TowerDebias: A Novel Debiasing Method based on the Tower Property

We consider prediction of a variable $Y$ from a feature vector X and a vector of sensitive variables $S$. The target $Y$ may be either numeric (in a regression setting) or dichotomous (in a two-class classification setting where $Y$ = 1 or $Y$ = 0). The $m$-class case can be handled using $m$ dichotomous variables with regards to the material presented here.

Consider an algorithm developed by a vendor, such as COMPAS, which was trained on data $(Y_{i},X_{i},S_{i}),i=1,\dots,n$. This data can be assumed to originate from some specific data-generating process, with the algorithm’s initial objective being to estimate $E(Y|X,S)$. However, the client who purchased the algorithm wishes to exclude $S$ and instead estimate $E(Y|X)$¹¹1Quantities of the form E(), P(), Var() and so on refer to the probability distribution of the data generating process..

In the regression setting, we assume squared prediction error loss to minimize error. In the classification setting, we define:

where the predicted class label is given by:

This approach minimizes overall misclassification rate by selecting the class with the highest predicted probability.

Several potential use cases for employing tDB can be identified:

• (a)

  The client has no knowledge of the internal structure of the black-box algorithm and lacks access to the original training data. Thus, the client will need to gather their own data from new cases that arise during the tool’s initial use.

• (b)

  Client has no knowledge of the inner workings of the black-box algorithm but is given the training data.

• (c)

  User of the algorithm, potentially even the original developer, knows the black-box’s details and possesses the training data. He/she is satisfied with the performance of the algorithm, but desires a quick, simple way to remove the influence of $S$ in its predictions.

In each setting, the individual aims to predict new cases using only $X$. The client either does not know $S$ or chooses to disregard it. In other words, although the algorithm provides estimates of $E(Y|X,S)$, the goal is to use $E(Y|X)$ instead as the prediction instead. In other words, even though the algorithm gives us an estimated $E(Y|X,S)$, we wish to instead use estimated $E(Y|X)$ as our prediction. In this paper, we present tDB as an innovative approach to modify the predictions of the original algorithm, bypassing the “black box” nature of the model.

Many commercial black-box models may include sensitive variables, which raises significant ethical and legal concerns. In the pursuit of fairness, a fundamental Fairness-Utility tradeoff emerges: a delicate balance between fairness and predictive accuracy (Gu et al., 2022; Sun et al., 2023). This tradeoff highlights that prioritizing fairness often leads to a reduction in accuracy. However, the extent of this tradeoff is influenced by the specific fairness metrics employed and their implementation details. This paper explores the impact of applying tDB on the fairness-utility tradeoff across different datasets, encompassing both regression and classification tasks.

The paper is organized as follows: Section 2 reviews prior literature on fair machine learning, focusing on relevant methods and proposed fairness metrics; Section 3 introduces the towerDebias algorithm and provides supporting proofs demonstrating fairness improvements; Section 4 presents empirical results of tDB on multiple datasets; and Section 5 concludes with a discussion and future directions.

Already, much literature has been published on different fairness measures, many of which are complex and require substantial domain-specific knowledge. In this section, we discuss some widely accepted fairness measures:

• (a)

  Demographic Parity: This fairness criterion evaluates group fairness by requiring that individuals from both marginalized and non-marginalized groups have an equal probability of being assigned to the positive class, regardless of group membership. For a binary $Y$ and $S$, it can be expressed as:

  $$P(\hat{Y}=1|S=0)=P(\hat{Y}=1|S=1)$$

  In this equation, the probability of being assigned to the positive class (1) is the same for both Group $S$ = 0 and Group $S$ = 1, indicating group fairness. Thus, demographic parity requires that the prediction be independent of the protected attribute, meaning that membership to a protected class should not be correlated with the outcome variable.

• (b)

  Equalized Odds: This criterion offers a more robust measure of group fairness compared to demographic parity. It requires that the predictor $\hat{Y}$ satisfies equalized odds with respect to the protected attribute $S$ and the outcome $Y$, if $\hat{Y}$ and $S$ are independent conditional on $Y$. For binary $Y$ and $S$, this can be expressed as:

  $$P(\hat{Y}=1|S=0,Y=y)=P(\hat{Y}=1|S=1,Y=y),\quad y\in\{0,1\}$$

  For the outcome $y=1$, this constraint requires that $\hat{Y}$ has equal true positive rates across the two demographics, $S=0$ and $S=1$. For $y=0$, the constraint ensures equal false positive rates. In this way, equalized odds requires that the model maintains consistent accuracy across all demographic groups, penalizing models that perform well only on the majority group (Hardt et al., 2016).

