Fair Kernel Regression through Cross-Covariance Operators
Authors: Adrian Perez-Suay
Paula Gordaliza
Jean-Michel Loubes
Dino Sejdinovic
Gustau Camps-Valls
Venue: Accepted by TMLR ()

Abstract: Ensuring fairness in machine learning models is a difficult problem from both a formulation and implementation perspective. One sensible criterion for achieving fairness is Equalised Odds, which requires that subjects in protected and unprotected groups have equal true and false positive rates. However, practical implementation is challenging. This work proposes two ways to address this issue through the conditional independence operator. First, given the output values, it is used as a fairness measure of independence between model predictions and sensitive variables. Second, it is used as a regularisation term in the problem formulation, which seeks optimal models that balance performance and fairness concerning the sensitive variables. To illustrate the potential of our approach, we consider different scenarios. First, we use the Gaussian model to provide new insights into the problem formulation and numerical results on its convergence. Second, we present the formulation using the conditional cross-covariance operator. We anticipate that a closed-form solution is possible in the general problem formulation, including in the case of a kernel formulation setting. Third, we introduce a normalised criterion of the conditional independence operator. All formulations are posed under the risk minimisation principle, which leads to theoretical results on the performance.
Additionally, insights are provided into using these operators under a Gaussian Process setting. Our methods are compared to state-of-the-art methods in terms of performance and fairness metrics on a representative set of real problems. The results obtained with our proposed methodology show promising performance-fairness curves. Furthermore, we discuss the usefulness of linear weights in the fair model to describe the behaviour of the features when enforcing fairness over a particular set of input features.

Published in Transactions on Machine Learning Research (06/2023)
Fair Kernel Regression through Cross- Pross-Covariance Operators
Adrián Pércz-Suay
Adrion. Perion.Perez@uien PerezOuv.es
Image Processing Laboratory (IPL)
Universitat de Valência
Paula Gordaliza
pgortaliza@boamathooog
Basque Center for Applied Mathematics Universidad Pública de Navorra
MicheMichel Loubes
Baches@@math.o www.-touuuusse..ff
Institut de Mathématiques Toulouse
University Toulouse 3
Dino Sejdinovic
dino. aeto.ssjddnnoviiccaalllaaaa eduu.
School of Computer and Muthematical Sciences University of Adelaide
Gustau Camps- Valls
Gustau. Camps Stmmps@ur.c.
Image Processing Laboratory (IPL)
Universitat de València
Reviewed on OpenReview: openreview, net/ forum? id- MyQ1ee11
Abstract
Ensuring fairness in machine learning models a complex problem from both a formulation and implementation perspective. One sensible criterion for achieving fairness Equalised Odds, which requires that subjects in protected and unprotected groups have equal true and false positive rates. However, practical implementation is challenging. This work proposes two ways to address this issue through the conditional independence operator. First, given the output values, it is used fairness measure of independence between model predictions and sensitive variables. Second, it is used as a regularisation term in the problem formu- lation, which scoks optimal models that balance performance and fairness concerning the sensitive variables. To illustrate the potential of our approach, we consider different scenar- ios. First, we use the Gaussian model to provide new insights into the problem formulation and numerical results its convergence. Second, we present the formulation using the conditional cross-covariance operator. We anticipate that a closed-form solution is possible in the general problem formulation, including in the case of kernel formulation setting. Third, we introduce a normalised criterion of the conditional independence operator. All formulations are posed under the risk minimisation principle, which leads to theoretical results on the performance. Additionally, insights are provided into using these operators under a Gaussian Process setting. Our methods are compared state-of-the-art methods in terms of performance and fairness metrics on representative set of real problems. The sults obtained with proposed methodology show promising mnrformance-fairnee curves. Furthermore, we discuss the usefulness of lincar weights in the fair model to describe the behaviour of the features when enforcing fairness over a particular set of input features.
1 Introduction
Machine learning systems are increasingly deployed in many applications with important socictal, economic, environmental implications. Individuals and society affected, handling algorithmic bias obtain

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fairness guarantees has become an important research topic. The term bias generally refers to inclination or prejudice for or against one person or group based on their characteristics and need not have negative connotations, In machine learning, bias can exist in many shapes and forms and can be introduced at any stage in the model development pipeline (Fazelpour and Danks, 2021; Pessach and Shmucli, 2022; Besse et al., 2021). When such bias is related lo the information in specific characteristics involving sensitive information, the concept of algorithmic fairness raised. In this sense, an algorithm is considered fair if it makes predictions that do not favour or discriminate against certain individuals or groups based on sensitive characteristics. This definition, however, needs to be properly formalised. Therefore algorithm design, implementation, and deployment need a formal and concrete definition of favouring or disadvontaging a certain population subgroup. Such quantification of favour or discrimination is one of the fundamental points of contention where ethical issues come into play. Artificial Intelligence (AT) regulation is beginning to be formalised, and it is vitally important that the scientific community is part of this. Indeed, the Europcan Union has recently released the "Ethies guidelines for trustworthy AI" report where it is stated that unfairness and biases must be avoided
The definition of fairness is an elusive question and the first cause of controversy in the field. In general, fairness objectives can be categorised into individual and group fairness. On the one hand, individual fairness (Dwork et al., 2012; Joseph et al., 2016; Kusner et al., 2017; Kim et al., 2018; Heidari et al., 2018; Joseph et al., 2018) try to achieve "similar treatment to similar individuals". On the other hand, group or statistical fairness focuses on reducing inequalities at a group level, where groups may be defined using sensitive variables such as race, gender, age, or disabilities. The objective is expressed through a measure of statistical independence between the variables involved in the learning process. This approach is the most popular in the literature because of its case of application to any data distribution (Calders and Verwer, 2010; Kamishima et al., 2012; Kleinberg el. al., 2016; Hardt et al., 2016; Chouldechova, 2017; Zafar et al., 2017; Donini et al., 2018; Agarwal et al., 2018; Williamson and Menon, 2019). The first proposal along this line is the demographic parity (DP) (Kamiran and Calders, 2009; Kamishima et al., 2011; 2012; Pérez-Suay et al., 2017), which requires the outcome to be independent of the sensitive variables. However, it can impair the ultimate utility we want to achieve (Zhao and Gordon, 2019). In light of this, when the ground truth is available, which is plausible in the supervised learning setup, this independence can be relaxed on the true value of the target. This metric is called Equalised odds (EO) (Hardt et al., 2016) and requires the truc positive and false posilive rates to be the same across different groups
Obtaining statistical fairness, either DP or EO has been mainly addressed by (i) preprocessing the data to remove sensitive dependencics explicitly (Kamiran and Calders, 2009; Luo et al., 2015; Ristanoski et al., 2013; Gordaliza et al., 2019) or (ii) modifying classification rules incorporating the fairness raints somehow (Pedreshi et al., 2008; Ruggieri et al., 2010). Furthermore, to a lesser extent, (iii) post-processing the outputs from the learning algorithm with additional minimisation constraints guarantee fairness can also be found in the literature (Wei et al., 2021). Most methods in the first group seek fair representation learning, i.e. achieving fairness through finding an optimal way to preprocess the data and map it into a latent, space where all information about the sensitive variables is removed. After such preprocessing, standard machine techniques are employed to build predictive models. Examples of these methods include Zemel et al. (2013); Kamiran and Calders (2012); Adebayo and Kagal (2016); Calmon et al. (2017); Madras et al. (2018); Song et al. (2018); Zhang et al. (2018). Down-weighting or removing sensitive fcatures has also been proposod (Zeng et al., 2016), yet it is often insufficient as related variables may still enter the model (Besse el al., 2021).
