A Multi-Objective Framework for Balancing Fairness and Accuracy in Debiasing Machine Learning Models

Abstract

Machine learning algorithms significantly impact decision-making in high-stakes domains, necessitating a balance between fairness and accuracy. This study introduces an in-processing, multi-objective framework that leverages the Reject Option Classification (ROC) algorithm to simultaneously optimize fairness and accuracy while safeguarding protected attributes such as age and gender. Our approach seeks a multi-objective optimization solution that balances accuracy, group fairness loss, and individual fairness loss. The framework integrates fairness objectives without relying on a weighted summation method, instead focusing on directly optimizing the trade-offs. Empirical evaluations on publicly available datasets, including German Credit, Adult Income, and COMPAS, reveal several significant findings: the ROC-based approach demonstrates superior performance, achieving an accuracy of 94.29%, an individual fairness loss of 0.04, and a group fairness loss of 0.06 on the German Credit dataset. These results underscore the effectiveness of our framework, particularly the ROC component, in enhancing both the fairness and performance of machine learning models.

1. Introduction

Machine learning algorithms have revolutionized decision-making in many high-impact industries like finance, healthcare, and safety. However, decisions made by such systems can have adverse effects, making it essential to develop new techniques that ensure that the decisions made are accurate and fair. Research in this domain aids the progress of Artificial Intelligence, emphasizing the importance of fairness in AI systems amidst the growing use of automated decision-making processes. Research has continuously [1] exposed that ML algorithms are often biased towards privileged groups and fail to ensure fairness for unprivileged groups. Modern ML algorithms prioritize high accuracy by focusing on a single, unconstrained objective, making them unable to account for the many real-world constraints that must be met when deploying an ML model. For instance, the COMPAS score, utilized in American courts to anticipate the likelihood of recidivism, is a widely recognized instance of algorithmic discrimination due to disregard of fairness constraints  [2]. It frequently predicts black defendants to be at significantly higher risk than white defendants, even when the former group does not re-offend. This could reinforce systemic bias through a negative feedback loop and result in increasingly biased predictions over time. As a result, there is heightened interest from academia and industry toward developing highly unbiased and precise ML algorithms.

To address bias in ML models, many existing pipelines suffer from a fairness-accuracy trade-off, where increasing fairness can lead to a decrease in accuracy and vice versa. For instance, an algorithm fine-tuned for maximum accuracy may inadvertently perpetuate bias against specific groups. Conversely, an algorithm designed to achieve fairness may compromise accuracy by making incorrect classifications for particular data points. The complexity of debiasing stems from its nature as a multi-objective problem rather than a single-objective problem, highlighting the necessity for refined and advanced AI methodologies equipped to tackle such challenges.

We investigate the potential of multi-objective optimization in improving model accuracy while simultaneously addressing multiple fairness metrics that may conflict with one another on – German Credit [3], Adult Income [4] and COMPAS dataset [5]. This study introduces a novel in-processing, multi-objective framework that utilizes the Reject Option Classification algorithm to balance fairness and accuracy without penalizing protected attributes such as age and gender. Our methodology demonstrates its effectiveness through comprehensive evaluations of German Credit, Adult Income, and COMPAS datasets. While existing literature often focuses on optimizing for group fairness alone, our framework integrates both individual and group fairness with accuracy, showcasing ROC’s superior performance across these datasets. This contrasts with prior work that predominantly addressed group fairness (e.g., [6,7,8,9]).

In summary, the key contributions of this research include:

• The development of a robust fairness framework employing multi-objective optimization to simultaneously optimize accuracy and fairness without penalizing protective attributes.
• A comprehensive experimental analysis conducted across various datasets and comparing our framework to baseline methods demonstrate its capacity to increase fairness while significantly improving performance.

This paper is structured as follows: Section 2 offers a literature review that delves into prior research on fairness and optimization. In Section 3, we articulate the background of our framework and fairness definitions used across our work. Section 4 provides the experimental setup of our framework along with the dataset particulars. Section 5 gives a comprehensive overview of our research findings, which focus on performance in terms of accuracy, individual fairness, and achieved group fairness for sensitive features like race and gender. In Section 6 encapsulates the conclusion of our paper. Lastly, in Section 7, we explore potential research directions for our work.

2. Related Work

2.1. Pre-Processing Fairness Techniques

The growing importance of fairness in ML has led to increasing research that seeks to address existing biases in decision-making [10] and statistical limitations of fairness definitions [11]. To comprehensively address these biases, fairness techniques are broadly classified into three categories: pre-processing, in-processing, and post-processing [12]. Within this, individual and group fairness emerge as two pivotal dimensions illuminating different facets of fair decision-making. Group fairness aims to ensure nondiscrimination across protected groups, optimizing metrics, such as disparate impact [13], and individual fairness implies that fair ML models should treat comparable users similarly [14].

Before training an ML model, pre-processing techniques modify the input data to remove biases. Techniques such as re-sampling, avoidance of sensitive data collection, and probabilistic data transformations have been employed to create fair representations of the data [15,16]. While these approaches have demonstrated promise, their focus on the input data can limit their ability to address biases that emerge during the training process [17].

2.2. In-Processing Fairness Techniques

In-processing techniques attempt to address biases by modifying the learning algorithm itself. These methods often involve directly incorporating fairness techniques into the optimization process [18,19]. One popular approach is to employ adversarial training, which seeks to minimize a classifier’s ability to predict sensitive attributes while maintaining high accuracy on the task of interest [20,21]. However, these methods tend to focus on single-objective optimization or concentrate on a single sensitive attribute, thus limiting their ability to achieve fairness across multiple objectives and attributes [22].

