Recent advancements have made significant progress in addressing fair ranking and fairness with continuous sensitive attributes as separate challenges. However, their intersection remains underexplored, although crucial for guaranteeing a wider applicability of fairness requirements. In many real-world contexts, sensitive attributes such as age, weight, income, or degree of disability are measured on a continuous scale rather than in discrete categories. Addressing the continuous nature of these attributes is essential for ensuring efective fairness in such scenarios. This work aims to fill the gap in the existing literature by proposing a novel methodology that integrates state-of-the-art techniques to address longterm fairness in the presence of continuous protected attributes. We demonstrate the efectiveness and lfexibility of our approach using real-world data.
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This work aims to fill the gap in the existing literature by proposing a methodology that integrates state-of-the-art techniques to address long-term fairness in the presence of continuous protected attribu.tes
The paper is structured as follows. Section 2 aims at placing our work within the existing literature on fair machine learning, focusing on applications of fairness in ranking and fairness with continuous sensitive attributes. In Section 3, we provide the essential technical background required to understand the details and significance of our approach, as it incorporates diferent state-of-the-art techniques and frameworks. Following this, we describe the specific aspects of our contribution in Section 4, where we present our methodology grounded on a specific use case. We outline the main results of our empirical evaluation in Section 5. Finally, we summarize our findings in Section 6 and highlight potential directions for future investigation.
To the best of our knowledge, no previous work has addressed the task of fair ranking with continuous sensitive attributes. Still, there has been a significant growth in publications over the last decade in the two distinct fields, both stemming from the broader domain of fair machine learning. Hereby, we summarize the key developments in these areas as a means to efectively frame our work within the current state of the art.
Mehrabi et al. [
In their survey [
Most fair ranking approaches employ top- proportional representations as a fairness metric.
Namely, they try to ensure an equal representation of protected groups in the first candidates.
For example, among the post-processing fairness methods for learning-to-rank, [
In the last few years, some works have proposed new metrics and computational methodologies
to address continuous sensitive attributes in fairness enforcement tasks. Among them, [
A parallel efort was undertaken by [
In this section, we provide a formalization of the ranking problem general enough to model
our case study and other similar applications. Next, we introduce FAiRDAS [
We focus on a process wherein a set ℛ of resources undergoes repetitive ranking guided by observable information arriving over time. For example, ℛ may contain students that need to be ranked based on predicted academic performance. The observable information, hereinafter referred to as batches, is seen as a stochastic process indexed by time, denoted as { }∞=1 . Each batch is a random variable characterized by a domain and probability distribution ( ). The ranking quality is characterized using a metric function defined in probabilistic terms, typically relying on expectations or event probabilities, namely: ∶ , ↦ [ ; ] (1) Here, ∈ Θ is an action vectorwhose values can be adjusted to control the ranking procedure behavior. For example, the action vectormight represent penalty or reward terms linked to sensitive groups. The vector [ ; ] ∈ ℝ denotes the values of metrics for a given batch and action vector . In real-world scenarios, these metrics will always admit a finite sample formulation, often derived by substituting theoretical expectations with sample averages.
Given that the ranking is performed for every batch, followed by an adjustment of the action vector, the ranking problem can be defined in terms of the tuple: ⟨{ }∞=1 , { }∞=1 ⟩ (2) where and are the batch and action vector at time respectively. The value of the metrics at time is determined given and (Equation (1)).
FAiRDAS [
∈Θ ℒ ( , ̄ ) = ‖ [ ; ] − ̄ ‖22.
The solution method for Equation (3) relies on the action space characteristics and the chosen distance function. A possible choice for ℒ ( , ̄ )is the Euclidean distance:
The exact evaluation of Equation (4) is often unfeasible, primarily due to the unknown distribution ( ); thus, metric values [ derived from historical data.
; ] are replaced typically with a Monte Carlo approximation FAiRDAS Grounding. To apply FAiRDAS efectively to a specific scenario, it is essential to delineate its core components: 1) the metrics of intere,stwhich establish the criteria for evaluating fairness and ranking quality; 2) the corresponding threshold vecto;r3s) the target dynamic system which define the ideal metrics behavior; 4) the set of action, sdelineating how metrics can be manipulated to enhance ranking fairness and quality; 5) the distance functio,n defining the metric for assessing the efectiveness of the target system’s approximation; and ifnally, 6) the optimization methodussed to address Equation (3), which heavily depends on the chosen set of actions and distance function.