The second group (ii) consists of empirical risk minimisation (ERM) approaches that incorporate fairness constraints in the learning process using restrictions or an additional penalty term. From a theoretical point of view, if we denote by L a loss function, a fair algorithm can be achieved by restricting the loss minimisation over a general class F of algorithms to a subclass Frair satisfying a chosen fairness criteria. Statistical consistency results for the quantification of the loss in the accuracy of an algorithm under fairness constraints have been the topic of many authors, including Donini et al. (2018) or del Barrio et al. (2020). In the latter, authors introduced the term price for fairness for the difference between the risk in the fair class and the general Bayes risk (achieved for the best, but possibly unfair, algorithm), namely E(Frir)) =
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inf JeFre, ((Cff)))iifferB(((())) It is essential to realise that awareness of sensitive information conditions the learning scenario and, consequently, the definition of such a fair class. We propose in section 3.2 bias- unaware kernel regression setting where the sensitive information is known and it is decided not to be an input to the algorithm and to be used only to impose fairness constraints. In this way, we obtain fair predictors both in the sense of DP (section 3.3) and EO (sections 3.4,3.5). Additionally, we propose an exlension to the bias-aware framework in the Appendix, where such sensitive characteristics of the individuals are not observed but assumed to be present in the input data and solve it for the particular case of fair Gaussian models in section 4.
The price for DP has been widely studied in the regression setting, e.g. including mini-max results for computing {(FP)) = inf B(C(S)) - seF B(C())), where FDP detones the fair class of algorithms satisfying the DP criteria (Gouic et al., 2020; Chzhen et al., 2020). However, there are few contributions to controlling such a quantity (FFD) in a general nonlinear setting. This is mainly due to the challenge of obtaining fair models, which could be characterised as the optimal model in the least-squares sense. Fortunately, the fairness regularisation setting (Kamishima et al., 2012) allowed to provide a closed-form solution for the general noulinear setting (Pérez-Suay et al., 2017). In this framework, we precisely obtain an upper bound (in Thm. 3.7) on the difference between the fair estimator and the unrestricted RKHS regressor. On the other hand, it has been shown that obtaining algorithmic outcomes accomplishing EO is generally challenging to formulate and, more notably, to optimise (Woodworth et al., 2017). That is, controlling {(FE)), where FBO denotes the fair class of algorithms satisfying the EO criteria, is not feasible in general. As an exception, it is possible to obtain a closed-form solution to the minimisation problem in the case of linear models. However, this becomes infeasible in a general nonlinear setting. To address this issue, we propose imposing EO through mcasure of conditional dependence via conditional cross-covariance operators defined on reproducing Kernel Hilbert spaces (Fukumizu et al., 2004) as additional regulariser of the risk minimisation problem (2). This allows us also to obtain upper bounds provided in corollary 3.9, and with a normalised version of the conditional cross-covariance operator in corollary 3.12.
Contributions. The formulation of the fair learning problem proposed here is distinct and differs from previous proposals. The paper provides four main contributions in bias-mnaware setting: (1) control of the loss between the best DP fair regressor (Pérez-Suay et al., 2017) and the unconstrained RKHS regressor; (2) the use of the conditional independence measure given by the conditional cross-covariance operator (Fukumizu et al., 2008) as an FO measurement in the general kernel regression setting; (3) a closed-form solution to the EO fair optimal predictor in such context; as well as (4) an upper bound for the loss between the best EO fair regressor and the best unconstrained one. It is also worth mentioning our proposal of a probabilistic treatment of this case with Gaussian Processes (GPs) in the Appendix. Moreover, in a bias-aware setting, two additional contributions are (5) a closed-form solution to the EO fair optimal predictor under the linear regression model for multidimensional X and S; (6) proposal to extend previous contributions (1-4) to the bias-aware kernel setting in the Appendix.
Structure of the paper. The remainder of the article is structured as follows. First, in section 2, we present the penalised empirical risk minimisation framework for the fair learning supervised problem. Then, the general kernel setting is studied in section 3 in this regularisation framework using two different fairness penalties to the loss minimisation problem: the HSIC operator as the DP penalty in section 3.3; the conditional cooas-cov operator as the EO penalty in section 3.4, as well as a normalised version of the latter in section 3.5. Later in section 4, we solve the fair minimisation problem under the standard regression model. Finally, we give empirical evidence of performance in both synthetic and real-life experiments 5. Appendices contain technical details, a probabilistic treatment based on GPs, and information about the data and implementation issues. A working implementation, demos and code snippets are available at https:/ wwwp//wwwwwwwww/ ww .es/ pesss//s/aaaa///e data/ code/ /002 FACIL.zip.
2 Problem setting
The crucial aim of fair learning is designing procedures to remove or control the influence of sensitive information on the forecast that could lead to unfair decisions. Let us denote a general class of ML algorithms by F. We consider a supervised fair learning setting where an algorithm f E F is designed to learn the
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relationships between imput variables and target variable Y R. Here consider both imput variables multidimensional, namely X E X C R2, S E S C IRC, with d, 2 1.
The learning setting is different depending on the algorithm's awareness of the sensitive information. On the one hand, in most fundamental problems, the input can be separated into a sensitive part S E S and a non-sensitive, but a proxy of the former, part X E X. In this bias-unaware scenario where J = ( X R), even if the algorithm f is trained only from X, it may be able learn the information in S, resulting in potentially biased outcomes y f(X) represented in Figure 1. The other possible formulation of the problem is consider bias-aware setting where the algorithm f(X) is trained from the complete imput (X, S). This framework is helpful in fields such Econometrios, where it is common to assume the presence of an unobserved sensitive variable S within the input data X, as described, for instance, by Fabris et al. (2022) or Bird et al. (2019), and references therein.
data X =
Figure 1: Bias-unaware scenario.
From a mathematical point of view, group fairness consists of statistical independence between the predictions Y and the sensitive variable S (del Barrio et al., 2020). The most stringent and common definition is demographic parity (DP), which requires the independence between the outcome and the protected attribute Y I S. When the ground truth is available, which is plausible in the supervised learning setup, the equalised odds (FO) criteria relaxes this independence on the true value of the target, namely Y 1 S Y. Hence, building fair algorithms through an ERMM-aased approach turns the minimisation of the risk B(C(ÿ)) = EI(Y - 8)a] into a multi-objective problem which requires a trade-off between the model's accuracy and fairness. While in some fields of application, is desirable ensure the highest possible level of fairness, in others, including health care or criminal justice, performance should not be decreased since the decisions would have serious implications for individuals and society. Establishing such trade-off has been one of the important challenges in the MT, community in the last years, and wide array of approaches has been proposed in the literature (Dwork et al., 2012; Chouldechova, 2017; Zafar et al., 2017; Gordaliza et al., 2019; Chiappa et al., 2020). The majority of the works focus on a bias-unaware learning setting where the risk minimisation problem is
inf B(C(f)) = inf (E - {(X)) F= == (5 X X
(1)
FEF
JEF
Set the learning sample as i.i.d. observations (æ1, 81, 31), ........ (In, 8n, Mn) of (X, Y) drawn from an unknown distribution P. Then, for a given model class F, consider Fn the best model that can be estimated by minimising over F,
Ia arg min- FEF (w: (x:) +=:)))))++0(()) }
(2)
where {))) acts regularizer of the predictive function and controls the smoothness and complexity of the model. We denote by '* the oracle rule
f E arg min Er((()) + XQ(J). JEF
which is the best (yet unknown) predictor that could be constructed if the true distribution were known.
