
Potassium, Sodium, Chloride, Glucose, Magnesium, Calcium, While Blood Cell Count, Platelets Count, Ricarbonate, Hemoglohin, Partial Thromboplastin Time (PTT), Prothrombin Time (PT), Arterial pHI, Arierial Blood Gas, Arterial Lactale, Blood Urea Nitrugen (BUN), Creatimine, Trunsaminase (SGOT). Serum Centumic-Pyrruvee Transaminase (SGPT),
Lab Values
Serum
Total Bilirubin, International Normalized Ratio (INR).
Heart Rate, Systolic Blood Pressure, Mcan Blood Pressure, Diastolic Blood Pressure, Respiratory Rate, Temperature (Celsius), FiO2, PaO2, PsCO2, PaO2/FiO2 ratio, SpO2,
Vitals
Machanical Ventilation, Fluid Intake (4 hourly), Fluid Intake (Total), Fluid Outpur hourly),
Intake/Output
Fluid Output (Total)
Action Space. We define a binary action for medical intervention based on intravenous (IV) fluid and maximum vasopressor (VP) dosage in given hour window, where a; = 1 represent either both interventions taken, and = 0 indicates no action taken.
Treatment Outcome. The ground truth treatment outcome in each time step evaluated using SOFA (measuring organ failure) and the arterial lactate levels (higher in septic patients). Specifically, the treatment outcome penalizes high SOFA scores and increases in SOFA and lactate levels from the previous time step (Raghu et al., 2017):
SOFA SOFA & SOFA 0) (5125(5 87 Y 0.0251 (st+1
SOFA SOFA)
61+1
2tanh( lactate
o6 lactate)
8++1
Behavioral Cloning. Our proposed framework should work with any imitation learning algorithm as long as predictions of action rewards are differentiable. For now, we will focus on behavioral cloning. The expert's s demonstrations are divided into i.i.d. state-action pairs. We train a neural network as described in the experimental setup to predict posterior action probabilities.
C ADDITIONAL EXPERIMENTS
In this section, we provide additional results to comprehensively evaluate our proposed methods, specifically:
1. DP and EO: While the main paper investigates fairness using EO, Appendix C.1 demonstrates application of F a roocco using DP and CAL notions of fairness.
2. Estimation convergence: Appendix C.2 evaluates the convergence of Fa Pairoooco score estimation on different mini-batch sizes on real datasets.
3. Statistical testing: Appendix C.3 demonstrates how the FairCocco Score can be employed
as test statistic in permutation-based testing for stronger faimess transparency.
4. Sensitivity: Appendix C.4 investigates performance sensitivities, specifically performance-faimess
trade-offs, according to varying numbers of sensitive attributes.
C.I ADDITIONAL RESULTS: EXPERIMENTS WITH DP AND CAL
To highlight FaorC'ss''s compatibility with fairness definitions other than EO, we apply it to demographic parity (DP) and calibration (CAL). We perform the same experiments on 1) binary clas- sification tasks, 2) regression task with multiple sensitive attributes. The experiments are performed using the procedures described in the experimental setup.
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Demographic Parity. DP requires statistical independence between predictions and attributes. Disparate impact (DI) is a metric frequently used to evaluate DP (Feldman et al., 2015):
P(Ý 1|A 1) DI P(Y = 1)
(16)
where and A denote respectively the discriminated and non- disermmminneted groups. The US Equal Employment Opportunity Commission Recommendation advocates that DI should not be below 80%, commonly known as the 80%-rule. 4 DI closer to 1 corresponds to lower levels of disparate impacts across population subgroups. We show the performance of Fa rCocCo for DP in Table 9 and 10, demonstrating superior performance on a benchmark of binary classification tasks as well as protection of multiple sensitive attributes in regression settings.
Table 9: Performance in binary setting. Accuracy (ACC) and DI under DP. NN an unregularised neural network that is used as base learner, the best results are emboldened.
