1 Introduction↩︎

Machine learning (ML) systems have infiltrated society in numerous industries including healthcare, finance, and education [1]. These models undoubtedly add value in decision-making as they can leverage large amounts of information and help scale processes. Ensemble and deep learning systems are amongst the most popular predictive ML methods due to their strong predictive accuracy [2]. While these methods optimize for accuracy, most lack interpretability, or the ability for a human to understand how the model produced its results. Furthermore, with the growing use of ML in high-stakes applications, there has been concern and evidence of bias within existing systems [3]. The combination of many popular models lacking both interpretability and fairness leads to mistrust amongst users.

Several methods have addressed bias in ML systems [1]. Specifically, some methods attempt to improve fairness by rebalancing the dataset concerning underrepresented groups, removing features that may be correlated with the protected attribute, or adding synthetic data to increase representation [4]. Others add a fairness-aware penalty or constraint to mitigate bias [5]. While these methods effectively enhance fairness, they often neglect the crucial intersection of fairness and interpretability.

While interpretability in machine learning, particularly in discerning important features, has been extensively explored [6], its application to fairness or boosting methods remains relatively underexplored. Traditional boosting techniques like AdaBoost and gradient boosting excel in predictive accuracy but typically overlook bias mitigation. Although fair boosting methods exist, they often lack interpretability regarding fairness [5], [7]. Our objective is to devise an interpretable approach that achieves high performance in both accuracy and fairness.

Inspired by both MP-Boost which offers feature interpretability [8] and fair adversarial boosting [7] which offers improvements in fairness, we propose a novel algorithm Fair MP-Boost which is a double stochastic gradient boosting technique based upon minipatches (MPs) [9]. Our algorithm adaptively learns MPs which are tiny subsets of features and observations, at each iteration that balance fairness and accuracy for both observations and features. Our learning scheme allows us to leverage traditional feature importance scores for trees like the MDI [10] and the recently proposed FairTreeFIS[11]. These learned feature probability distributions allow practitioners to interpret the predictions of Fair MP-Boost. The adaptive sampling of observations further ensures the most relevant minipatches are being sampled for training.

Contributions. In this work, we develop a fair and interpretable boosting scheme. Our contributions consist of developing a novel algorithm with the major novel components being an adaptive learning scheme for observation and feature probabilities that balance fairness and accuracy. The result of this is an interpretable boosting model that allows us to understand which features and observations are useful for balancing fairness and accuracy. Through empirical analysis, we validate interpretations of our feature subsampling scheme and we demonstrate that Fair MP-Boost outperforms state-of-the-art bias mitigation algorithms.

2 Fair MP-Boost↩︎

Motivated by MP-Boost [8], we aim to devise a boosting approach that maintains fairness and interpretability without compromising accuracy. We begin by defining group bias metrics and introducing helpful notation. Let \(z_i \in \{0,1\}\) for \(i \dots n\) denote an indicator variable for the protected attribute (e.g., gender, race). Group fairness entails uniform predictions across all demographic groups [12]. Specifically, Demographic Parity (DP) assesses whether predictions \(\hat{\boldsymbol{y}}\) are independent of the protected attribute \(\boldsymbol{z}\). We formalize DP as follows.

While we focus on implementing DP in this study, our approach readily extends to other fairness criteria, such as Equality of Opportunity [12]. We represent our dataset as \((\boldsymbol{X}, \boldsymbol{y}, \boldsymbol{z}) \in (\mathbb{R}^{N \times M}, \mathbb{R}^N, \mathbb{R}^N)\). Concentrating on binary classification tasks, each sample \(x_i \in \mathbb{R}^M\) corresponds to a label \(y_i \in \{-1,+1\}\) and a protected attribute \(z_i \in \{0,1\}\).