2.3. Post-Processing Fairness Techniques

Post-processing techniques modify the predictions of a trained ML model to ensure fairness. They include threshold adjustment and calibration techniques, which aim to align the classifier’s output distribution with the desired fairness criteria [23]. While these methods can be effective in certain scenarios, their reliance on a fixed model and a large amount of sensitive data can limit their ability to optimize group and individual fairness and accuracy simultaneously [24]. Therefore, there is a need to capture individual fairness, which indirectly improves group fairness by reducing information on protected attributes [25]. The post-processing algorithms can be viewed as a single-objective graph that preserves the desired “treat similar individuals similarly” interpretation [26].

2.4. Multi-Objective Optimization Techniques

Although single-objective approaches have been commonly employed in fairness research, multi-objective optimization has emerged as a promising alternative to address the inherent trade-offs between fairness and accuracy [27]. Approaches such as the use of gradient-based optimization have been proposed to achieve this balance [28]. However, these methods often focus on single fairness objectives or single sensitive attributes, which may not be sufficient for addressing fairness in complex scenarios [29].

2.5. Recent Advances in Multi-Objective Optimization

The multi-objective optimization approach proves to be the most effective in ensuring fairness, as seen in [30,31,32,33]. The [34] multi-objective optimization framework can ensure results are fair to various demographic groups by using a combination of statistical and ML techniques. This approach can be used to solve the Pareto fair-dimensionality-reduction problem by adopting the adaptive gradient-based algorithm [35]. Additionally,

Table 1. Summary table.

Related Research	Advantages	Disadvantages
Ashktorab et al. [36]	Provides insights into practitioner perspectives on various fairness definitions and evaluations	Limited sample size may affect the generalizability of findings.
Julia et al. [37]	Evaluates the balance between accuracy and fairness in predictive models for recidivism.	Highlights the challenge of achieving both accuracy and fairness simultaneously.
Alessandro et al. [38]	Provides empirical evidence of unfair practices in algorithmic decision-making in insurance.	Focused on a specific case study, limiting broader applicability of findings.
Zemel et al. [39]	Offers a method to learn representations that satisfy fairness constraints while maintaining utility.	Complexity in ensuring that representations remain useful while enforcing fairness.
Pierson et al. [40]	Explores how demographic factors influence perceptions of fairness in algorithms.	Findings vary significantly based on the demographic context of the study.
Veale et al. [41]	Identifies design needs for ensuring fairness in high-stakes decisions made by algorithms.	Highlights implementation challenges in bureaucratic setting.
Fleisher et al. [42]	Discusses the philosophical underpinnings of individual fairness, highlighting its limitations.	Individual fairness does not adequately address group disparities.
Caton et al. [43]	Summarizes diverse definitions and metrics of fairness, contributing to a clearer framework for understanding fairness.	Practical applications of various metrics vary widely.

summarizes and highlights the challenges of achieving a balance between fairness and accuracy in machine learning, underscoring the relevance of our framework.

2.6. Bayesian Approaches in Machine Learning

In addition to multi-objective optimization techniques, most recent advancements in Bayesian approaches have shown promise in various machine learning downstream tasks, including positional data analysis. For instance, Bayesian model-assessment algorithms have been developed for diffusing particle data, as illustrated in recent works [44,45]. These methods focus on probabilistically assessing model performance and handling uncertainty in complex datasets, which could be instrumental to balance fairness and accuracy when integrating fairness metrics in a multi-objective optimization framework. This study [46] enhances machine-learning techniques for decoding anomalous diffusion in single-particle-tracking data by incorporating uncertainty estimates using a Bayesian deep-learning approach, thus providing accurate predictions and insights into the diffusion models. The probabilistic nature of these approaches aligns with the broader goal of ensuring fair decision-making across various sensitive groups by accounting for uncertainties inherent in real-world datasets.

Despite the significant strides made by existing methods in addressing fairness in machine learning, there remains a need for a comprehensive framework that can concurrently optimize multiple fairness objectives while considering multiple protected attributes [47,48]. The proposed approach in this study builds upon the strengths of in-processing techniques and multi-objective optimization, aiming to fill this gap and offer a more versatile fairness framework. We introduce a multi-objective framework utilizing the Reject Option Classification algorithm to optimize fairness and accuracy in machine learning models, addressing critical trade-offs inherent in high-stakes decision-making. Unlike traditional methods that rely on weighted summation to balance these objectives, our approach optimizes fairness without compromising model performance. This distinguishes our work from conventional techniques that typically apply fairness constraints during pre-processing, in-processing, or post-processing stages and often struggle with setting subjective weights for conflicting objectives, our approach balances accuracy-fairness tradeoff into the optimization process to provide a more effective and nuanced solution. Empirical evaluations on datasets such as German Credit, COMPAS, and Adult Income demonstrate our framework’s ability to maintain high accuracy while significantly enhancing fairness metrics. Our approach performs better in managing the fairness-accuracy trade-off than existing methods like stochastic multi-objective approaches to optimize fairness and accuracy [49] and fairness through unawareness [50].

3. Background

3.1. Fairness Definitions

Model fairness in supervised learning aims to use labelled training data to predict classification $Y\in 0,1$ using feature vector X. The fairness of the prediction $\widehat{Y}$ is evaluated with sensitive groups (e.g., gender or race) identified by sensitive attributes A (e.g., female or black). The sensitive attributes are binary $A\in 0,1$, where $A=0$ represents the unprivileged group (e.g., female) and $A=1$ represents the privileged group (e.g., male).