COMPAS
German
Drug
Adult
Method
ACC
ACC
ACC
ACC
Donini at. (2018)
0.70 + 0.02
0.81 + 0.03
0.70 + 0.06
0.93 0.07
0.74 0.03
0.75
0.01
0.72
0.84
NN
0.90 + 0.02
0.39 + 0.32
0.74 + 0.07
1.26 0.54
0,80 = 0.08
0.12 0.22
0.81
0.22
Mary al. (2019)
0.87 + 0.04
0.76 0 0.07
0.71 + 0.08
0.96 0.25
0.80 + 0.06
0.73
0.79
0.83
Steinherg (2020h)
0.03
0.05
0.06
0.13
0.03
0.05
0.77
FCcCoo
0.88 1: 0.03
0.90 1: 0.06
0.73 1: 0.06
1.02 1: 19
0.78 1: 0.02
0.84 1: 0.07
0.83
0.97
Table 10: Protection of multiple attributes. Level of protection provided to individual attributes when all attributes are simultaneously protected under DP. Lowest MSE Fa EcocCo scores are emboldened. (left) C&C dataset, (right) Students dataset.
Join:
macePetllack 00000
meePetWWitte COCCO
racePctAsian COCCO
racePcillisp COCCO
Joint
ape
SEX
Method
MSE
00000
Methad
MSE
COC00
COCCO
CO000
AW
= =101
0.22
0.20
+.0.5
0.16
+.066
0.24
0.03
0.03
0.01
0.09
+00.3
NW
-1.05
0.25
#:0.5
0.16
=1.03
0.13
+-0.07
0.11
0.53
0,09
I007
0.15
0.85
0U07
0.30
1.0%
0.34
6.03
FAGL
=: 000
0.0.2
2.001
0.04
0,03
#: 0.002
FACL.
-102
4: 1:0.0
= 11011
0.02
FARMI
0.60
=: 0.07
0.12
+ 0.033
0.15
+.0.2
0.15
0.02
0.04
= 0.00
0.06
+ 0.03
FARMT
0.35
0.00
+ 0.003
0.11
0.00
0.00
+.00
0.05
Fa1100000
0.49
1L06
1,08
+ 0.02
RM8
0.01
0,07
IU02
0.13
0.01
RJ14
* 0.01
Fa1rC0020
0.33
1..02
+ 0.02
1.06
0.03
11.1
0.02
0.04
Calibration. CAL requires conditional independence between target and sensitive attributes given predictions. As the conditioning variable is continuous, we report the Fa i rCOcCO score on the same experiments. We see in Table 11 and 12 that Fai =cocco achieves superior fair and predictive outcomes under different definitions of fairness when compared to other methods.
Table 11: Performance in binary setting. Accuracy (ACC) and FairCocco (COCCO) under CAL; the best results are emboldened.
COMPAS
German
Drug
Adult
Method
ACC
COCCO
ACC
COCCO
ACC
COCCO
ACC
COCCO
Donini (2018)
0.03
0.05
0.13
0.78
0.16
NN
0.90 0402
0.07 0.02
0.74 : 0.07
0.07= 0.03
0.80 :: 0.08
0.2 0.08
0.84
0.18
Mary al. (2019)
0.87 + 0.12
0.07 0.03
0.71 + 0.11
0.06--0.02
0.79 0.03
0.08 0.03
0.81
0.15
(Stcinberg al.,
6.79
Fa1r00000
0.89 d: 0.02
0.02 1: 0.02
0.71 0.05
0.02 1: 0.01
0.78 179::0.00 0.06
0.11 = 0.06
0.83
0.11
C.2 FAICCOCCO ESTIMATION
In this section, we provide additional results convergence of FairCoCCO Score estimation as a function of batch size, similar to the experiment performed in the main paper. We show convergence on Adult and German dataset in Figure 2. We note that while convergence of estimation depends on properties of different datasets, the estimation of Fa rcocco Score stabilizes at batch sizes 256.
uni formgui de ins..com com.