Now we want to additionally impose group fairness constraints. For this, we include an additional regular- isation term S) into the general problem (2) accounting for independence between the predictive function if and the sensitive variables in S as follows
(f) = Σ(s f(x,)) + AQ(S) + +I[((())) st),
(3)
1=1
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where A. 2 O. The solution fu,X,4 argmingeF (f) is the optimal fair predictor. Notice that the above formulation (3) allows to impose the two notions of group fairness. Indeed, T(f(z))):: 8:) would be a measure of dependence between f and S in the case of imposing DP, or fIY and S in the case of EO.
In this work, we propose a regularisation approach for (1) to obtain a solution for a bias bias- fair predictor in general non-linear setting under both fairness criteria. To do this, we measure the independence via two different covariance operators: the well-known Hilbert-Schmidt Independence Criterion (HSIC) for DP and the conditional covaricove operator (Fukumizu et al., 2008) for EO. Additionally, extension to the bias-aware setting where minimisation (1) would be done from functions f {f X' x S + RR). In particular, we address this case for Gaussian models and obtain a closed-form solution for the optimal fair predictor and & lower bound for the minimal excess risk.
propose in
the Appendix
3 Fair kernel regression through dependence regularisation
Several authors have considered the framework of penalised regression either to achieve fairness or to provide guarantees for independence. In Kamishima et (2012), the authors propose use a logistic loss and a version of the mutual information estimator. In the framework of RKHS regression, Pérez-Suay al. (2017) quantified the dependence using the Hilbert-Schmidt Independence Criterion (HSIC) based the norm of the particular covarianv operator the corresponding RKHS. In seminal paper (Gretton et al., 2005), closed-form solutions to the kernel ridge regression are obtained even when the sensitive variable is multidimensional.
3.1 Notations and preliminaries
We shall first introduce some notations on kernel theory that will be used in the following sections.
Definition 3.1 (Metric spaces, kernels and adjoint operators). Let X be metric space and assume K X x X + R is a measurable kernel with associaled RKHS HK and canonical feature map á X + HK- Suppose bounded, i.e. supzex K(x, x) =: x² < co. By So : Hx + (2 Px) we denote the inclusion. The adjoint operator 5 (XX HK is given by
5 / ((=))(=)dPx))
for any g E (x),,P))
We further introduce some relevant operators needed in the following:
Definition 3.2 (Covariance operator). The non-centred operator To == HK + Hx that is given by
TØs / (5, ({,(())) dPx(x),
and the integral operator Lo = SoS3 (x), + L2 Px), given by
Log= = | g(z) dPx(x).
It can be shown that both operators To and Le are positive and trace class, hence compact, satisfying |T S tr(T)) ? || S tr(L)) < x²
where tr(-) denotes the trace of an operator. Note that for any f Hk we have
|TTofll [|Sof]l(P(P))
For given data x = (f11,.., xta) drawn i.i.d. from Px we define the sampling operator by
So : Hx
R S Sof (.,

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One easily verifies that the adjoint operator Sz R Hx is given by
Soy ,50(xz))
for any = (y1.-.,,h)) Un) ) E R With these definitions, we additionally introduce To Hx + Hx as
:= S
n Σ
Φ(zz) ) @ p(z,)
and T Rm -> I as :R"
L. = S KK
11
where K = (K (K (1))) denotes the kernel matrix.
3.2 unaware fair kernel regression setting
Definition 3.3. Nonparametric model, feature maps and reproducing kernels Let X and S be standard Borel. Consider the non-porametric regression model
Y fx)
(4)
where E|E[X, S] = and (X,S,Y) x 1- M M] are random variables, distributed according to some unknown measure P... We consider the corvesponding feature maps
6 Hk, : Hks-
into seporable RKHSs (Hk, (Kx,()))) (x) and (HKs. (s)))) with associated reproducing kernels
K( (b(x), (()))) Ks(s,s')
see e.g. (Steinwart and Christmann, 2008, Ch. 4).
In bias-aware framework, the sonsitive information given in conjunction with the other characteristics of the individuals X in the data. Therefore, the problem consists in minimising (2) over functions f + R. It well-known that the regression function f*(x) E x] E T,2 Px) is the unique minimiser of L. Now, given sample (x, : ((I1, $1, 1/), ..., (Ii, Sn; mn)), we consider the empirical minimisation problem (2) with the usual penalty P(f)) [| Using the representer theorem (Schölkopf et al., 2001),
the minimiscr fn,x = arg min (f) can be written as
fax())= = Σ <,)
where model weights ån are estimated by minimising the new loss in the dual arg min [yy Kall + Xa Ka, which has closed-form solution â, = (K + X1)-ly.
To impose group fairness constraints, we include an additional regularisation term into the general problem (2) accounting for independence between the predictive function and the sensitive variables in S:
Ln)x(())= = Σ (u - ((x))) MIII + (I((
(5)
where 7, 1 2 0, and the predictive function is the result fr,A, arg miny (f). For the sake of readability, in the following, we omit subindex n and denote the minimiser of (5) by fDP or fRO depending the fairness penalty T.

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Remark 3.4. This formulation of the problem could be extended to bias-aware framework where the sensilive information. is ossumed to exist bul. not observed; that is, il is hidden in the complete inpul. X = (X, from which f is trained. See more details at Appendix A.1
In the following sections, we obtain solution for fair predictor in each case by measuring the independence via two different covariance operators, precisely the well-known Hilbert-Schmidi Independence Criterion (HSIC) for DP and the Conditional Cross-covariance Operator (Fukumizu et al., 2008) for EO.
3.3 DP-fair kernel regression through HSIC operator
Mcasuring statistical independence using reproducing kernels has been investigated in Gretton et al. (2005; 2007). We recall some basic definitions and facts that we need the sequel. Tn particular, the Hilbert- Schmidt Independence Criterion (HSIC) mcasures dependence between random variables X on a domain and S on domain S with joint distribution Pxs. Roughly speaking, HSIC measures the distance wetween embedding of Pxs and the product of the marginals Px & Ps into an appropriate RKHS.
To this end, let us introduce useful operator in this work:
Definition 3.5. Cross-covariance operator Csx: Hx + HKs
Csx == (sx((xx(((SS ps) @ (6(X) - px)] = ( - ps @ px.
Given sample sx = ((x1, ..... (Ins sn)), the empirical cross-covariance operator Cox Hk HKs is defined as
Cax =
1 Σ
x(t)) 2 v(tz) - As & pxx
==1
where
P. = - Σ
((8g)
Ax
1 = Σ ((z n
Then HSIC(Psx HK, HKs) ) = |Csxx ll:ss
We solve the minimisation problem by regularised empirical risk minimisation. In particular, fairness as DP is incorporated by adding the additional penalty term [(((z)), 8:) = / which can delect statistical independence f IL S. A short calculation, after applying an extension of the representer theorem, shows that the minimiscr of the above problem (5) is given by
DD = -f + AI -
(6)
Remark 3.6. The assumptions required for the representer theorem (Schölkopf et al., 2001) are met in our selling. Firslly, we employ slandard training sample {(x4, v:)) C Rd x R. Secondly, the function used over the regulariser is the square function g(Ilfll) = I[ll which is a strictly increasing real-valued function. Thirdly, since we utilise norms over kernel matrices, the choice of the cost function can be any arbitrury error function.