• (c)

  Correlation $\rho(\hat{Y},S)$: Correlation-based measures are commonly used in fairness evaluation (Deho et al., 2022; Mary et al., 2019). Baharlouei et al. (2019) extends fairness measurement to continuous variables by applying the Renyi maximum correlation coefficient. Lee et al. (2022) introduces a maximal correlation framework for formulating fairness constraints. Roh et al. (2023) explores the concept of correlation shifts in the context of fair training. In general, correlations between predicted $\hat{Y}$ and $S$ can be expressed as:

  $$\rho(\hat{Y},S)=\text{Correlation between predicted }\hat{Y}\text{ and }S$$

  In this paper, we apply correlation-based measures to evaluate fairness between our predicted $\hat{Y}$ and the sensitive attribute $S$. Specifically, we use the Kendall’s Tau correlation coefficient to examine the relationships between these variables. Kendall’s Tau is a non-parametric measure that captures both the strength and direction of the association, making it well-suited for data on an ordinal or categorical scale. We choose Kendall’s Tau over other correlation measures, such as Pearson’s correlation, because it does not require the assumption of linearity, allowing us to assess relationships even when $\hat{Y}$ or $S$ are categorical or binary (KENDALL, 1938).

  For categorical $S$, we apply one-hot encoding to create dummy variables for each category, then compute Kendall’s Tau for each category separately to assess the individual reduction in correlation. Lower values (closer to 0) of Kendall’s Tau indicate reduced association between the predictions from the black-box model and $S$, which reflects higher fairness. Since interpreting the direction of association is not the primary focus of our fairness analysis, we use the absolute value of Kendall’s Tau to evaluate the overall reduction in correlation between predictions and $S$.

Assessing accuracy is generally more straightforward than evaluating fairness. In regression tasks, we measure accuracy using the Mean Absolute Prediction Error (MAPE), which calculates the average absolute difference between the predicted and true values. For binary classification tasks, we measure accuracy using the overall misclassification rate to represent the proportion of incorrect predictions among all predictions made.

The Tower Property of conditional expectation (Wolpert, 2009) states:

The Tower Property of conditional expectation expresses that the conditional expectation of $Y$ given $X$ can be decomposed into the conditional expectation of $Y$ given both $X$ and a sensitive attribute $S$, then conditioned again on $X$. Note that conditional expectations themselves are random variables. For example, $E(Y|X)$—as a function of the random variable $X$—is itself a random variable.

To illustrate this property intuitively, consider the following example using a dataset that will be referenced later in the paper. This dataset includes: $Y$, a score on the Law School Admissions Test; $X$, family income (five quintile levels); and $S$, race. What does equation (3.1) mean at the “population” level of the data generating process?

For instance, $E(Y|4,Black)$ is the mean LSAT score for Black test-takers in the fourth income quintile. Now, consider the expression:

This quantity averages the mean LSAT score for income level 4 over all races. (3.1) then states that this average will equal the overall mean score for income level 4, regardless of race. This aligns with our intuition: conditioning on income (level 4) and then averaging over all races gives the same result as directly conditioning on income alone. Note that the averaging is weighted by the probabilities of the various races.

In simple terms, our method can be summarized as follows:

To remove the impact of the sensitive variable $S$ on predictions of $Y$ from $X$, average the predictions over $S$.

(3.1) is, of course, an idealization at the population or data-generating process level. However, in practical situations, the vendor’s black-box algorithm can only estimate $E(Y|X,S)$ from training data, $(Y_{i},X_{i},S_{i})$, $i=1,n$. In such cases, what additional factors must be considered?

Let’s revisit the LSAT example, now adding the features undergraduate GPA and cluster (an indicator of the law school’s ranking). Now $X=(income,GPA,cluster)$, and suppose we want to predict a new case where $X=(1,2.7,1)$. Using tDB, we would take the average LSAT scores for all cases in our training data that match this exact value of $X$. However, it turns out that there are only 7 such cases in our dataset—hardly enough to obtain a reliable average. Furthermore, if income were measured in dollars rather than quintiles, or if GPA were available to two decimal places, there might be no cases at all matching this exact $X$ value.