3.1.1. Group Fairness Loss

Group fairness loss ensures a predictive model performs equitably across different demographic groups. While the literature commonly discusses several definitions of group fairness, including statistical parity, equal opportunity difference, and predictive parity, our study focuses exclusively on statistical parity as the group fairness loss objective.

Statistical Parity quantifies the deviation in predictive outcomes between different groups. Specifically, it measures how much a model’s predictions differ between groups defined by a sensitive attribute. The statistical parity loss of a binary predictor $\widehat{Y}$ is defined as:

$$\mathrm{Statistical}\mathrm{Parity}\mathrm{Loss}=|Pr(\widehat{Y}=1|A=1)-Pr(\widehat{Y}=1|A=0)|$$

(1)

A lower statistical parity loss indicates more equitable treatment across groups, aligning with the principle of group fairness. Our study did not incorporate other group fairness measures, such as predictive parity or equalized odds, focusing solely on minimizing statistical parity loss.

3.1.2. Individual Fairness Loss

Individual fairness loss ensures that similar individuals receive similar predictions, irrespective of their group membership or other characteristics. For this study, we utilize Counterfactual Fairness to evaluate individual fairness loss.

Counterfactual Fairness is achieved if the model’s prediction for an individual remains consistent across counterfactual scenarios where only the sensitive attribute changes. Formally, an individual x with features X and sensitive attribute A is counterfactually fair if:

$$\mathrm{Counterfactual}\mathrm{Fairness}\mathrm{Loss}=\left|\widehat{Y}(x,A)-\widehat{Y}(x,A^{\prime })\right|$$

(2)

where $\widehat{Y}$ is the model’s prediction, x denotes the features, A represents the current sensitive attribute, and $A^{\prime }$ is the counterfactual sensitive attribute. A lower loss value indicates a higher degree of counterfactual fairness.

3.2. Multi-Objective Optimization

Multi-objective optimization (MOO) involves solving problems that require the simultaneous optimization of multiple conflicting objectives. Unlike single-objective optimization, which focuses on maximizing or minimizing a single metric, MOO considers multiple objectives and seeks a set of solutions that offer the best possible trade-offs among the objectives. These solutions are known as the Pareto front, where no other solutions are strictly better in all objectives.

In our study, the objectives include accuracy, individual fairness loss, and group fairness loss. The trade-offs among these objectives are crucial since improving one might degrade the others. Instead of reducing the MOO problem to a single-objective problem by aggregating the objectives into a weighted sum, we handle them separately to maintain the multi-objective nature of the problem.

The multi-objective optimization problem can be formulated as:

$$\underset{\mathit{h}}{min}{\mathit{O}}_{\mathrm{fair}}(\mathit{h},\mathit{\mu })=\left[O_{\mathrm{accuracy}}\left(\mathit{h}\right),O_{\mathrm{individual}\_\mathrm{fairness}}(\mathit{h},\mathit{\mu }),O_{\mathrm{group}\_\mathrm{fairness}}(\mathit{h},\mathit{\mu })\right]$$

where $\mathit{h}$ represents the decision variables (in this case, the model parameters), and $\mathit{\mu }$ represents the weights or penalties associated with fairness constraints.

We proposed a novel approach in which these objectives are optimized using the Reject Option Classification (ROC) method, combined with Particle Swarm Optimization (PSO) and NSGA-II. These algorithms help find the Pareto front, allowing for the exploration of the best trade-offs between accuracy and fairness.

4. Experiment

4.1. Data

In our study, we conducted a comprehensive experimental analysis using the German Credit, COMPAS, and Adult Income datasets. The German Credit dataset consists of 1000 instances and 20 attributes, including credit amount, duration, and payment status, with protected attributes such as gender and age. The dataset exhibited a significant class imbalance, which we addressed using the SMOTE-Tomek technique. This method combines the Synthetic Minority Over-sampling Technique to oversample the minority class with Tomek Links to remove majority class samples closest to the minority class, thereby effectively balancing the classes. The Adult Income dataset contains 14 features, encompassing both categorical and numerical variables such as age, education, and occupation, which were used to predict whether an individual earns more than $50K annually.The COMPAS recidivism dataset includes information on over 10,000 criminal defendants in Broward County, Florida. It features attributes such as race, sex, age, criminal history, and charge degree, with a binary target variable indicating whether the defendant recidivated within two years.

We performed an 80–20 split following pre-processing for all three datasets to create training and testing sets, ensuring robust evaluation across different fairness metrics.

4.2. Models Explored

4.2.1. NSGA-II Algorithm

The Non-dominated Sorting Genetic Algorithm II (NSGA-II) is a popular evolutionary algorithm designed for multi-objective optimization problems. NSGA-II employs genetic operators, such as selection, crossover, and mutation, to evolve a population of candidate solutions towards the Pareto front.

Mathematically, a non-dominated sorting algorithm can be described as:

Let P be the population of the n candidate solutions, each with m objectives. For each solution i, define the sets $S_i$ and $n_i$ as follows:

$$\begin{array}{c}{S}_i=\{j\in P\mid i\mathrm{dominates}j\}\\ n_i=\left|\{j\in P\mid j\mathrm{dominates}i\}\right|\end{array}$$

In the above equation, $S_i$ is the set of solutions dominated by solution i, and $n_i$ is the number of solutions that dominate solution i. Next, we defined the first front $F_1$ as the set of the solutions that are not dominated by any other solution in P, $F_1$ can be expressed as:

$$\begin{array}{c}{F}_1=\{i\in P\mid n_i=0\}\end{array}$$

Once we identified the first front, we used a similar procedure to find the next front $F_2$, which contains the solutions that were dominated by exactly one solution in $F_1$, $F_2$ can be expressed as:

$$\begin{array}{c}{F}_2=\{i\in P\mid n_i=1,j\mathrm{in}{F}_1\mathrm{such}\mathrm{that}i\mathrm{dominates}j\}\end{array}$$

We then continued this process to identify all the fronts in the population, and the final result is a set of fronts, where the first front contains the non-dominated solution.