2.1 Algorithm Framework↩︎

Our primary goal is to develop a classifier that is fair, accurate, and interpretable. Our key idea to do this is to leverage a double stochastic gradient boosting technique where we adaptively learn both features and observations as the algorithm proceeds. We propose to do this by utilizing minipatches, where we adaptively sample features and observations that contribute to fairness and accuracy [9]. Formally, a minipatch is defined as \((\boldsymbol{X}_{\mathcal{R},\mathcal{C}},\boldsymbol{y}_{\mathcal{R}})\), where \(\mathcal{R}\) denotes a subset of examples with size \(n\), and \(\mathcal{C}\) denotes a subset of features with size \(m\). Furthermore, we define \(\mathcal{H}\) to be the class of decision trees, where each \(h \in \mathcal{H}\) is a function \(h: \mathbb{R} \rightarrow \{-1, +1\}\).

The main novel components of our algorithm are the adaptively learned observation and feature sampling schemes. As outlined in Algorithm 1, we first sample an MP based on adaptive observation and feature probability distributions \(\boldsymbol{p}\) and \(\boldsymbol{q}\) initially set to be uniform \(\left(\mathbf{U}_{[N]}\right)\) and \(\left(\mathbf{U}_{[M]}\right)\), where \([N] = 1,2 \dots, N\) and \([M] = 1,2, \dots, M\) respectively. Then, we fit a tree to the minipatch, and lastly, we update the output given the ensemble function \(E: \mathbb{R}^M \rightarrow \mathbb{R}\). The inspiration for this algorithm structure is taken from MP-Boost [8], however, the novel component is how we learn and adaptively update the observation probability \(\boldsymbol{p}\) and the feature probability \(\boldsymbol{q}\).

Additionally, and similar to other stochastic boosting regimes that utilize out-of-bag samples for internal validation, we propose to leverage out-of-patch samples for internal validation including early stopping and possibly hyperparameter tuning. We adopt a scheme similar to MP-Boost [8] to compute the out-of-patch error. Specifically, we aggregate the output of weak learners that do not include sample \(i\) it in their minipatch: \(G^{(t)}(\mathbf{x_i}) = G^{(t-1)}(\mathbf{x}_i) + h^{(t)}\left(\mathbf{x}_{i_{\mathcal{C}^{t}}}\right) \forall_i \notin \mathcal{R}^{(t)}\). The out-of-patch update is defined as \[\begin{align} \label{eqn:oop} \text{oop}^{(t)} = \frac{1}{i \notin \mathcal{R}^{(t)}}\sum_{i=1}^{i \notin \mathcal{R}^{(t)}}{{\mathbb{1}}}_{\{sgn(G^{(t)}(\mathbf{x}_i)) = y_i\}}. \end{align}\tag{1}\] This out-of-patch output serves as a conservative estimate of test accuracy. Our early stopping rule is straightforward: if the current out-of-patch performance increases beyond a certain margin, the algorithm requires more time to refine; otherwise, it is deemed to generalize well to the test set and learning can cease.

2.2 Adaptive feature and observation sampling↩︎

The main novel component of our Fair MP-Boost algorithm is the development of a novel adaptive scheme to learn observation and feature sampling probabilities, \(\boldsymbol{p}\) and \(\boldsymbol{q}\) respectively. The key idea of our proposal is that \(\boldsymbol{p}\) and \(\boldsymbol{q}\) should be based on a linear combination of functions that trade off accuracy and fairness. Specifically, we define a hyperparameter, \(\alpha\), that controls this tradeoff. When \(\alpha = 0\), our Fair MP-Boost algorithm solely focuses on accuracy, which reverts to the MP-Boost algorithm introduced by [8] as a special case. When \(\alpha = 1\), Fair MP-Boost solely focuses on fairness and represents a novel fairness-focused boosting algorithm. When \(\alpha \in (0,1)\), Fair MP-Boost balances both accuracy and fairness. The same tradeoff parameter \(\alpha\) is used in both the feature and observation sampling probabilities.