The Generalized Disparate Impact (GeDI) was first introduced in [
This choice is inspired by the copula transformations of the Hirschfeld–Gebelein–Rényi (HGR) maximum correlation coeficient. However, one major diference between GeDI and HGR lies in the absence of a second mapping function on the output feature ∈ , which prevents it from measuring non-functional dependencies of the type ↦ , akin to the DIDI. In addition to that, instead of leveraging the original definition of Pearson’s coeficient, the formulation of GeDI is slightly altered to make the indicator sensitive to scale variations. This ensures that reductions in unfairness are proportionally translated to diminished disparate impacts even if the shape of the unfair behavior is not modified, and also guarantees compatibility between GeDI and DIDI since both metrics yield identical results when the input attribute is binary. Finally, the mapping function () is restricted to a linear combination over a polynomial kernel. This allows one to frame the computation as a linear optimization problem, thus keeping a low computational burden although retaining high approximation capabilities thanks to the inherent non-linearities. Additionally, it serves the dual purpose of reducing overfitting while maximizing user-configurability and interpretability of the metric. (3) (4)
Formally, () is defined as the vector product V ⋅ , where V is the polynomial expansion matrix built from the input vector – i.e., the Vandermonde matrix –, while ∈ ℝ is a coeficient vector that weighs the contribution of each polynomial order. GeDI is eventually computed as: GeDI(, ; V ) =| cov(V ⋅ , ) var(V ⋅ ) | s.t. ‖‖ 1 = 1 (5) where the constraint on the L1 norm of the coeficient vector is intended to replace the absence of the scaling factor on the output term. An important detail to note is that the order of the polynomial expansion is part of the specification of the indicator, as it appears in its notation and aims to ofer users a simple way to balance the bias-variance trade-of.
As a demonstration of our approach, we focus on ranking students by their predicted academic performance to identify those at risk of dropping out. The real-world data is provided by the Canarian Agency for Quality Assessment and Accreditation (ACCUEE)1, which gathers information to assess the performance of their educational system through regular diagnostic reports. The data spans four academic years (2015-2019) including (1) the evaluation of students’ academic proficiency in subjects such as Mathematics, Spanish, and English and (2) context questionnaires completed by students, school principals, families, and teachers to collect sociodemographic background information. In our test case, we rank students based on their Mathematics proficiency measured by a normalized score. The protected attribute considered is the Economic, Social, and Cultural Status (ESCS) [24], namely a continuous indicator that serves as a proxy for the socioeconomic status of students. Ensuring long-term stability is crucial in this context: although consistently high accuracy and fairness are desirable, it is essential to maintain stable actions over time to prevent negatively afecting students’ academic progress.
In addressing the task at hand, we define two distinct groundings of FAiRDAS framework.
The first grounding, inspired by [
Set of Actions. Inspired by [
Metrics of Interest. In our case study, we are interested in decreasing socioeconomic
discrimination while preserving ranking accuracy; thus, we need 1) a fairness metric able to deal
with the continuous protected attribute ESCS and 2) an accuracy metric to measure the drop in
ranking performance due to the application of the action vector . As a fairness metric, we rely
on GeDI, whereas to assess the system’s drop in performance we measure the sum of absolute
diferences between the original and modified scores, namely:
(6)
(7)
1
where is the number of students in a batch,
∈ [
> 0 for some to mitigate discrimination. Given that the action vector must have a unit L1 norm, the trivial solution of = 0 for all , which would nullify both metrics, is not allowed.
Target Dynamic System. As we aim to meet the metric thresholds while maintaining longterm stability, we define our desired behavior by means of following dynamic system, which defines a smooth evolution of the target metrics toward the thresholds:
+̄1 = ⊙ ( ̄ − ) + ̄ , where ̄ represent the metric values in the target system, is the vector of thresholds, ∈ (0, 2) , and ⊙ refers to the Hadamard (element-wise) product. Given that we are focusing on two metrics (GeDI and SAE), is a 2-dimensional vector, with its values determined through a preliminary experiment detailed in Section 5.