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Table 12: Protection of multiple attributes. Level of protection provided to individual attributes when all attributes are simultaneously protected under CAL. Lowest MSE and Fair roCCO score are emboldened. (left) C&C dataset, (right) Students dataset.
Join COCCO
racePetBlack COCCO
racePerWhite COCCO
racePetAssion COCCO
racePctllup COCCO
Joint
Method
MSC
Method
MSE
COCCO
COCCO
COCCO
0.22
1. 0:00
0,16
0.003
0.16
0.004
0.13
1.000
0.07
1.0.08
1. 1.0.3
NN
1:0.55
0:25
1:0.5
0.31
0.99
0.19
0.11
0.00
4.0044
FACI
1.0.3
1.002
1.0022
1.002
0.05
0.13
0.12
0.05
+.033
PARMT
+.006
0.36
0.35
0.001
+
Falecoccu
0.05
0.07
0,03
PatxO0cco
1 0.05
0.37
1. 0.02
0.04
1. 0.00
0.03
02
==
0.3
0.25
A- 0.5
0.15
0.2
FarcOCCO
Falcocc
0.15
DED
0.1
0.1
0.00
0.05
Size
Baltch Bize
(a)
(b)
Figure 2: Estimation of FairCOCCO Score. (a) Adult dataset, (b) German dataset.
C.3 STATISTICAL TESTING
We demonstrate how the proposed fairness measures can be employed as a test statistic to perform statistical tests, resulting in stronger guarantees and transparency (Fukumizu et al., 2007; Gretton et al., 2005). We highlight that while other fairness measures (MI and MCC) can be developed as test statistics, the empirical estimation these measures involve multiple levels of approximations, and is unclear whether the approximated statistics still retain the theoretical properties. Figure 3 shows the distributions of predictions with fairness regularization. Notably, EO only requires statistical independence between predictions and sensitive attributes given true outcome, whereas DP enforces "strict" independence between predictions and attributes.
Table 13: Statistical testing. Eecuracy-fairness trade-offs under different fairness notions and corresponding test of statistical significance. (left) EO setting, (right) DP setting.
ACC
DEO
COCCO
p-value
ACC
Dr
COCCO
p-value
0.0
78.33
0.66
0.21
0.00
0.0
78.33
3.05
0.07
0.00
0.2
76.67
0.39
0.14
0.14
0.2
72.56
1.54
0.03
0.04
0.5
70.36
0.07
0.03
0.45
0.5
69.33
1.77
0.01
0.09
1.0
67.78
0.03
0.02
0.74
1.0
67.38
1.13
0.01
0.14
2.0
60.57
0.00
0.01
0.90
2.0
64.60
0.92
0.00
0.27
As the null distribution is not known (Fukumizu et al., 2007), permutation testing is performed. Table 13 reveals the accuracy-fairee fairness trade-offs and p-values under different regulation strengths. The p-values indicate the probability of observing the test statistic under null hypothesis of (conditional) independence. As we expect, stronger faimess regularization leads to lower levels of unfairness as measured by DI and DEO, well as stronger guarantees in statistical tests. For example, at À 2.0, we can say with 90% chance that predictions are conditionally independent of sensitive attributes (under EO) or 27% chance that predictions are independent of sensitive attributes (under DP).
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LHb Brapet
Goaghd
Tabe SOAEZZ
301921
0ub
6.2
6.r
30
155
100
030
1.00
0.0
Figure 3: Visualizing FairCOCCO regularization. (Top) distribution of predictions for label 1 of different group memberships under EO. (Bottom) distribution of predictions for different group memberships under DP. Predictions are produced by regularized logistic regression model with = 0, 0.5, 1.0, respectively, across each row.