Let us first dive into the specifics of the observation sampling scheme. The general idea is to prioritize observations that are more difficult to learn. For accuracy, misclassified samples are more difficult to learn, so we propose to increase their probabilities to be sampled more frequently. Formally, let \(\mathcal{L}_A: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}^+\) be a function that measures the similarity between the ensemble outputs and labels. To ensure misclassified observations are sampled more, we assign an accuracy-based probability proportional to \(\mathcal{L}_A(y_i, E(x_i))\) to the \(i^{th}\) observation.

The fairness-focused observation update is a bit trickier because the fairness of a single observation cannot explicitly be measured using common group fairness measures. Inspired by Fair Adversarial Gradient Tree Boosting (FAB)[7], at each iteration, we propose to use an adversarial classifier \(h^{(t)}_F\) that predicts the protected attribute based on the output of \(h^{(t)}\). Using the function \(h_F:\mathbb{R}\rightarrow \mathbb{R}^+\), observation probabilities are iteratively updated to upsample those with a higher probability of predicting the protected attribute based on \(h^{(t)}_F\) outputs. Leveraging \(\mathcal{L}_F: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}^+\), which measures the similarity between the protected attribute and the probability of observing it given the adversarial model’s output, we update the fairness-based probabilities to be proportional to \(\mathcal{L}_F\left(z_i, h_F^{(t)}\left(h^{(t)}\left((x_i)_{C^{(k)}}\right)\right)\right)\) for observations \(i \in \mathcal{R}^{(t)}\). Combining both \(\mathcal{L}_A\) and \(\mathcal{L}_F\), the observation probability update can be defined as: \[\begin{align} \label{eqn:obs95probs} \tilde{\boldsymbol{p}} = (1-\alpha)\mathcal{L}_A+ (\alpha)\mathcal{L}_F. \end{align}\tag{2}\] The exponential or cross-entropy functions are suitable loss functions for \(\mathcal{L}_A\) and \(\mathcal{L}_F\). As an example and in this paper’s empirical studies, we use the exponential loss for both \(\mathcal{L}_A\) and \(\mathcal{L}_F\). Lastly, we normalize the probabilities to ensure they sum to one: \(\boldsymbol{p}^{(t+1)} = \tilde{\boldsymbol{p}}/\sum_{i=1}^N\tilde{\boldsymbol{p}}_i\).

Inspired by the MP-Boost algorithm which adaptively learns features, we seek to do so in a manner that balances both accuracy and fairness. To this end, we need a measure for each tree of how features contribute to both fairness and accuracy. For accuracy, we rely on what we call TreeFIS(MDI) [10]; for fairness, we rely on the recently proposed FairTreeFIS[11] which uses an MDI-like metric based on group fairness. By leveraging TreeFIS and FairTreeFIS, we dynamically select features that optimize both accuracy and fairness and facilitate a clearer interpretation of our ensemble model’s outcomes.

Formally, let \(\boldsymbol{q}\) be the probability distribution of features. Additionally, let \(I_j^{h^{(t)}}\) be accuracy feature importance scores computed using TreeFIS(MDI) and \(F_j^{h^{(t)}}\) be the fairness feature importance score computed using the FairTreeFIS for features \(j \in C^{(t)}\). Different from TreeFIS, FairTreeFIS can be both positive and negative. Positive values correspond to features that contribute to fairness and negative values indicate features contributing to bias. This means that the FairTreeFIS values are not on the same scale as TreeFIS which are non-negative. To account for this, we propose to use a ReLu function \(R(x) = max(0,x)\) that zeros out biased features. Furthermore, to ensure we do not establish feature sampling probabilities too early in learning, we include a slow learning term \(\mu \in (0,1)\), which determines the ratio of exploration versus exploitation. The feature probability update can be defined as \[\begin{align} \label{eqn:feat95probs} \tilde{\boldsymbol{q}}_j = (1-\mu)q_j^{(t)} + \mu \left(\alpha*R(F_j^{h(t))})+ (1 - \alpha)*I_j^{h(t)}\right), \end{align}\tag{3}\] for every \(j \in C^{(t)}\). Note that \(\tilde{\boldsymbol{q}}_j = max(\tilde{\boldsymbol{q}}_j, \epsilon)\). Following inspiration from multi-armed bandits, this ensures that feature sampling probabilities are never set to exactly zero and hence always open for exploration. Analogous to the observation probabilities, we then normalize the probabilities to ensure they sum to one: \(\boldsymbol{q}^{(t+1)}_j = \tilde{\boldsymbol{q}}_j/\sum_{i=1}^N\tilde{\boldsymbol{q}}_j\).