Moreover, the extension of the representer theorem, which allows for the addition of any penalty I(f), is valid if such penalisation depends only on point values H,. Indeed, sollowing the usual proof in Schölkopf et al. (2001), the solution to the minimisation problem can be projected onto the space generated by (K(,,1),.... [(,,xa)) Thus, we have f = fx + h, where fK = E, (,,KK,,,, and the function h is orthonormal to the span of the above space. In particular, since h((x;) = (h(-), K(,x;;), utilizing the repro- ducing properly, we find h(x;)) =0. O.
This implies that if the penalty [(fK+ Z 2(KK) is equal or smaller than that of functions where h O, and consequently, for those functions as well. Based on this, it can be deduced that the solution the minimisation problem can be expressed in terms of functions generated by K K(,
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We want to emphasise that our use of the regulariser |C enables obtain the optimal estimator in closed-form solution, which differs from the HSIC definition. 'hhe distinction between these two approaches is discussed in Section 3.3 of the manuscript Li et al. (2022). This enables us to obtain closed-form solutions for the laiulas--e optimisation (EO) formulations derived in the subsequent sections.
The following Theorem controls the fair penalised estimator and the unrestricted RKHS regressor. Theorem 3.7. For any À > 0 and μ O, we have almost surely
P fxollC((P)) S MM 1 41
The proof of this Theorem is provided in Appendir A.2.
The following lemma, identified as Lemma 3.8, demonstrates the relationship between the loss incurred by the fairness regressor and that of the unconstrained regressor.
Lemma 3.8. Let be I the best predictor then for any À 2 0 and μ 2 0, we have almost surely
r - ADP
(x(P)) s $||5~ Ba 1P(Px) + +11520- 12. fhP (x((P))
To prove this lemma, add and subtract the term fΛ and subsequently apply the triangle inequality.
3.4 EO-fair kernel regression through conditional cross-covar operator
The hardness of implementing EO as the fairness constraints into statistical methods usually derives from relaxations of that condition, such Equalised correlations equality of opportunities. In this work, we directly impose such a fairness condition by considering conditional independence measure as penalty term to the usual risk minimisation problem.
Since it is well-known that the optimal fair predictor in the sense of EO, namely arg minjeFPo (f,) is not Teasible to compute (Woodworth al., 2017), we propose alternatively quantify the fairness regularizer I(f, S) through the conditional cross-covariance operator
-
- -CyyCr
(7)
The above operator was introduced in Fukumizu et al. (2004) as measure of conditional independence between random variables. The measure was further formalised later in Fukumizu et al. (2008). This work introduces the conditional independence operator as a fairness regulariser. The EO can expressed in terms of an extended version of (7) as
YISY & [Cyssy-llPs 0.
Therefore, given a sample (x = ((21, 81, v1), (En, Sn, Un)), EO is impose by adding 1(fiX,8) = |i empirical conditional cruss-covariance operator Craly HK - HKg is defined as
Eilyy = C5 - C
A short calculation shows that the minimiser of the fairness regularisation problem is given by
ABO (f, AI + Crir))
The justification for the above formula is similar to the previons section, and the Remark 3.6 remains valid in this context. Consequently, we can state the following result that controls the loss of the best fair constrained predictor and the best (possibly unfair) unconstrained predictor.
Theorem 3.9. For any X > 0 and μ > 0, we have almost surely
ILF fxoll1a(P2) MMRlC 4X
The proof of this theorem follows the proof of the Theorem 3.7.

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Furthermore, we introduce martially-normaliise EO regressor in 3.5.
The following lemma, identified as Lemma 3.10, demonstrates the relationship between the loss incurred by the fairness regressor and that of the unconstrained regressor.
Lemma 3.10. Let be f the best predictor then for any X 2 0 and p 2 0, we have almost surely
-
A80 (((Px) $ fAol + +16800- fro (x(x))
fxol (((x)
To prove this lemma, add and subtract the term fx, and subsequently apply the triangle inequality.
We will use the same notation for the matricos associated with the random variables to improve readability. Specifically, let X E R be the matrix of the complete input data (samples x dimensionality), S E R the sensitive variable (S is treated as a subset of X), and Y contains the observed true labels, which can be considered in the multi-output setting being a matrix outputs. Recall that S is not limited to the one-dimensional case; rather than that, it can consider an arbitrary, finite number of sensitive attributes (s 1).
We expand our formulation to the use of kernels in the nonlinear framework. Then, map the input data to an RKHS X + L, recall that 4 = K; also map the sensitive attributes as S - v, where T = Ks; and the representer theorem (Schölkopf et al., 2001) states that w = x'aa a. The solution of the so-called FACIL (Fair Conditional Independence Learning) kernel mothod can be written as
a = (K Al a11 - 11,)" Ks(1 11,))) "Y,,
(8)
where H1 = -( is the orthogonal projection; hyperparametors À and μ need to be tuned by cross-validation or fixed priori to achieve a sensible fairness level.
Remark 3.11. The solution of the FACIL in its linear formulation can be written as:
n,)55 (T
w =
3.5 EO with normalised conditional independence measures
In this section, we formalised a framework for including the cross-covariance operators in the cost function and provided a cross-covariance interpretation of fair learning. We follow this motivation to obtain models through the normalisod conditional independence operator
Vsxly C55 (C5x Csy
(9)
The above operator (9) codifies the conditional independence relation of S 1 X|Y. However, this normalised operator does not allow us to obtain closed-form solution in the kernel case (we want to note that it can be reached in the linear model formulation). However, a closed-form solution can be achieved in the kernel case by considering the partially normalised operator as follows:
VsxlY C/ (Cax CsxCy)lyx),
(10)
and introdueing it in the fairness regularisation term as I(S,S,) X) VsXYYS. By considering the corre- sponding empirical estimator D. it can be shown that the minimiscr of the fairness regularisation problem is given by
PEo = (f + AT + WD )rsly) Corollary 3.12. For any X >> and μ > 0, we have almost surely
5 fxolleP(PS) & MrllV Proty 4X
The proof of this corollary follows the proof of the Theorem 3.7.
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The following lemma, identified Lemma 3.13, demonstrates the relationship between the loss incurred by the fairness regressor and that of the unconstrained regressor.
Lemma 3.13. Let be f the best predictor then for any X 2 0 and p 2 0, we have almost surely
280 Aum
$ $115~ fxo (2(Px) + -11 fEO - (P(x)) LA(Px)
To prove this lemma, we add and subtract the term fx and subsequently apply the triangle incquality.
By following this, and through the use of kernel feature maps and empirical estimators of the operator, we obtain the dual weights solution of the so-called NFACIL (Fair normalised Conditional Independence Learning):
a= (K AI u(I - IIy) Ks(nel + Ks) (1)
Since the existence of Kg is not ensured, we included the regularisation (w + nel)-1, c cho- sen in Smola and Schölkopf (2001), and applied the Woodbury-Morrssn matrix inversion lemma D)
(A+ BCD) BC= A-C
Remark 3.14. The solutions of both (8) and (11) require dala centring in Hilbert spaces, which can be done implicitly by kernel matrir centring through. H = L1T
Remark 3.15. The solution of the NFACIL, in its linear formulation, can be written as:
(X X + AI AI /V Vexlu) y.-
4 Bias- aware Gaussian models with EO fairness constraints
We first focus on the particular setting of Gaussian regression models under EO constraints. In this case, estimating an optimal model which satisfies the fairness constraint can be expressed in closed form. To facilitate the understanding of the subsequent experiments in section 5, and without loss of generality, we stale this section with 5 one-dimensional.