We use Equation (4) – Euclidean distance – as the distance function, optimizing it with the scipy implementation of Sequential Least Squares Programming (SLSQP) optimizer.
Set of Actions. To avoid the discretization of the protected attribute ESCS, we define the set of
possible actions as a family of polynomial functions parameterized by ∈ ℝ +1 , where is
the order of the polynomial2. The functions map each value of ESCS to a real number, which is
then used as a multiplicative discount factor to modify the student’s score. First, we rescale ESCS
into the domain [
As in Discrete Action,swe use GeDI as fairness metric to deal with the continuous nature of ESCS. The ranking performance is measured by the mean squared error where is the ESCS value of the k-th student, and ( )the weighting polynomial function evaluated on . As in Discrete Action,sthe two metrics of interest conflict since to be close to the constant function = 1 while GeDI forces to deviate from it. MSE pushes
We rely on the same dynamic system in Equation (7), as our goal is to stably evolve the two metrics of interest below the predefined thresholds.
As before, we use Equation (4) – Euclidean distance – as a distance function. However, when optimizing it, we rely on the scipy implementation of the Trust Region Method (trust-constr), as it proved to be more reliable in the solution, although at the expense of a slightly higher computational time.
This section outlines the empirical evaluation performed on the case study described in Section 4. We first define the evaluation procedure and then report the numerical results 4 .
We compare each of the two groundings with a baseline method focusing on metrics of interest and action smoothness(m
) described below. For each approach, we report the mean and standard deviation of the metrics across batches to assess performance and stability over time. 2In our application, we choose = 4 as it provides a suficient trade-of between the expressiveness of the function and the known numerical instability of polynomial kernels, along with their higher computational workload. 3We rely on MSE and not on SAE to avoid the computation of an absolute error. 4The source code to reproduce the experiments can be found at https://github.com/EleMisi/FairRanking under MIT license.
Action Smoothness. To evaluate the stability of the chosen actions over time, we compute the cosine distance between actions performed on consecutive batches. For the Discrete Actions grounding, m is defined as follows:
where is the number of incoming batches and is the action vector of the -th batch. For
the Continuous Actiongsrounding, m is computed by evaluating the weighting polynomial
functions on a fine-grained discretization of the interval [
Baseline Approach. We compare FAiRDAS in its Discrete Actiongsrounding against a baseline approach that focuses on finding the optimal action vector that minimizes:
As a preliminary step, we examine how the eigenvalues of the FAiRDAS dynamic system influence action smoothness to determine their optimal values for the experiments. We conduct multiple runs with a fixed threshold while varying the eigenvalues (Table 1). As expected, based on the theoretical characteristics of the dynamic state under consideration, lower eigenvalues result in more stable actions in both groundings. For our experiments, we select the eigenvalues corresponding to the inflection point of the action smoothness metric.
Next, we compare the performance of FAiRDAS and the baseline under diferent pairs of thresholds for the metrics of interest. For both Discrete Actionasnd Continuous Actionssettings, the threshold pair {0, 2} represents an extreme scenario where fairness is prioritized over ranking performance. Subsequently, we examine a loose threshold pair, {0.7, 0.7}, and a stringent pair, {0.5, 0.5}. Finally, we investigate a pair of thresholds, {0.2, 0.2}, that cannot be reached. ℒ () = max (GeDI(), )+ max (SAE(), ) where and are the metrics’ thresholds. The action vector is the same described in Section 4.1, and it is optimized via the SLSQP method, as for FAiRDAS. For the Continuous Actionsgrounding, the baseline approach searches for the optimal polynomial function that satisfies the constraints described in Section 4.2 and minimizes: ℒ () = max (GeDI(), )+ max (MSE(), ). with and are the metrics’ thresholds. As for FAiRDAS, we rely on the Trust Region Methods to tackle the optimization problem. Mean and standard deviation of the action smoothness computed over the batches for FAiRDAS. We analyse the results for 5 diferent eigenvalues ( ) with a fixed threshold pair {0.5, 0.5}. For each eigenvalue, we run eight experiments. We select = 0.2 as the elbow of the curve for both groundings (in bold). metrics throughout 100 batches for both baseline and FAiRDAS approach in Discrete Actions setting. Across all threshold pairs, the two methods achieve comparable levels of the metrics of interest (GeDI and SAE). However, the baseline exhibits notably higher levels of instability in the chosen actions (higher m
) compared to FAiRDAS, especially with stringent thresholds.