C.4 SENSITIVITY ANALYSIS: ACCURACY-FAIRNESS TRADE-OFFS
One of the key contributions of this study is the introduction of a differentiable fairness penalty that naturally extend multiple sensitive attributes. In this section, generate the frontier of possible values on three experiments to better evaluate the sensitivity our proposed methods to different numbers of sensitive attributes:
0.165
0.250
0.160
0.225
0.20
0.200
0.175
0.5
0.100
0.125
0.075
1:50
0:16
0.50 0.85
0.70 Accuracy
0.75 0a55
Figure 4: Fairness-accuracy trade-off. (left) C&C dataset with four sensitive attributes; (middle) students dataset with two sensitive attributes; (right) drugs dataset with three sensitive attributes.
Regression on C&C with 4 attributes: racePctBlack, aacePctAsian. racePctWhie and racePctHisp.
Regression on Students with 2 attributes: age and gender, Binary classification task on Drugs with attributes: age, gender, and ethnicity.
As Figure illustrates, similarly, fairesss and prediction outcomes are achieved at various number of sensitive attributes.
20


Reviews:
[{"review_id": "r-ppv_tj3e1", "reviewer": "Reviewer_iTEE", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The paper presents a new fairness metric that can measure fairness for multiple sensitive attributes of any type. The author also developed a learning algorithm by using their fairness criteria as a regularization term.", "summary_of_the_review": "See the comments in the \"weakness\" section. \n\n..........................................\nI thank the authors for explaining some of the concerns. However, I still think that the novelty of the paper is limited and not enough for ICLR. However, after reading other reviewers comments and authors feedback, I've decided to increase my overall rating.", "strength_and_weaknesses": "The paper aims to solve an important problem which is ensuring fairness for any number and any type of protected attributes.\n\nBut the paper suffers from lots of weaknesses:\n- The multiple protected attribute-based fairness notions are not new. There are several existing works on this as follows: Multicalibration (U. Hebert-Johnson et al, 2018), subgroup fairness (M. Kearns et al, 2018), and intersectional fairness (J. Foulds et al, 2020). The paper doesn't motivate enough why we need a new multiple attribute-based fairness metric?\n- The proposed learning algorithm is also not new. Adding fairness metric as a regularization term with a trade-off parameter is the standard approach in many existing fair ML work. For example, Foulds et al, 2020 developed a backpropagation-based learning algorithm with the regularization term, designed using intersectional fairness criteria. \n- Regarding the discrete and continuous outcomes. Agarwal et al 2018 proposed a reduction-based constrained optimization method that can ensure fairness in classification task for single or multiple protected attributes. Furthermore, they extended their reduction based method for regression task (Agarwal et al 2019) with fairness metric for continuous outcomes.\n- I am very surprised that authors didn't compare/analyze their proposed metric and learning algorithms with any of the multiple protected attribute-based existing fair ML works in their experiments. This is the biggest weakness of the paper.          \n\n ", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "T5ISjojNuw", "reviewer": "Reviewer_yfN2", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The paper addresses the extension of existing fairness notions in ML such as Demographic Parity, Equal Opportunity, and Calibration to settings where the both the target and sensitive attributes are potentially multivariate and continuous.\n\n It proposes a metric (FairCOCCO) based on the cross covariance operator over reproducing kernel Hilbert spaces that can be used to both measure and regularize existing ML systems for fairness-aware learning, and shows comparable or better results over existing methods on multiple real-world datasets.", "summary_of_the_review": "I think the paper has merit, and proposes a well-motivated and grounded application of existing literature to easily extend existing fairness definitions into multivariate target and attribute scenarios. The main drawback of the paper seem to be the relatively equivalent results to existing methods, and a potential lack of technical novelty.", "strength_and_weaknesses": "Strengths:\n\nThe paper does a good job on explaining the merits of the cross covariance operator as a viable tool to extend DP, EO and Calibration to multivariate and potentially continuous target and sensitive attributes. The presentation is solid, the regularization objective is well motivated, and the experimental results  show improvement over the examined baselines.