4.2.2. Particle Swarm Optimization

Particle Swarm Optimization is a global optimization algorithm designed to find the optimal value of a multi-dimensional objective function. In our case, PSO is used to optimize hyperparameters for the XGBoost classifier, aiming to enhance accuracy and AUC without compromising fairness.

Mathematically, let $x_i^t$ be the position of particle i at time t, with $v_i^t$ as its velocity. The best position of particle i is denoted by $pbes{t}_{ij}$, and the best overall position by $gbes{t}_j$.

Key updates in PSO are:

• Global Best Position:

  $$\mathrm{g}\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}j=\left\{\begin{array}{cc}\mathrm{p}\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}ij\hfill & \mathrm{if}f\left(\mathrm{p}\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}ij\right)>f\left(\mathrm{g}\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}j\right)\mathrm{g}\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}j\hfill \\ \mathrm{otherwise}\hfill \end{array}\right.$$
• Velocity Update:

  $$vijt+1=wvijt+c_1{r}_1(\mathrm{p}\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}ij-x_{ijt})+c_2{r}_2(\mathrm{g}\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}j-xijt)$$
• Position Update:

  $$x_{ijt+1}=x_{ijt}+v_{ijt+1}$$

Here, i and j index the particles and dimensions, respectively, with N as the swarm size and d as the problem’s dimensionality.

4.2.3. Differential Evolution

The Differential Evolution algorithm is a population-based optimization method used to solve complex multi-objective problems. DE works by evolving a population of candidate solutions through mutation, crossover, and selection. Each candidate solution, called a vector, is updated iteratively based on the difference between randomly selected pairs of vectors.

Mathematically, the mutation operation is defined as:

$${\mathbf{v}}_i^{(t+1)}={\mathbf{x}}_{r1}^{\left(t\right)}+F·({\mathbf{x}}_{r2}^{\left(t\right)}-{\mathbf{x}}_{r3}^{\left(t\right)})$$

where ${\mathbf{v}}_i^{(t+1)}$ is the mutant vector, ${\mathbf{x}}_{r1}^{\left(t\right)},{\mathbf{x}}_{r2}^{\left(t\right)},{\mathbf{x}}_{r3}^{\left(t\right)}$ are randomly selected vectors from the population at iteration t, and F is a scaling factor.

The crossover step generates a trial vector ${\mathbf{u}}_i^{(t+1)}$:

$$u_{ij}^{(t+1)}=\left\{\begin{array}{cc}{v}_{ij}^{(t+1)}\hfill & \mathrm{if}{\mathrm{rand}}_j\le C_r\mathrm{or}j=j_{\mathrm{rand}}\hfill \\ x_{ij}^{\left(t\right)}\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

where $C_r$ is the crossover probability, and $j_{\mathrm{rand}}$ is a random index ensuring at least one component is inherited from the mutant vector.

Finally, in the selection step, the trial vector ${\mathbf{u}}_i^{(t+1)}$ is compared with the target vector ${\mathbf{x}}_i^{\left(t\right)}$, and the better one is selected for the next generation:

$${\mathbf{x}}_i^{(t+1)}=\left\{\begin{array}{cc}{\mathbf{u}}_i^{(t+1)}\hfill & \mathrm{if}f\left({\mathbf{u}}_i^{(t+1)}\right)\le f\left({\mathbf{x}}_i^{\left(t\right)}\right)\hfill \\ {\mathbf{x}}_i^{\left(t\right)}\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

Here, f denotes the objective function to be minimized.

4.2.4. Genetic Algorithm

The Genetic Algorithm is a bio-inspired optimization technique that mimics the process of natural selection to find optimal solutions to complex problems. GA operates on a population of candidate solutions called chromosomes, which evolve over successive generations through selection, crossover, and mutation.

Mathematically, the selection process is often based on fitness proportionate selection, where the probability of selecting an individual i is:

$$P\left(i\right)={\displaystyle \frac{f_i}{{\sum }_{j=1}^N{f}_j}}$$

where $f_i$ is the fitness of the $i\mathrm{th}$ chromosome, and N is the population size.

The crossover operation combines two parent chromosomes to produce offspring. For single-point crossover, if ${\mathbf{p}}_1$ and ${\mathbf{p}}_2$ are two parent chromosomes, then the offspring ${\mathbf{o}}_1$ and ${\mathbf{o}}_2$ are given by:

$${\mathbf{o}}_1=[{\mathbf{p}}_1^{1:k},{\mathbf{p}}_2^{k+1:n}],{\mathbf{o}}_2=[{\mathbf{p}}_2^{1:k},{\mathbf{p}}_1^{k+1:n}]$$

where k is the crossover point, and n is the chromosome length.

Mutation introduces diversity by randomly altering the gene values of offspring with a small probability. If ${\mathbf{o}}_{ij}$ represents the $j\mathrm{th}$ gene of the $i\mathrm{th}$ offspring, the mutation operation can be represented as:

$${\mathbf{o}}_{ij}^{\mathrm{new}}=\left\{\begin{array}{cc}\mathrm{random}\mathrm{value}\hfill & \mathrm{if}\mathrm{rand}\le P_m\hfill \\ {\mathbf{o}}_{ij}\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

where $P_m$ is the mutation probability.