The solution is to instead average over all cases near the given value of $X$. To do this, we introduce a new parameter, $k$, which represents the number of training cases whose $Y$ values will be averaged—specifically, the $k$ cases nearest to the $X$ value we want to predict.²²2This parameter is analogous to $k$ in the K-Nearest Neighbors method.

Choosing the right $k$ is crucial to balancing the influence of the sensitive variable. In particular, $k$ defines the number of neighboring rows used to compute the average LSAT score. A small $k$ might lead to an overly narrow selection of cases, resulting in minimal reduction in correlation with the sensitive variable. Conversely, a larger $k$ may include rows that are too distant and not representative of the new data point being debiased. Thus, choosing an appropriate $k$ is extremely important as it must balance the need for a sufficiently large sample size while maintaining proximity to the target data point. As we will see in the empirical section, this choice of $k$ has a considerable impact on the fairness-utility tradeoff.

The next two parts of this section present supporting proofs for the reduction in correlation when applying tDB. Section 3.3 provides an expression for Pearson correlation reduction in a simulated trivariate normal case. Section 3.4 extends this concept to more general cases, using vector spaces to indicate fairness improvements.

Suppose we are predicting $Y$ using a single numeric feature $X$ and a numeric $S$. We assume that (X,S,Y) follows a trivariate normal distribution. This implies that the joint distribution of $(X,S,Y)$ is normal, and therefore, each individual variable $X$, $S$, and $Y$ also has a marginal normal distribution (Johnson and Wichern, 1993).

Our goal is to derive a closed-form expression for the reduction in Pearson correlation between the predicted $Y$ and $S$ when predicting $Y$ from both $(X,S)$ (Case I) versus predicting $Y$ from just $X$ (Case II). In this simulation, we are specifically interested in assessing the reduction in the Pearson correlation, as opposed to using Kendall’s Tau correlation used in the tDB method.

The choice of Pearson correlation in this context is motivated by the nature of the trivariate normal distribution. The Pearson correlation coefficient quantifies the strength of the linear relationship between two numeric variables (Faizi and Alvi, 2023). Its inherent linearity properties across both the numerator and denominator of its formula helps facilitate simplification for our expressions. Specifically, the numerator of Pearson’s correlation involves computing $\text{Cov}(U,V)$ and exhibits bilinearity in the random variables $U$ and $V$. This bilinearity property allows us to simplify expressions by factoring out any constants and leveraging additivity.

In contrast, Kendall’s Tau lacks such linearity properties, which makes it less conducive to similar simplifications. Thus, Pearson correlation is better suited for this particular example, as it enables the derivation and analysis of a closed-form expression for the reduction in correlation between the two cases. This choice provides a clear and interpretable framework for examining the impact of predicting $Y$ from both $X$ and $S$ versus predicting $Y$ solely from $X$.

Given our application, the general formula for calculating the Pearson correlation coefficient³³3This general $\rho(\hat{Y},S)$ will be used to compute the correlations in both Case I and Case II. $\rho(\hat{Y},S)$ between the predicted $\hat{Y}$ and $S$ is as follows (Mukaka, 2012):

To proceed, let’s first define key quantities using regression coefficients to represent the relationships among $X$, $Y$, and $S$. These will be important in calculating correlation reductions between Case I and Case II.

In Case I, where we predict $Y$ using $(X,S)$, the linear relationship of the predicted $Y$—denoted as $Y_{1}$—can be represented as follows:

Here:

• •

  $\beta_{0}$ is the intercept term,

• •

  $\beta_{1}$ is the coefficient for $X$,

• •

  $\beta_{2}$ is the coefficient for $S$.

For Case II, we may express the linear relationship to predict $Y$ solely from $X$—denoted as $Y_{2}$—as follows:

Similarly, we define:

• •

  $\delta_{0}$ is the intercept term,

• •

  $\delta_{1}$ is the coefficient for $X$.