\n\nWeaknesses:\n\nI think the novelty of the work is somewhat limited. The main theoretical results presented in the paper are taken directly from Fukumizu 2007, though the application to fairness metric is, to my knowledge, novel.\n\nSome key implementation details are shown in the appendix, rather than discussed in the main paper, such as the closed form expression for FairCOCCO in Appendix A, which also seems taken from Fukumizu 2007. \n\nUpon seeing the discussion on Appendix C.2 on the impact of batch size on the FairCOCCO measure, I would like to ask the authors if comparable hyper parameter optimization was also performed for the competing baseline methods. I perhaps missed a similar analysis on how important the regularization strength term $\\lambda$ was to the final results.\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "QlZ-cnldsfD", "reviewer": "Reviewer_stjQ", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The paper proposes a new fairness metric, FairCOCCO SCORE, that can quantify unfairness among groups are distinguished based on multiple sensitive attributes of different types. It measures the unfairness in reproducing kernel Hilbert space based on cross-covariance operators. Based on this measure, the paper further proposes a regularization term that can be used for learning fair predictors.\n", "summary_of_the_review": "While the paper proposes a new fairness measure that is applicable to settings where groups are distinguished by (continuous) multiple sensitive attributes, it is not clear to me why the proposed measure is better than the existing unfairness measures that are also applicable to continuous sensitive attributes.   ", "strength_and_weaknesses": "Strength:\n1. Unlike most existing studies that focused on measuring/remedying unfairness against a single binary sensitive attribute, this paper focuses on the more refined (sub-)groups that are potentially distinguished based on multiple, continuous sensitive attributes  \n2. The paper is well-organized, and the literature review seems to be extensive.\n3. The paper conducted extensive experiments on multiple datasets and compared the performance of the proposed method with many existing algorithms.  \n\nWeakness: \n1. The paper proposes a new kernel method-based fairness measure. However, there is no theoretical support to justify why the proposed measure is better than the existing measures. Its performance is only evaluated empirically in experiments. \n3. On the comparison in experiments: the authors compared the proposed FairCOCCO with other methods on various datasets. Many results are competitive, and it is not obvious that FairCOCCO outperforms other methods. For example, in Table 9, the bolded results are not the best, and the accuracy and fairness are very similar under different algorithms. ", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "FtCNch-dFjN", "reviewer": "Reviewer_V8La", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The paper proposes to practical method called FairCOCCO that can incorporate multitype/multivariation sensitive attributes for fairness-aware learning. The method simply develops the kernel-based measure of fairness and uses it as a regularizer for learning a predictive model. The experimental results seem promising, but the comparing baselines are relatively old. \n", "summary_of_the_review": "Overall, while the proposed method is simple, the experimental results seem promising. The actual computational complexity comparison would be helpful for making the final decision. ", "strength_and_weaknesses": "Strength\n- The paper proposed a novel in-processing based fairness-aware learning method based on FairCOCCO, which can be computed via using kernel based conditional cross covariance operator. \n- The experiments show that the method can be applied in various setting that involve multi-variate, continuous-valued sensitive attributes. \n- It outperforms several baselines on multiple settings. \n- Detailed analyses are given in Supplementary. \n\nWeakness\n- Some recent baselines are missing, e.g., \n [Jung et el.,  Learning fair classifiers with partially annotated group labels, CVPR 2021]\n[Quadrianto et al., Discovering fair representations in the data domain. CVPR 2019]\n[Jiang et al., Identifying and correcting label bias in machine learning. AISTATS 2020]\n- Result on COMPAS is a bit dubious -- DEO is 0.00??\n- While the result on the relaxation on computing COCCO score is given in the appendix, what is the actual computational complexity? Fairly comparing with the CPU time would be also beneficial. \n- Some more results on vision dataset such as UTKFace would be also helpful. \n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}]

Author rebuttal:

Decision: Reject