The new population is formed by selecting the best chromosomes from the current population and the newly generated offspring, ensuring that the population evolves towards better solutions over time.

4.2.5. Nelder Mead Algorithm

The Nelder-Mead algorithm is a derivative-free optimization technique used to find the minimum of an objective function in a multidimensional space. It operates on a simplex, which is a geometric figure with $n+1$ vertices for an n-dimensional problem. The algorithm iteratively updates the simplex by performing reflection, expansion, contraction, or shrinkage, depending on the function values at the vertices.

Mathematically, the reflection step is defined as:

$${\mathbf{x}}_r={\mathbf{x}}_c+\alpha ({\mathbf{x}}_c-{\mathbf{x}}_h)$$

where ${\mathbf{x}}_r$ is the reflected point, ${\mathbf{x}}_c$ is the centroid of all points except the worst point ${\mathbf{x}}_h$, and $\alpha$ is the reflection coefficient.

If the reflection is successful, the algorithm may perform an expansion:

$${\mathbf{x}}_e={\mathbf{x}}_c+\gamma ({\mathbf{x}}_r-{\mathbf{x}}_c)$$

where $\gamma$ is the expansion coefficient.

In the case of failure, a contraction may be attempted:

$${\mathbf{x}}_c={\mathbf{x}}_c+\rho ({\mathbf{x}}_h-{\mathbf{x}}_c)$$

where $\rho$ is the contraction coefficient.

This process is repeated until convergence is achieved, with the simplex gradually moving towards the function’s minimum.

4.2.6. Reject Option Classification

The Reject Option Classification technique allows us to simultaneously optimize multiple objectives, such as accuracy and fairness, without sacrificing one for the other. The basic idea is to introduce a third class of “rejected samples” in addition to the “good” and “bad” score classes. By incorporating ROC, we adapt the objective function to penalize misclassifying rejected samples, allowing us to fine-tune the balance between accuracy and fairness. Intuitively, let us say that we have a multi-objective problem with two main objectives: maximizing the accuracy and minimizing the percentage of the misclassified instances from the rejected class; our optimization can be formulated as:

$$\begin{array}{cc}\hfill \mathrm{maximize}& f_1\left(\mathit{x}\right)=\mathrm{accuracy}\left(\mathit{x}\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathrm{minimize}& f_2\left(\mathit{x}\right)={\displaystyle \frac{\mathrm{misclassified}\mathrm{rejected}\mathrm{instances}\left(\mathit{x}\right)}{\mathrm{total}\mathrm{rejected}\mathrm{instances}\left(\mathit{x}\right)}}\hfill \end{array}$$

subject to:

$$\mathrm{total}\mathrm{rejected}\mathrm{instances}\left(\mathit{x}\right)\le \alpha \times \mathrm{total}\mathrm{instances}$$

wherein, the

$$\mathit{x}\in X$$

Note that this is a two-objective optimization problem, where $f_1\left(\mathit{x}\right)$ is the accuracy of the classifier and $f_2\left(\mathit{x}\right)$ is the misclassification rate of rejected instances. The optimization is subject to the constraint that the total number of rejected instances should be less than or equal to $\alpha$ times the total number of instances in the dataset, where $\alpha$ is a pre-specified parameter. The decision variable $\mathit{x}$ belongs to the feasible set $\mathcal{X}$.

4.3. Training Setup

The focus of this study is to simultaneously optimize the accuracy and fairness objectives while considering two sensitive features: gender and age. We employed tree-based classifiers, such as XGBoost, Decision Trees, and Random Forest, to find the best baseline model. Next, we employed the NSGA-II algorithm with a population size of 5000 and a maximum of 1000 generations, setting the crossover probability to 0.9 and the mutation probability to 0.1. For Particle Swarm Optimization, the training ran for 10 epochs with 2 particles and 3 dimensions. The Genetic Algorithm was configured with 10 generations and 10 solutions per population, optimizing learning rate, max depth, and number of estimators to balance accuracy and fairness. The Differential Evolution algorithm was used with bounds set for learning rate, max depth, and the number of estimators, aiming to minimize a weighted objective combining accuracy and statistical parity. Finally, the Nelder-Mead optimization was run with an initial guess and adjusted for learning rate, max depth, and the number of estimators to find the optimal configuration, focusing on improving fairness without significantly compromising accuracy. For Reject Option Classification, we defined a reject option threshold of 0.65 to compute accuracy, individual fairness loss, and counterfactual fairness loss for rejected and accepted instances.

4.4. Evaluation Metrics

We assessed model performance using accuracy, recall, precision, and F1-Score metrics. Accuracy gauges overall correctness, while recall emphasizes true positive identification. Precision focuses on minimizing false positives, and F1-Score balances precision and recall. Additionally, we employed the AUC metric to evaluate classifier performance, measuring the trade-off between sensitivity and specificity across classification thresholds, with a higher AUC value indicating better classifier performance.

5. Results

5.1. Baseline Results

We evaluated several classification algorithms to establish robust baselines for our fairness-aware framework, focusing on tree-based methods like Decision Trees, Random Forest, and XGBoost, known for handling complex, multi-attribute data. These algorithms were selected for their ability to capture intricate data relationships and align with multi-objective optimization. We also included non-tree-based methods like AdaBoost and Naive Bayes to compare broadly. This diverse set of baselines ensures a comprehensive assessment of our framework’s performance and fairness improvements.