Building fair Gaussian models with known have already been studied in the literature. Precisely, the majority of the works (Tan et al., 2020; Li et al., 2022) considered the linear normal model Y= f3, (X) + e, where the errors En i.idd ~ (0 1) are such that B(E XX = 0, and the predictor is fāg (X) = 35 where the parameter BS E Rpx1 is different for each group. In other words, S different linear models are built.
However, in some contexts including econometrics (Wooldridge, 2002; Stock, 2002), it natural to assume that certain sensitive information exists but is not observed. To address this case, we rather consider S unknown and set the following model Y = f0o.0 (X, +e,
(11)
where the predictor
f S) B,S 8 X, 3o R. B Rpx1
(12)
is a linear combination of the sensitive and non-sensitive attributes (and the errors are as above). Then, the joint distribution SYY is (p + 2) dimeensioa normal, and we denote the vectors of mcans and the covariance matrices as follows
μx
Σx X
Exs
Σxy
N
ps
Σs
Esy
by
Σ
Σg
Σy
We note that EO requires the linear fair predictor to be independent of S conditionally given Y, that is
(X,S) Y
which under the normal model assumption is equivalent to the second-order moment constraint
Co Y) 0.
(13)
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Hence, seeking for fair linear predictor amounts to obtaining conditions on the coefficients Su. 3 for (13) to hold. If we denote by Asxy E [px1 the veclor of correction, for fairness
Asxy = ExsEx - -Esy
(14)
EgDiy 25y
then the optimal fair equality of odds predictor under can be exactly computed as follows, whose proof is in the Appendix A.
Proposition 4.1. Under the normal model (11), the optimal FOOfaair linear predictor of the form (12) is given as the solution to the following optimisation problem to obtain Pu.fair Bfair
E [( -),,,(X,
arg min ()
among the class of parameters FEO = () E R such that (8 (ExxSxx - Esy Exy) +
3o (EsEx - 20))) = If moreover Y and S are non-linearly dependent, can be exactly computed as
far Baair Asxy
3jair = Σn Ezy,
where
Σz Ex + EgAsxY Abxr Es ExsA
Ezr Exy + Egy Asxr
We consider the case where bias is only due biased observations while the task itself is fair, and therefore Y and S are not linearly dependent.
Remark 4.2. Note that the previous result can be extended to the case where the sensitive variable S is multidimensional by considering in (14) the matrix of correction for fairness
Asxy (ExsEy Esy Exy) (EsLy -ss
It is of interest to quantify the price for fairness E(FEo) when imposing (Bo, B) FEO. This will be done in section 5 comparing with the general loss associated with the minimiscr
() arg (o.. B (xxx)))(XX
(15)
We observe that condition (13) is generally weaker constraint in broader setup, where achieving EO conveys computational challenges as discussed in Woodworth et al. (2017). Authors showed that even in the restricted case of learning linear predictors, assuming convex loss function, and demanding that only the sign of the predictor needs to be non-disrimiinaoor, the problem of matching false positive (FPR) and false negative (FNR) rates requires exponential time to solve in the worst case. Motivated by this hardness result they also proposed a relaxation of the criterion of EOO by more tractable notion of non-discrimination basod on sccond-order moments. In particular, they introduced the notion of Equalised correlations, which indeed is generally weaker condition than (13), but when considering the squared loss and when (X, Y) are jointly Gaussian, it is equivalent (and, subsequently, equivalent to EO). In this case, they provided a characterisation of the fair lincar predictor as solution to a minimisation problem solved in the case of the binary target Y, where they also computed the price for such a fair predictor.
Since linear prediction can be seen as the most suitable framework for Gaussian processes, the relaxation of (13) could be justified the appropriate notion of fairness when we restrict ourselves lincar predictors. Furthermore, linear predictors, especially under kernel transformations, are used in various applications. They thus form a practically relevant. family of predictors where one would like to achieve non-discrimination.
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E(Foo)
log(size)
Figure 2: Boxplots of the computations {FFE) of the price for fairness under a normal lincar model.
In Proposition 4.1, we restrict ourselves to the standard framework in which the computation of the optimal fair predictor still feasible. Yet, for many distributions and hypothesis classes, there may not exist a non- constant, deterministic, fair predictor. can be a challenging task. Thus, we stress that one can include non-linear basis functions to ensure greater expressivity. Some examples of such basis functions include Random Fourier Features (Rahimi and Recht, 2007) other known basis functions (Yu et al., 2016; Le et al., 2013). Hence, in the next section, propose an alternative approach for ensuring EO fairness in a broader setup.
worth noting that generalising non-lincar Gaussian processes
5 Experiments
In this section, we provide empirical evidence of the performance of the proposed methods in set of experiments. Firstly, numerical evidence of convergence of the loss bound the EO linear regression setting is provided over simulation set. Secondly, we study the trade-off between error rates and fairness in the proposed cross-covariance metrie; the results cover six databases. Thirdly, empirical comparison of the weights behaviour in the linear model evaluation.
5.1 Loss bound EO linear regression
We show results in simulations aimed at estimating the minimal excess risk when imposing EO in the normal linear regression framework, that is, the price for fairness
E(Fro) (( c ( (F(66))))
We here consider S ~~0000 10) and X R2, such that
X N [[]][80))
and Cou(Xi,,) Cov(X2, 8) 0.1. To simulate the true values of the target Y, have chosen in (12) true parameters B5 0.1 and 3* (2, 1).
The results of 1, 000 replications of the experiment, therefore 1, 0,000 computations of the empirical price for fairness {FFro, are shown in Fig. 5.1 taking particularly a log-linearly spaced sequence between 2 and of length 10. In all cases, we varied parameters to the maximum range possible such that the models did not go into numerical errors. In addition, the average minimal excess risk {FFoo) and its standard deviation the sample size increases are included in Table We observe that the estimation converges as the averaged value stabilises and the standard deviation numerically decreases.
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Table 1: Average minimal excess risk (FFo)) and standard deviation sd(E(FBo)) as sample size increases
in logarithmically spaced grid ranging from 10a to 10 samples.
Size
100
167
278
464
774
1292
2154
3594
5995
10000
((FEo)
1.02
0.75
0.71
0.69
0.67
0.67
0.65
0.65
0.65
0.65
sd ((((E))))
0.81
0.54
0.35
0.26
0.20
0.16
0.12
0.09
0.07
0.05
5.2 EO in open datasets
The second set of experiments uses four real datasets (over six considered protected variables) demonstrate the empirical behaviour of the developed methods. The dataset selection has been made based the relevance in the field and the broader studies using them. In particular, we consider: 1) the Adult income dataset (Dua and Graff, 2017), 2) the Communities and Crime (Redmond, 2009) (C&C), 3) the National Longitudinal Survey of Youth (Burcau of Labor Statistics, 2019) (NLSY), and 4) the Compas recidivism risk score data (Larson al., 2016).
Experimental results reveal the trade-off between error rate and fairness through the root-mean-squared error (RMSE) and in EO terms through the conditional independence mcasure. All the reported results are average of 25 independent trials. We split data into training, validation and test independent sets. We fix the size of the training set to N = 600 samples, the size of the validation set to 100 samples, and the test set to 2000 samples, or the remainder available. Rocall that our methods can handle multiple, possibly continuous, protected attributes. We have included Gaussian Process formulations FACIL and NFACIL methods to provide a complete comparison. GP versions are used to work closer to risk minimisation methods; curves are not too long due to numerical problems reported with the μ parameter. For this reason, Figure 3[top row] shows the results of the Adult dataset by using = 1 considering the ethnic origin and gender the sensitive variables independently, as well as deemed logether (s 2) in the last column. In the comparison, we have also considered other state-of-the-art methods that handle multiple protected attributes, particularly those proposed in Pérez-Suay et al. (2017); Tan et al. (2020). In the second row, the results of the remaining datasets (2-4) illustrate the good behaviour of our proposal. Specially paradigmatic is the C&C case, where the normalised FACIT-GP consistently reduces conditional independence without increasing. Our approach generally achieves greater accuracy for a given level of fairness.