Note that since each case uses a different set of predictors to estimate $\hat{Y}$, the resulting predictions—$\hat{Y_{1}}$ and $\hat{Y_{2}}$—will be different. We are interested in finding a closed-form expression for the difference in correlations between Case I and Case II for $\rho_{\text{reduc}}(\hat{Y},S)$, which can be defined as:

Furthermore, given the assumption of a trivariate normal distribution, the linear relationship between $X$ and $S$ allows us to model $S$ as a function of $X$. Note that we are focusing on modeling $S$, instead of $\hat{S}$, as in the previous examples for $\hat{Y_{i}}$. The inclusion of the $\epsilon$ term accounts for any errors. Thus, this relationship can be expressed as:

In this model:

• •

  $\alpha_{0}$ is the intercept term,

• •

  $\alpha_{1}$ is the coefficient for $X$,

• •

  $\epsilon\sim\mathcal{N}(0,\sigma^{2})$ is the error term, normally distributed with mean zero and variance $\sigma^{2}$, and is independent of the predictor terms $X$ and $S$ (Kutner et al., 1974).

Now that we have defined the necessary quantities to model the relationship between the given variables, we can proceed with finding $\rho_{\text{reduc}}(\hat{Y},S)$.

To begin, we may define $\rho(\hat{Y}_{1},S)$ as:

Similarly, we may define $\rho(\hat{Y}_{2},S)$ as:

Thus, $\rho_{\text{reduc}}(\hat{Y},S)$ is:

This expression quantifies the reduction in Pearson correlation between $\hat{Y}$ and $S$ between Case I (with both $X$ and $S$ as predictors) and Case II (with only $X$).

Fairness improvements can be established more broadly, extending beyond the simulated trivariate normal case, through the use of $L_{2}$ function spaces. This section will demonstrate the proof for this scenario.⁴⁴4More details to be added.

The SVCensus dataset is a subset of U.S. Census data from the early 2000s, focusing on income levels across six engineering occupations within Silicon Valley. Each person’s record in the dataset includes attributes such as occupation, education level, number of weeks worked, and age. Our goal is to predict wage income ($Y$) with respect to gender as the sensitive attribute ($S$). Note that this is regression problem, and gender contains two categories: male and female.

The graphs above illustrate the fairness-accuracy tradeoffs from applying towerDebias method to the baseline model predictions. Figure (1) shows the improvement in fairness—the reduction in correlation between predicted wage income and gender—and Figure (2) displays the comparative losses in accuracy.

The tDB method demonstrates substantial reductions in correlations across different values of the tradeoff parameter $k$. When applied to the predictions of traditional machine learning algorithms, tDB achieves a notable reduction in correlations, with decreases of up to 50% compared to baseline results. For instance, the initial correlations for Linear Regression, K-NN, and Neural Networks were around 0.25, while tDB reduced these correlations to below 0.1, even at smaller values of $k$. In contrast, the fairML methods had already incorporated fairness in their baseline predictions to achieve smaller initial correlations, but tDB still provided further improvements. However, in Zafar’s Linear Regression model, which started with a relatively low initial correlation of 0.1, tDB did not display any further reductions.

Regarding accuracy, tDB shows modest losses in comparative MAPE results for both traditional and fair machine learning algorithms. For traditional models, the baseline accuracy averaged around $25,000, with only a slight decrease in accuracy after applying tDB. Even at higher values of $k$ (k $\geq$ 30), the relative loss in accuracy was less than 5% as compared to the original results. In fact, for XGBoost, the error appears decrease by $2,000—indicating a simultaneous improvement in both fairness and accuracy measures. For the fairML algorithms, which initially had a higher MAPE of approximately $35,000, the impact of tDB was even less significant, resulting in less than a 1% reduction in accuracy.

Both Figures (1) and (2) highlight the effect on the fairness-utility tradeoff on the SVCensus data. Overall, applying tDB on this dataset across various baseline model predictions results in considerable correlation reductions between predicted income and gender, $\tau(\hat{Y},S)$, to highlight improvements in fairness with minimal loss in predictive accuracy. This underscores the effectiveness of tDB in controlling the influence of the sensitive variable $S$ and producing fairer predictions.

The Law School Admissions dataset⁷⁷7Note that the data pertains to students who were already admitted to law school, so despite the title, it does not relate to the admissions process itself. is a survey of law school students in the United States from 1991. It includes various demographic and academic details, such as age, grade decile score, undergraduate GPA, family income, gender, race, and more. The dataset is used to predict LSAT scores ($Y$), with race considered as the sensitive attribute ($S$). This is a regression task, where race is categorized into the following groups: Asian, Black, White, Other, and Hispanic. This setup broadens the scope of fairness measurement beyond binary sensitive attributes, such as gender, as shown in the previous example with the SVCensus data.