Finally, our novel adaptively learned observation and feature sampling probabilities also yield intrinsic interpretations of leverage points and feature importance scores respectively. Specifically, we propose to average the observation and feature sampling probabilities after a burn-in phase when the probabilities have stabilized. In this paper, our empirical work focuses on validating our feature sampling probabilities as a measure of intrinsic feature importance.

2.3 Hyperparameter Tuning↩︎

The minipatch size plays a pivotal role as a hyperparameter in our algorithm. Inspiration from boosting suggests having a series of weak learners. If the minipatches are smaller, they are going to be weaker learners, hence \(n = \sqrt{N}\) and \(m = \sqrt{M}\). Another critical hyperparameter in our algorithm is the learning rate \(\mu\), for which we set as a default to \(\mu = 0.2\). Additionally, we expect that a user may specify the level of \(\alpha\) they want, but if not this tuned using the Fairness-AUC [13]. It is also worth noting that these hyperparameters can be selected in a data-driven manner using our out-of-patch criterion.

3 Empirical Studies↩︎

3.1 Simulation Setup and Results↩︎

We design simulation studies to validate our proposed Fair MP-Boost method in terms of interpretability, accuracy, and fairness. We utilize four feature groups: features within \(G_1\) and \(G_2\) exhibit correlation with the protected attribute \(z\) and thus exhibit bias; features within \(G_1\) and \(G_3\) are indicative features linked to the outcome \(y\); and features within \(G_4\) solely contribute noise. We simulate the protected attribute, \(z_i\), as \(\mathbf{z}\stackrel{i.i.d}{\sim} Bernoulli(\pi)\) and take \(\pi = 0.2\). Then, the data is generated as \(\mathbf{x}_{i,j}\stackrel{i.i.d}{\sim}N(\alpha_j*z_i, \boldsymbol{\Sigma})\) with \(\alpha_j = 2\) if \(j \in G_1\) or \(G_2\) and \(\alpha_j = 0\) if \(j \in G_2\) or \(G_4\). Hence, all features in \(G_1\) and \(G_2\) are strongly associated with \(z\) and should be identified as biased. Furthermore, we consider a non-linear additive scenario where \(f(x_i) = \beta_0 + \sum_{j=1}^p\beta_jsin(x_{ij})\). We also let \(\beta_j = 1\) for \(j \in G_1\) or \(G_3\) and \(\beta_j = 0\) for \(j \in G_2\) or \(G_4\). We employ a logistic model with \(y_i \stackrel{i.i.d}{\sim} Bernoulli (\sigma( f(x_i))\), where \(\sigma\) is the sigmoid function.

We begin by validating the feature sampling probabilities of Fair MP-Boost. In Figure 2, we present average feature sampling probabilities of Fair MP-Boost on the simulated dataset with \(\alpha = 0.1\) in the top panel and \(\alpha = 0.9\) in the bottom panel. In the top panel, when accuracy is prioritized, features in \(G_1\) and \(G_3\), or the features associated with the response \(y\) have higher sampling probabilities than the features correlated with the protected attribute \(z\) and the noise features. Note that while features in \(G_2\) and \(G_4\) have lower sampling probabilities, they are not zeroed out and are still sampled between 3-5% of the time. Similarly, in the top panel where fairness is prioritized, the features in \(G_3\) which are uncorrelated with the protected attribute are sampled the most. Looking at these average feature sampling probabilities allows us to interpret important features and validate our Fair MP-Boost feature subsampling scheme.