5.1.1. German Credit Dataset

Table 2. Comparison of Accuracy, Precision, Recall and F1-Score of various classifiers for German Credit.

Classification Model	Accuracy	Precision	Recall	F1-Score
Adaboost	78.5%	85.5%	83.68%	84.58%
Decision Trees	70.5%	80.1%	77.3%	78.7%
Naive Bayes	72.0%	82.4%	76.5%	79.4%
Random Forest	80.0%	83.1%	90.7%	86.7%
XGBoost	82.0%	85.7%	89.3%	87.5%

compares five different ML classifiers—Adaboost, Decision Trees, Naive Bayes, Random Forest, and XGBoost classifier—based on their accuracy, precision, recall, and F1-score. XGBoost emerged as the top performer with an Accuracy of 82.0%, a Precision of 85.7%, and a Recall of 89.3%, resulting in an F1-Score of 87.5%. Random Forest also demonstrated strong performance, achieving an Accuracy of 80.0% and the highest Recall of 90.7%, leading to an F1-Score of 86.7%. While AdaBoost, Decision Trees, and Naive Bayes performed well, they were outpaced by the Random Forest and XGBoost ensemble methods. These results indicate that XGBoost is highly effective at balancing precision and recall, making it an intuitive choice as our primary baseline model.

In Figure 1, AdaBoost, Random Forest, and XGBoost exhibit relatively higher AUC values, ranging from 0.82 to 0.83, compared to Decision Trees, which have a moderate AUC value of 0.69. The blue dotted represents the ROC curve for a random classifier which implies it would have an equal chance of correctly classifying true positives and incorrectly classifying false positives. Therefore, the diagonal line represents a model that is essentially guessing randomly. Any model that performs better than random guessing will have its ROC curve above this line.

5.1.2. COMPAS Dataset

In

Table 3. Comparison of Accuracy, Precision, Recall and F1-Score of various classifiers for COMPAS dataset.

Classification Model	Accuracy	Precision	Recall	F1-Score
Adaboost	66.40%	62.52%	61.96%	62.24%
Decision Trees	63.08%	59.09%	56.52%	57.78%
Naive Bayes	64.05%	60.23%	57.61%	58.89%
Random Forest	64.21%	60.00%	59.78%	59.89%
XGBoost	66.72%	63.43%	60.33%	61.84%

, XGBoost emerged as the top-performing classifier with an accuracy of 66.72%, precision of 63.43%, and an F1-Score of 61.84%, demonstrating a strong balance between precision and recall. Adaboost also performed well, closely following XGBoost in accuracy and precision. In contrast, Decision Trees exhibited the lowest accuracy at 63.08% and the lowest recall at 56.52%, indicating its comparatively lower performance across these metrics. These results highlight the effectiveness of ensemble methods like XGBoost and Adaboost over simpler models such as Decision Trees in predicting recidivism with the COMPAS dataset.

In Figure 2, XGBoost, Random Forest, and AdaBoost have relatively higher AUC values, ranging from 0.71 to 0.73. The blue dotted represents the ROC curve for a random classifier and any model that performs better than random classifier will have its ROC curve above this line.

5.1.3. Adult Income Dataset

As shown in

Table 4. Comparison of Accuracy, Precision, Recall and F1-Score of various classifiers for Adult Income dataset.

Classification Model	Accuracy	Precision	Recall	F1-Score
Adaboost	83.80%	65.91%	68.05%	66.96%
Decision Trees	81.38%	60.98%	63.27%	62.11%
Naive Bayes	79.66%	65.97%	32.34%	43.40%
Random Forest	84.83%	70.09%	64.74%	67.31%
XGBoost	86.35%	72.67%	69.57%	71.09%

, XGBoost achieved the highest performance on the Adult Income dataset with an accuracy of 86.35%, precision of 72.67%, and an F1-Score of 71.09%, making it the most effective model. Random Forest followed closely with 84.83% accuracy and an F1-Score of 67.31%. Adaboost also performed well, achieving 83.80% accuracy and a balanced F1-Score of 66.96%. In contrast, Naive Bayes showed the weakest performance. These results highlight the effectiveness of ensemble methods, especially XGBoost, for this classification task.

In Figure 3, AdaBoost, Random Forest, and XGBoost have relatively higher AUC values, ranging from 0.90 to 0.92. This indicates that these algorithms perform well in distinguishing between good and bad credit instances, compared to Decision Trees, which have a moderate AUC value of 0.75.

5.2. Multi-Objective Optimization

In our study, we performed multi-objective optimization experiments across three datasets—German Credit, Adult Income, and COMPAS—to explore our approach to minimizing model error while simultaneously addressing group and individual fairness loss objectives. These experiments aimed to achieve fairness without negatively impacting sensitive attributes such as gender and age. We evaluated the model’s individual fairness loss (counterfactual fairness loss) and group fairness loss (statistical parity loss).

5.2.1. German Credit Dataset

Table 5. Performance results of different multi-objective optimization algorithms by sensitive subgroup (Gender—Male or Female) compared to the ROC on the baseline XGBoost model—GERMAN CREDIT dataset.