5.3 Linear model sensitivity in weights terms
In this section, we explore the behaviour of the fair DP against EO models unveiled in this work. In particular, we focus on the linear models of the linear-Fair Learning (Pérez-Suay et al., 2017) and the linear-FACIT. method.
Fairness conditions have been set to enforce independence along ethnic origin (9-th feature) in the Adult database. Figure 4 illustrates the averaged absolute differences between linear-Fair Learning weights' model (WFL) and our linear-FACIL weights' model (WFA) proposal against least squares (LS) weights' models (wLS). As can be seen, the change most notably remains in the enforced feature in both methods. Moreover, the seems to have a more extensive (not significant) difference against the LS weights model.
In light of this, Table 2 summarises a t-lest of the hypothesis that both distributions come with equal means. The null hypothesis ("men are equal") cannot be rejected the 5% significance level for 9 of 14 features. The remaining five appear in the table and indicate that the null hypothesis can be rejected at the 5%
Table 2: Features from Adult database where a t-test of l7Ls wFrl and |WLS wFal rejects equal means
at the 5% significance level.
# feature
3
13
p-value (< 0.05)
1.70e-04
3.78e-05
5.79c-04
8.70e-04
1.48e-05
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Ethnic origin
Gender
Ethnic & Gender
0.02
0.52
0.02
FAL PM
FRL FAGE-IM FACL NFACIL
FAL AM FAGE-NN
0.015
FP &P NFACIL GP
Conditinnl Indepeddene
0.015
Condifnnllin Inddeeennddeeeeeeeee
indeperdence
0.013
FA MFACL OP LFR-PP
LFR-DP
LFRGP
0.01
0.01
0.005
8x
By
0.7
0.75
0.5
0.85
0.7
0.75
0.8
0.85
0.7
0.75
0.8
0.85
RMSE
RMSE
C & C
NT.SY
Compas
x10
FAL-PM
Independencs
10
---ACIL GP LPR-OP
FACH CP
10
NEACI
FACIL RM
FACIL-OP
NFACIL FM
Conditonal independencee
1.5m
NFACIL FM FACI-OP
FAGL-IM
10
10
Conditional
0.5
10
0.1
0.15
0.2
0.25
0.3
0.35
1.2
1.4
1.8
1.8
2.2
0.82
0.84
0.88
0.88
0.9
0.92
0.54
RMSE
Figure 3: Results on Adult income dataset considering single and multiple sensible variables (first row). Communities and Crime (C & C), National Longitudinal Survey of Youth (NLSY), and the Compas dataset are the second row.
Ls
Averaged absolute value
10
10
3
5
7
9
11
13
Feature number
Figure 4: Averaged absolute value of weights subtraction between lincar models in the Adult dataset when enforcing fairness with 9-th feature (ethnic origin). Involved models weights are: Least Squares (WLs), Fair Learning (WFL), and linear-FACII. (WFA).
6 Conclusions
We introduced and theoretically motivated method for learning fair regression models different settings. Firstly, we focused the more straightforward case, but also valuable for many practical applications, of linear Gaussian model. An optimal fair linear predictor satisfying EO is obtained as function of the model's parameters in this sotting. Morcover, we provided simulations that verify the price convergence for lairness. Secondly, an empirical risk minimisation family of methods and the most. notable solutions in the linear and kernel cases. Furthermore, the Appendix provides additional extensions to the Gaussian Process regression details. Along with the proposed methods, two operators have been used to give an EO model. Firstly, the conditional cross-cvaariance operator is used as an EO fairness measure and regularisation parameter aarameter-secondlyyy partially-normalie version of the conditional cross-covariance operator. The use of
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those operators reaches closed-form solutions in all cases, providing a straightforward way to develop novel fair methodologies.
We apply our approach to obtain fair models, demonstrating competitive empirical performance on several datasets relative to seate-of-thearr methods. Regarding the experimental setup, the fairness parameter p allows systemically vary along the aairues--ccuuracy trade-off. An optimal hyperparameter problem-specific and should be selected priori. Importantly, this strictly needs comparable models, which is ensured by introducing the partial normalisation of the operator.
Future work involves developing more scalable algorithms using randomised approximations and providing fairness conditions to other models like neural networks. This could, unfortunately, trade accuracy for the essential properties of the linear kernel and the probabilistic GP trealment. In the future, we also plan lo study the effect of the regulariser on the confidence intervals for the predictions and the feature ranking (oventually through ARD kernels) derived from our FACIL-GP models. Finally, there are intrinsic links between fairness, causal inference and physical consistency, and these connections can offer new insights about algorithmic solutions in these fields.
Acknowledgments
Adrián Pérez-Suay and Gustau Camps- Valls thank Fundación BBVA for their support through the project "Causal inference in the human-biosphere coupled system (SCALE)".
Paula Gordaliza work was supported by the Elkartek program under the Grants 3KIA project (KK- 2021/00123) and also by the Basque Government through the BERC 2022-2025 program and by the Min- istry of Science and Innovation: BCAM Severo Ochoa accreditation CEX2021-001142--) MICIN AEI / 10. 10.13039/500 150100001000033
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A Appendix
Remark A.1. The problem could be considered in more general framework where S does not belong to X. Assuming ExlyK(X) (elyK(XxX)) < oo and Esly < 00, the lensor kernel K @ Ks expressed as (K @ (s)((a,)) (x', = ((z,,')Ks(s,,* defines on RKHS HKØKs which con compactly embedded into (2 x Pxs).
More precisely, we let
T: : Toso HKøKs 4 L2 x S, Pxs)
(16)
f 4
with
(T) & (2) s), ()) KoKs
In this case, the problem is to minimize over functions : X x S + R
((g) = B[( ())
It is well known that the regression function g" (z, 8) = -X = (x, | 12 x S, Pxs) is the unique minimiser of C. In addition, we will assume our model to be well-specified, that is, 9 can be wrillen as
9" (5", & (s)((a,8)) (x)
for some f H
Given sample (x,s,y) = (121, vi), -.. (Iin Snx Un)) we consider the empirical minimization problem
Cax(f)== = E((e,)))))))) - w;)² + Allffl
j=1
where X is the regularization parumeter. Using the properly of the RKHS, the solution this ménimmiaa- tion problem has a solution which can be wrilten as
Ba(x,5) = Σ (),
5=1
with.