This finding confirms the ability of FAiRDAS to maintain both performance efectiveness and fairness over time while also avoiding drastic actions that may raise ethical concerns. The increased stability of FAiRDAS approach is demonstrated in Figure 1, which shows the action vectors selected by both approaches in an experiment with stringent thresholds. This figure provides a component-wise comparison of the baseline and FAiRDAS action vectors across all 100 batches. As detailed in Section 4.1, each component of the action vectors afects students from the corresponding ESCS level and acts as penalizing factors on their scores, potentially altering their ranking. Higher values indicate more significant penalization, while values near zero mean the student’s score remains untouched. The baseline method tends to favor rapid and drastic interventions, indicated by 1) the sudden color change between batches and 2) action components close to one (lighter color). In contrast, FAiRDAS exhibits a more moderated and balanced behavior, with action vectors evolving smoothly over the experiment (gradual color changes along x-axis) and similar penalization across groups (uniform color along y-axis). (a) (b)
Results with Continuous Actions. In Table 3 we report the mean and standard deviation of
the metrics over 100 batches for both baseline and FAiRDAS approach under Continuous Actions
setting. As with Discrete Action,sthe numerical results confirm FAiRDAS’s capability to maintain
both performance efectiveness and fairness over time while avoiding drastic actions. FAiRDAS
and the baseline achieve similar levels for the metrics of interest (GeDI and MSE) across all
thresholds, but FAiRDAS reaches lower values of action smoothness (m ). This result is
exemplified in Figure 2, where we present an example of the polynomial functions selected by
FAiRDAS and the baseline throughout 100 batches. Each column displays the function chosen
for the corresponding batch, evaluated over the ESCS domain [
We introduced a novel approach that integrates state-of-the-art techniques to address
longterm fairness in the presence of continuous protected attributes. This is achieved by pairing
FAiRDAS [
To the best of our knowledge, this is the first work that tackles long-term fairness and stability in ranking with continuous attributes. Thus, we believe that it could lay the groundwork for further research and applications in several domains where handling continuous attributes and stability are of key importance, yet currently understudied.
The work has been partially supported by the AEQUITAS project funded by the European Union’s Horizon Europe Programme (Grant Agreement No. 101070363), and by PNRR - M4C2 Investimento 1.3, Partenariato Esteso PE00000013 - “FAIR - Future Artificial Intelligence Research” - Spoke 8 “Pervasive AI”, funded by the European Commission under the NextGeneration EU programme5. 5Disclaimer: This paper reflects only the authors’ views. The European Commission is not responsible for any use that may be made of the information it contains. https://proceedings.mlr.press/v202/giuliani23a.html. [23] S. Aghaei, M. J. Azizi, P. Vayanos, Learning optimal and fair decision trees for nondiscriminative decision-making, in: The Thirty-Third AAAI Conference on Artificial Intelligence, AAAI 2019, The Thirty-First Innovative Applications of Artificial Intelligence Conference, IAAI 2019, The Ninth AAAI Symposium on Educational Advances in Artificial Intelligence, EAAI 2019, Honolulu, Hawaii, USA, January 27 - February 1, 2019, AAAI Press, 2019, pp. 1418–1426. URL: https://doi.org/10.1609/aaai.v33i01.33011418. doi:10.1609/aaai. v33i01.33011418. [24] F. Avvisati, The measure of socio-economic status in pisa: a review and some suggested improvements, Large-scale Assessments in Education 8 (2020). URL: http://dx.doi.org/10. 1186/s40536-020-00086-x. doi:10.1186/s40536-020-00086-x.