Algorithm	Acc.	Ind. Fairness Loss—Gen	Group Fairness Loss—Gen
NSGA-II	80.01	0.03	0.04
Particle Swarm Optimization	83.03	0.02	0.04
Differential Evolution	84.59	0.02	0.08
Genetic Algorithm	76.10	0.01	0.06
Nelder Mead	79.50	0.02	0.06
Reject Option Classification
Accepted Instances	94.29	0.04	0.06
Rejected Instances	75.38	0.01	0.06

compares the performance metrics—accuracy, individual fairness loss, and group fairness loss—for the German Credit dataset across different multi-objective optimization algorithms. The NSGA-II algorithm achieved an accuracy of 80%, slightly lower than the baseline XGBoost model, with an individual fairness loss of 0.03 and a group fairness loss of 0.04, indicating a modest disparity in predictions. In contrast, PSO outperformed NSGA-II with an accuracy of 83%, accompanied by a lower individual fairness loss of 0.02 and a consistent group fairness loss of 0.04, suggesting more stable predictions across sensitive attributes. Differential Evolution further improved the accuracy to 84.59%, with an individual fairness loss of 0.02, though it exhibited a higher group fairness loss of 0.08, indicating a trade-off between accuracy and fairness metrics. The Genetic Algorithm, while achieving a lower accuracy of 76.10%, demonstrated the lowest individual fairness loss at 0.01, but with a slightly higher group fairness loss of 0.06. Nelder-Mead showed balanced performance with an accuracy of 79.50%, and individual and group fairness losses of 0.02 and 0.06, respectively. Lastly, the Reject Option Classification (ROC) applied to the XGBoost model significantly enhanced the accuracy to 94.29% for accepted instances, with an individual fairness loss of 0.04 and a group fairness loss of 0.06. Interestingly, the rejected instances exhibited a reduced accuracy of 75.38%, with an individual fairness loss of 0.01, underscoring the effectiveness of ROC in minimizing disparities.

5.2.2. COMPAS Dataset

Table 6. Performance results of different multi-objective optimization algorithms by sensitive subgroup (Gender—Male or Female) compared to the ROC on the baseline XGBoost model—COMPAS Dataset.

Algorithm	Acc.	Ind. Fairness Loss—Gen	Group Fairness Loss—Gen
NSGA-II	68.60	0.18	0.29
Particle Swarm Optimization	66.80	0.27	0.19
Differential Evolution	65.59	0.01	0.06
Genetic Algorithm	65.10	0.01	0.06
Nelder Mead	67.77	0.24	0.06
Reject Option Classification
Accepted Instances	71.43	0.41	0.01
Rejected Instances	64.00	0.14	0.11

summarizes the performance of various multi-objective optimization algorithms on the COMPAS dataset. The NSGA-II algorithm achieved an accuracy of 68.60%, with an individual fairness loss of 0.18 and a group fairness loss of 0.29. In comparison, Particle Swarm Optimization (PSO) showed slightly lower accuracy at 66.80% but improved individual fairness loss to 0.27, though this resulted in a decrease in group fairness loss to 0.19. Differential Evolution and the Genetic Algorithm both achieved similar results, with accuracies of 65.59% and 65.10%, respectively, and both showed minimal individual and group fairness losses at 0.01 and 0.06. Nelder-Mead performed moderately, with an accuracy of 67.77%, while showing a higher individual fairness loss of 0.24 and maintaining a low group fairness loss of 0.06. Finally, the Reject Option Classification (ROC) improved accuracy to 71.43% for accepted instances, with a significant reduction in individual fairness loss to 0.41 and a strong reduction in group fairness loss to 0.01. However, the rejected instances displayed a lower accuracy of 64.00%, with a reduced individual fairness loss of 0.14 and better group fairness loss of 0.11, illustrating ROC’s effectiveness in balancing accuracy and fairness on this dataset.

5.2.3. Adult Income Dataset

Table 7. Performance results of different multi-objective optimization algorithms by sensitive subgroup (Gender—Male or Female) compared to the ROC on the baseline XGBoost model—ADULT INCOME dataset.

Algorithm	Overall Acc.	Ind. Fairness Loss—Gen	Group Fairness Loss—Gen
NSGA-II	84.08	0.27	0.39
Particle Swarm Optimization	84.57	0.26	0.02
Differential Evolution	75.05	0.01	0.06
Genetic Algorithm	75.59	0.01	0.03
Nelder Mead	76.88	0.02	0.06
Reject Option Classification
Accepted Instances	84.3	0.25	0.02
Rejected Instances	90.3	0.01	0.02

highlights the results for the Adult Income dataset. The NSGA-II algorithm achieved an accuracy of 84.08%, with an individual fairness loss of 0.27 and a group fairness loss of 0.39, reflecting a balance between accuracy and fairness. PSO marginally outperformed NSGA-II with an accuracy of 84.57%, slightly better individual fairness loss at 0.26, and a significant improvement in group fairness loss at 0.02, indicating more equitable treatment across gender subgroups. Differential Evolution and Genetic Algorithm demonstrated lower accuracies of 75.05% and 75.59%, respectively, but with minimal individual fairness losses of 0.01, and moderate group fairness losses. Nelder-Mead showed a balanced accuracy of 76.88%, with both fairness losses remaining relatively low. For the ROC method, the accepted instances maintained a high accuracy of 84.3%, with improved fairness metrics, while the rejected instances exhibited a substantial accuracy increase to 90.3%, with individual fairness loss dropping to 0.01 and group fairness loss remaining at 0.02, underscoring ROC’s effectiveness in reducing disparities.