K( 8), (1,,, 8,)) = K(F,,,) @ Kg(8,88),
where the wj² 's constitute the solution of the minémization
min flly Kw 122 AK wK}
This leads up us the closed-form solution
== (K (X=
(17)
A.1 Proofs of section 4
We start recalling some facts about Gaussian random variables. Proposition A.2. If (U,V,W) are jointly Gaussion, then
Conditional expectation E(UVV) is linear in V and is given by E(U TEU) Euy E -V- -E()))
19

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Published in Transactions on Machine Learning Research (06/2023)
Conditional covariance L(v.yy does not depend on W and is given by
E(u.yi Ev.v Ev.w Ew 2U.w
Proof of Proposition 4.1. In the particular normal model, this independence means that the elements in positions (1, and (2, of the covariance matrix of random vector (f(Xu, S),5 Y) are exactly zero. Therefore, the class of fair predictors is written as
FEO == (f X' X' + Rd: Cou(f(X,, Y) = 0}
(18)
More precisely, the previous condition can be written in terms of the covariances of (X,, SY and the coefficients (Bo, B) of the linear model (11). Observe that the joint distribution of the random vector S), is
SoS + XT
Bojis + 2.TTPO pXx. us
1):
Σ E12 -1 Σy
N
NY
where
Σ1
SO2s + Ex 230
Ex.s
BoL's + ExT
Es R2xx
BoEs + +8² Ex,y
Bo + 3 Σx.Y Rx
E2
Esy
Hence, from Proposition A.2, we know that
() E1 Ey 212
Substituting the expressions above for En and Σg, obtain that gg..9 FEO if and only if
(BoEs + 3 Ex, = Esy (BoEst + 33 Ex.y)
Then the optinal EO-fair predictor in this setting is the solution to the following optimization problem:
(Bo.fotr. = arg min (B.
E [Y f... (X S))
(19)
FEO =
(2x Esy Ex.y) + 3o (Es2i >2r)) = -}.
We note that the Schwarz inequality and the assumption that Y and S nonlinearly dependent ensure EsEy - -gy > O. Then we obtain that the class of EO-fair predictors (Bo, B) E FBO are such that 3n = 3d Cs.x,,Y, where
Cs.x.,y = EX..S Esy
E R
ΣsEy - Σzyy
Hence, the optimal EO-fair predictor (19) can be obtained equivalently
Brair = arg min (Y - 3T (X,, + SCS,..,)))) SERP
Now if we denote Z: = X1, + SCs,x..Y, it is easy to check that the optimal EO-fair predictor can be exactly computed as
Bjair Σz,, where
Σn Σx, y S.X.. Cs..., Ex,s S.X..y
Exy Σx,r + Esy Esy
20

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Published in Transactions on Machine Learning Research (06/2023)
A.2 Proofs of section
Proof of Theorem 3.7. For any two bounded invertible operators A, B acting some Hilbert space have the relation
A = AB-
This can be proved by expanding the right term as follows A (B - A- (AA (T) BT) From this, we casily deduce:
(T& + AI + p Cox) (1
-f +1I + (ax) -
Cex (T x1)
Morcover, the spectral theorem yiclds (Theorem 2.7-8 in Kreyszig (1978), page 96), through the polar decomposition S4 - U V/S (Theorem 2, in Vito et al. (2005)) the following
1
II A1) 2VX
and, we define the cross-covariance operator as Eg = K = Sy and S. is the sampling operator
through v. As both Sq, So are bounded operators, we recall the polar decomposition of So = U1 =
Vl Then,
[( Al + Cox)
SSl |1 X (5 "Sll |1(15 + + 18 11 |Sglll
1
sup :c(0, | [E.0.115+11
f2 + + ul222
2vA
||(T + XI - 2V
Hence, since ||Syy|| I MR we get using (21)
I1f f & II(f AI u Cxx -- (xx A1)-+))||Sy|ll S MrllCaa 41
A.3
Summary of the proposed methods
Table 3: Summary of the proposed methods based on the frequentist ERM principle. Method name, operator
acting as a regularizer and weights' solution in its corresponding linear (w) or kernel (a) counterpart.
Method
Operator
Solution
FACIL
Esglv
w (X + XI + μX (1 11, (I X1,))) X
K-FACIL
Egjly
a KK + XI + p(I - 1I, Ks( - K)
NFACIL
Vsxly Egg' (Esxlp)
Di
1/2
w (XT X + + 1 SxIV S 1-12 -
1-1755T
K-NFACIL
P Psx|2 = Egg E-1//2 (Esx)))
a (K + XI p - Ks + Ks) Ks)-'(I 'II -II
A.4 EO with Gaussian Processes
We can derive a probabilistic version of FACIL using Gaussian processes (Rasmussen and Williams, 2006). Standard regression approximates observations {y 1a as the sum of some unknown latent function f(x) of the some covariates {a; Rdya plus consland power (homoscedaslic) Gaussian noise, i.e. y1 -x;;)
21

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Essentially GP prior is placed on the latent function f() and Gaussian prior is used for each latent noise term e, f()) GP(O, ( H)),, where &*) is covariance function parameterized by 6, and of is is Published in Transactions on Machine Learning Research (06/2023)
where o.). GP regression proceeds in Bayesian, non-parametric way, to fit observations. For convenience, let us assume the transformed data X = S (T- x collectively contains all transformed points č = (1- Likewise, we can define the relation explicitly in Hilbert spaces: & = (1-1l)) C$ so we can compute the corresponding kernel matrix (without explicitly mapping the data or applying the transformation) as K T =KC
hyper-parameter that specifies the noise variance so that the vector containing all the 0 == [2)
Essentially, a GP is a stochastic process whose marginals are distributed as multivariate Gaussian. In particular, given the prior GP, samples drawn from f(ž) at the set of locations { follow a joint multivariate Gaussian with zero mcan and covariance matrix K with [K], K& (s( R2). Specifically, considering y WI,..., and X [TI,...;áN]] with GP formulation we can obtain analytically P(f(#)), = (() where the posterior mean and variance are:
f(ø) K(E T +o v. (x) = ((ā,t) - R(a, 8)( o2IN)-
The PACIL-GP uses the kernel K CKCT, which can be readily computed and optimized with marginal log-likelihood maximization. GP regression and the kernel ridge regression solution for the predictive mean are identical (Kanagawa et al., 2018). However, with a GP formulation, we can directly control the matrix K by the choice of the kernel function (z( žz), infer the model parameters by maximum log- log-likelhhood, and obtain the posterior predictive variance (2)), not just point-wise wise estimates for the prediction.