Additionally, we consider the performance of Fair MP-Boost to various boosting and bias mitigation strategies. We directly compare our method with two relevant approaches: Fair Adversarial Gradient Tree Boosting (FAB) [7] and the FairGBM method [5], as well as an Adversarial approach [14]. Furthermore, FAB has a hyperparameter that controls the fairness-accuracy tradeoff, hence we show results for two values. We use default hyperparameter tuning in the available software for all of these approaches. In Table 1, we compare Random Forest, Gradient Boosting, FAB, the Adversarial approach, and FairGBM to Fair MP-Boost and report their accuracy and fairness metrics on the simulated dataset. We observe that while Random Forest, Gradient Boosting, and FairGBM have comparable accuracy, Fair MP-Boost achieves higher accuracy when accuracy is prioritized (\(\alpha = 0.1\)). Note also that even when accuracy is prioritized, Fair MP-Boost achieves equal if not higher fairness than all other methods. When fairness is prioritized (\(\alpha = 0.9\)), Fair MP-Boost significantly outperforms all other methods in terms of fairness, while still achieving comparable accuracy in comparison to other methods. While the fairness-accuracy tradeoff is evident in our results, Fair MP-Boost still outperforms other methods in terms of accuracy given it achieves the highest fairness. These results strongly validate Fair MP-Boost’s performance in terms of accuracy, fairness, and interpretability.

3.2 Case Study on Real Data↩︎

To align our work with existing fairness literature, we evaluate our method on two popular benchmark datasets. We examine the Adult Income dataset containing 14 features and approximately 48,000 individuals with class labels stating whether their income is greater than \(\$50,000\) and Gender as the protected attribute [15]. We also consider the Law School dataset, which has 8 features and 22,121 law school applicants with the class labels stating whether an individual will pass the Bar exam when finished with law school and Race as the protected attribute [16].

We first interpret our \(\mathrm{\small Fair MP-Boost}\) algorithm on the Adult Dataset with Gender as the protected attribute. Part A of Figure 3 displays the average feature sampling probabilities for \(\mathrm{\small Fair MP-Boost}\) when prioritizing accuracy (left) and fairness (right). Part B shows feature importance metrics using \(\emph{TreeFIS}\) and \(\emph{FairTreeFIS}\) for fair and unfair models. Notably, when accuracy is prioritized, the feature Married is sampled frequently but also exhibits a very negative \(\emph{FairTreeFIS}\) score, indicating its contribution to model bias. Conversely, when prioritizing fairness, Married is utilized less. The \(\emph{FairTreeFIS}\) metric (blue bars) facilitates the interpretation of fairness-contributing features, while the red bars show features impacting accuracy. When \(\alpha = 0.9\), none of the features significantly contribute to bias in the model. Moreover, Married is notably underutilized, resulting in a fairer model. The MDI feature importance (red bars) closely aligns with our sampling probabilities, affirming the validity of using average feature sampling probabilities as an intrinsic metric for feature importance.

Lastly, in Table 2, we compare accuracy and fairness results on several boosting and bias mitigation techniques using the Adult Income and Law School datasets. For the Adult dataset, Fair MP-Boost achieves comparable if not better performance in terms of accuracy in comparison to the other methods. For fairness, Fair MP-Boost outperforms all other methods. Similarly, for the law school dataset, Fair MP-Boost achieves comparable accuracy and outperforms other methods in terms of fairness. These results strongly validate our work.

4 Discussion↩︎

In this work, we introduce an interpretable boosting method aimed at enhancing group fairness while maintaining accuracy. A significant advantage of our approach lies in its interpretability, as it allows direct insight into the features influencing both accuracy and fairness.

References↩︎