5.2.4. Proposed Methodology

The proposed framework outlined in Algorithm 1 and shown in Figure 4 below, aims to address the fairness-accuracy trade-off in a machine learning model. This is demonstrated using XGBoost with a seed value of 32, a maximum depth of 8, a learning rate of 0.1, 100 estimators, and a binary logistic objective in a two-class classification context. It begins by initializing parameters and iterating until convergence is achieved. During each iteration, the dataset is balanced using the SMOTETomek technique to address the class imbalance and flips sensitive attributes in the test data for fairness considerations. The algorithm adjusts model weights based on a defined threshold and minimizes fairness objectives while training an XGBoost classifier. Finally, it sorts predicted probabilities, ultimately optimizing fairness objectives, to achieve a balance between fairness and accuracy in model predictions.

Algorithm 1: Fairness-Accuracy Trade-off with ROC on baseline (XGBoost).

Data: data X, target Y, sensitive attributes A, initial weights $\mu$, rejection threshold $\alpha$, multi-objectives $O_{fair}\left(\right)$

Result: Optimized fairness objectives $O_{fair}(h,\mu )$

Initialize:$\alpha \leftarrow 0.65$

repeat

until convergence;

5.2.5. Time Complexity Analysis

Our proposed framework involves multiple steps. For instance, filtering operations exhibit a time complexity of $O\left(n\right)$, where n is the number of rows. One-hot encoding of categorical features introduces additional columns, resulting in a complexity of $O(n·k)$, with k as the number of categorical features. Training the XGBoost model typically has a complexity of $O(n·m·log(n\left)\right)$, where m is the number of features, reflecting the computational effort of decision tree growth. Prediction complexity is $O(n·m·T)$, with T as the average number of trees. Fairness calculations, including accuracy and statistical parity, have a complexity of $O\left(n\right)$, while counterfactual fairness, involving additional predictions and sorting, has a complexity of $O(n·T+nlogn)$. Overall, the training process is $O(n·m·log(n\left)\right)$, prediction is $O(n·m·T)$, and fairness calculations combine to time complexity of $O(n·T+nlogn)$.

5.2.6. Running Time Analysis

For the time analysis, we compared the duration taken by the model in different stages, excluding data balancing time. The results, visualized in Figure 5, reveal that the COMPAS dataset requires significantly more time for model training, which could be attributed to the complexity and size of the dataset. In contrast, the German Credit dataset exhibits relatively low computational demand across all stages. This comparison allows us to understand the computational efficiency of the proposed method in real-world scenarios.

6. Conclusions

This study presents a robust framework to address the significant challenge of balancing fairness—individual and group—while maintaining high accuracy in machine learning models. By leveraging a multi-objective framework and Reject Option Classification, we have demonstrated that it is possible to mitigate bias related to sensitive attributes such as gender and age without sacrificing model performance. Our extensive experiments across three datasets—German Credit, COMPAS, and Adult Income—highlight the adaptability and robustness of the framework. Integrating various multi-objective optimization algorithms, such as NSGA-II, Particle Swarm Optimization, Differential Evolution, and Genetic Algorithms, revealed that the ROC method consistently outperforms these baselines across all datasets. For the German Credit dataset, ROC achieved the highest accuracy of 94.29% for accepted instances while maintaining a low individual fairness loss of 0.04 and group fairness loss of 0.06. This demonstrates the framework’s ability to handle complex fairness-accuracy trade-offs, as even the rejected instances showed acceptable performance (75.38% accuracy). On the COMPAS dataset, ROC improved accuracy to 71.43% for accepted instances, significantly reducing individual fairness loss to 0.41 and group fairness loss to 0.01, reflecting a major improvement in fairness. For the Adult Income dataset, ROC maintained a strong accuracy of 84.3% with fairness metrics improved across the board (individual fairness loss: 0.25, group fairness loss: 0.02), underscoring the scalability of the method across diverse domains. This study contributes to ongoing research in mitigating biases in predictive models and offers a reliable solution to balance fairness and accuracy in real-world applications.

7. Future Work

We plan to explore and analyze the integration of additional optimization algorithms to enhance the framework’s performance and robustness and investigate the impact of different fairness definitions and constraints. We plan to incorporate advanced optimization techniques, such as reinforcement learning-based methods, to enhance the balance between fairness and accuracy. Additionally, dynamic weight adjustment strategies within the multi-objective optimization process could be explored to make the fairness-accuracy trade-off more adaptive and responsive to varying contexts. Another critical direction is expanding the definition of fairness used in the study. While our work has focused on counterfactual fairness and statistical parity, there is potential to investigate how other fairness metrics, including equal opportunity, disparate impact, and demographic parity, influence the model’s performance. Exploring these alternative definitions could provide deeper insights into fairness dynamics across different domains and datasets. Given the promising results of static datasets used this study, future research would extend this framework to real-time decision-making environments, such as credit lending or judicial systems, where fairness considerations are crucial. This would involve assessing the computational efficiency of the proposed framework in dynamic, real-time settings and exploring fairness constraints in streaming models to ensure fairness is maintained in evolving real-time streams of datasets. Furthermore, this work has focused primarily on fairness concerning individual sensitive attributes like gender and age. A natural extension would be to examine intersectionality—how fairness holds up when multiple sensitive attributes intersect, such as gender and race. This would provide a more comprehensive understanding of the framework’s capabilities in ensuring fairness across diverse subgroups. Finally, we plan to integrate explainability into our proposed framework as it would be another important direction for future research. Providing transparent insights into how decisions are made, especially regarding which instances are rejected or accepted, would imply the model’s trustworthiness and support regulatory compliance. This is particularly relevant in high-stakes domains like healthcare and criminal justice, where explainability and fairness are critical.

The code for our paper can be found here: https://github.com/RN0311/Multiobjective-Optimization-For-Debiasing-Credit-System/tree/main (accessed on 15 September 2024).