Given sample (x ((11, 81,
(1,1,81,111 $1,81).... (Fay 8n, Un)) consider minimising
min L()) FEHK
(20)
with
- (f(z,) v,) + AllfI + pl1Cexf11
Ł(f) = 1 n1 1=1
where À > 0, / > are the regularisation paramcters. A short calculation shows that the minimiser of the above problem given by
fAp= (Tg Al pxxx
(21)
22


Reviews:
[{"review_id": "7X6FEW1i2l", "reviewer": "Reviewer_AZ5T", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "(S1) The paper leverages the tractability of RKHS' to obtain clean results that compare the L2(P)-difference between DP and EO constrained predictors and the best unconstrained predictors. \n(S2) The Gaussian process implementation leads itself to a tractable estimation procedure as a simple constrained loss minimization problem.\n(S3) The procedure appears to work well in the numerical experiments.\n\n(W1) Could the main theorems be stated directly in terms that compare the loss of the best fairness constrained predictor against the best unconstrained predictor?\n(W2) The implementation in terms of a linear CEF is simple. Could this be generalized to allow for possible non-linear Gaussian processes? This may complicate how exactly the fairness constraints are imposed (it doesn't seem like simple constraints would be enough). At the same time, you could also incorporate arbitrary non-linear basis functions b(X) into the linear model for generality. This should be remarked on in the paper.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper studies the problem of imposing demographic parity (DP) and equalized odds (EO) fairness criteria in a general kernel regression problem. To do so, the authors propose to incorporate a penalization term in the loss function that captures the degree of dependence between the predictor and sensitive attributes (DP) or the predictor and sensitivity attributes conditionally on the outcome (EO). For DP, the authors use the Hilbert-Schmidt independence criterion and for EO, the conditional cross-covariance operator (which measures independence using reproducing kernels). \n\nBy modeling the problem as a nonparametric regression problem in an RKHS, the authors are able to obtain closed-form expressions for the best fair prediction function satisfying the constraints -- see equation (5) and (7). Furthermore, they can obtain closed-form bounds on the ``cost of fairness' as measured by the L2(P) distance between the best fair predictor and the best unconstrained predictor (Theorem 3.2 and Theorem 3.3). For estimation, the authors implement their procedures with a gaussian process model with linear functional form assumption on the CEF.\n\n", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Please see the weakness I mentioned above. ", "broader_impact_concerns": "I have no concerns about the broader impact of this paper.\n\n", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "jTsYtfpefE", "reviewer": "Reviewer_EGvW", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "[Strengths]\n1. The paper includes a comprehensive summary of related work.\n2. The ideas of the paper are clearly presented, fair regression is achieved by introducing regularization that induces independence between the response and the protected attributes.\n3. The authors consider two independence inducing regularizations to achieve fairness, one uses the well-known HSIC as the regularization and the other adapts the conditional covariance operator. For both formulations, the authors provide closed-form estimators for the model. However, it should be noted that the independence conditions are not new, and the real contribution is the adaptation of the conditional covariance operator in fair regression.\n\n[Weaknesses]\n1. The presentation could be improved by compressing the background context (the main contribution starts at Sec 4).\n2. The contribution of the paper seems incremental --- both the HSIC and the conditional covariance operator have been widely studied in literature, and the HSIC based regularization has already been proposed for fairness applications, e.g., Pérez-Suay et al., 2017 proposed the HSIC for fair kernel regression. Additionally, the independence criterion based on conditional covariance operators (Eq 6 and Eq 9) was introduced by Fukumizu et al., 2004, 2008.\n3. Regarding the theory, adding another regularization based on the conditional covariance operator can violate the representer theorem, i.e., the solution may no longer have a nice linear form. I think this issue is not addressed in the paper.\n4. The impact on the fairness-accuracy tradeoff has not been discussed, In particular, it is trivial to obtain a fair estimator, e.g., a predictor that outputs random noise, choosing the covariance kernels for the protected attributes and the response is subtle. Additionally, the regularization coefficient in the paper can have a significant impact on the tradeoff. I think this needs to be discussed in more detail.\n\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "In this paper, the authors propose a method for fair kernel regression through regularization based on the conditional independence operator. In particular, Equalized Odds (EO) is used as the fairness criterion, and multivariate protected attributes are considered. The authors also provide closed-form estimators for the resulting models and demonstrate empirical performance on several benchmark fairness datasets.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. Does adding the additional independence inducing regularization violate the representer theorem? In particular, why the optimal estimator still has a linear form?\n2. How does the regularization coefficient impact the fairness-accuracy tradeoff?\n3. For multivariate S with discrete values, the RKHS kernel for S becomes more complicated. How does the method address this setting and how does it perform in practice?", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "nNDV2qSQSc", "reviewer": "Reviewer_yBfZ", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths: \nI think the main strength of the paper is that it derives the actual closed form solution of the problem they consider in various settings. A lot of the in-processing methods come up with an optimization method for solving the unfairness regularized empirical risk minimization problems, but having a closed form solution gives rise to easy computation of these fair models.  \n\n\n\nWeakness: \n(1) The presentation of the paper can be improved significantly. \n\nFirst, it was very hard to understand the results of the paper in a structured manner. For instance, in the abstract, it tries to describe things in terms of the different operators (conditional independence, normalized conditional independence operators) without explaining these concepts, and I don’t think the readers would be familiar with them. \n\nI think it would be helpful to initially describing these results in a much more succinct manner without necessarily trying to go into low-level details.  \n\n(2) Notations\nI had a hard time with the notations. Some notations were never actually defined and/or some notations were overloaded multiple times. Although some are very minute examples and can be sometimes understood from the context, I think the paper should be thorough in actually formally defining them. \n-There are bar{\\epsilon} and \\hat{\\epsilon} used to describe the price of fairness term in Section 5 that are never defined, although I suppose there are empirical estimates? \n-\\mathcal{F}_{EO} (in section 3.4) is actually never defined.\n-\\hat{f}^EO_{\\lambda, \\mu} is used in both equation (7) and in section 3.5 again to describe the closed form solution under different regularization terms (either unnormalized or partially normalized). \n-Similarly, (9) and (10) used to describe V_{SX|Y} again. Overloading of the notations in this manner make it somewhat hard to follow. \n\n\n(3) How good of a proxy are these regularization terms for the original fairness goals?\n-My understanding is that the main highlight of this paper is the regularization terms that work to incentivize independence between two random variables. The only discussion regarding the appropriateness of these regularization terms in terms of the original fairness goal seems to be found in equation (13): i.e. it’s 0 if and only if they are independent. However, the actual problem has a penalty parameter mu which controls how big this value is, meaning it will most likely not be 0. In the case they are not 0, are these still good proxies for the original fairness goals (i.e. making sure the difference in false positive rates be small across groups).\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper considers the problem of imposing fairness constraints to the empirical risk minimization problem. \n\nThey indirectly do so by penalizing the model via proxies that measure how independent appropriate values are: e.g. in the case of demographic parity, independence between the predicted value hat{y} and the group membership s. \n\nIn the kernel regression setting, they are able to derive a closed form solution that is close to the the true optimal solution for the above fairness regularized problem for demographic parity and equalized odds (Section 3).\n\nIn section 4, they study the setting where the model can take the demographic group information as an input; they call this Bias-aware models.\n\nFinally, they have some experiments that complement their theoretical results above. ", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "I will describe the changes that would be helpful in light of the weakness described above\n(1) Presentation: I think the writing can be improved by initially trying to describe the high level goal/motivation of the paper. Also, it would be helpful to describe the connection between the results at a high level so that the readers can more easily organize the results in their mind as they are reading.\n\n(2) I think definition environments should be used to formally state important definitions. Also, trying to avoid overloading notations and making sure all the notations are always defined before using them would greatly improve the paper. \n\n(3) In section 3 & 4, it would be helpful to have a little longer discussion about why the suggested fairness regularization term are good proxies for the original fairness goal, even when they are not exactly 0. For instance, for equalized odds, can it be said that the the smaller this regularization term, the smaller the difference in the true/false positive rates always? Similarly, instead of plotting the conditional independence in the Pareto curves, it would be more helpful to trace out the difference in true/false positive rates for equalized odds and the difference in the positive prediction rates for demographic parity. If the answer to the question is true, then it seems like the shape of these plots should stay the same.", "broader_impact_concerns": "I don’t have any broader impact concern for this paper. ", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper has been found interesting by the reviewers that still had some concerns. The replies to the reviewers were precise and on point and the paper was judged acceptable by majority of the reviewers but some still had some concern. The main concerns were about some claimed contributions that were already existing and need to be toned down and another one is the existence of the representer theorem. Those concerns seem to be minor but that must be addressed before final acceptance, which is why Accept with minor revision has been selected

More details below:
+ The authors need to clarify their contributions. For instance many ideas (HSIC, Cross-Covariance) have been proposed in other papers and are clearly stated as such. But the use of conditional independence that is claimed as a contribution (contribution 2)  was already proposed by Fukumizu et al. which btw is stated in the main paper.  There are enough novel results in the paper to justify acceptance without this contribution. 
+ The authors must recall the assumptions for applying the represented theorem and details how it can be used on equation (3). The response to the reviewer about this question was clear but it should appear in the paper. maybe the notation $f(X)$ is unclear and should appear as $\{f(x_i)\}_i$ ?


Decision: Accept (poster)