Similar to the results in the SVCensus data, Figure (3) demonstrates improvements in fairness—reduction in the correlation between predicted LSAT scores and each individual racial category—and Figure (4) presents the relative increase in MAPE in comparison to the baseline predictions.

In terms of fairness, we present individual results for each racial category: White, Black, Hispanic, Asian, and Other. Figure (3) illustrates the effectiveness of applying towerDebias across various baseline algorithms to produce considerable reduction in correlations between different racial groups. Notably, each racial group exhibited different baseline correlations. For instance, in the neural network baseline model, the correlation between predicted LSAT scores and “African American” and “White” was 0.3, while for “Asian” and “Other” it was below 0.1, and for “Hispanic”, it was 0.15. Similar correlations results were observed in the other traditional machine learning model predictions. Concerning the fairML functions, the baseline results showed smaller initial correlations due to the fairness constraints integrated into the model, but there was still some slight variation between the racial groups. In applying to towerDebias, we see an decrease of more than 50% across all of the racial groups starting at moderate values of the trade-off parameter $k$.

The accuracy results are similar to those observed in the SVCensus data. For example, the initial algorithms have a mean absolute prediction error of approximately 3.5, and applying towerDebias leads to an accuracy loss of less than 5% across different values of $k$. Interestingly, when towerDebias is applied to XGBoost, it actually improves accuracy, suggesting enhancements in both fairness and accuracy—warranting further investigation. The selection of $k$ rows showed a slight decrease in error, indicating potential improvements in predictive power. The fair machine learning algorithms initially performed similarly to conventional algorithms. Notably, the application of towerDebias resulted in a controlled increase in accuracy, with Zafar’s model experiencing a slight decrease and the Scutari method showing less than a 1% improvement.

Both the Law School Admissions and SVCensus datasets underscore the effectiveness of applying tDB to the fairness-utility tradeoff in regression settings. Our results consistently demonstrate this method effectiveness in reducing the influence of sensitive variables across multiple models and sensitive groups. This highlights the capability of tDB on improve fairness with only a modest reduction in utility. The next phase of experiments will focus on extending this method to classification scenarios.

The COMPAS dataset, which contains data on criminal offenders screened in Florida during 2013-14, is being used to predict the likelihood that a defendant will recommit a crime in the near future (probability of recidivism—denoted as $Y$). Race is treated as the sensitive variable ($S$), with the data pre-processed to include three racial categories: White, African American, and Hispanic. The task is now a binary classification problem where race is considered a categorical variable with three possible values.

An analysis of tDB’s impact on individual algorithms in the COMPAS dataset reveals nuanced outcomes. Logistic Regression, K-Nearest Neighbors, and Neural Networks showed similar fairness improvements, with tDB consistently reducing correlations between predictions ($\hat{Y}$) and sensitive attributes ($S$) across racial groups. Although baseline correlations $\rho(\hat{Y},S)$ varied by race, they consistently decreased with tDB; for example, as $k$ increased, $\rho(\hat{Y},S)$ for African Americans dropped from 0.23 to 0.2, with comparable reductions for Caucasians and Hispanics. KNN and Neural Networks echoed these fairness gains, though with some tradeoff in utility. In contrast, tDB applied to XGBoost showed an unexpected increase in $\rho(\hat{Y},S)$ for each race as $k$ increased, suggesting a need for further investigation.

Analysis of tDB’s fairness outcomes across fairML functions is similar. Ridge Regression models and Zafar’s Logistic Regression initially had lower correlations with sensitive attributes than traditional models, and tDB further enhanced fairness. In contrast, Scutari and Komiyama methods, like XGBoost, showed increased correlations with tDB, while Zafar’s Logistic Regression reduced correlation further for Hispanic and African American groups, maintaining a stable misclassification rate without increase.

In terms of accuracy, we observe a slight increase in misclassification rates. For example, Logistic Regression’s initial misclassification rate of 0.27 saw less than a 10% increase as tDB was applied to more rows. Overall, the utility loss remains controlled—with notable fairness gains—underscoring tDB’s effectiveness in improving fairness within the COMPAS dataset in classification settings.

The Iranian Churn dataset is used to predict customer churn, with “Exited” as the response variable and “Gender” and “Age” as the sensitive variables. In this example, we incorporate a continuous sensitive variable to extend the analysis beyond categorical or binary $S$.

Figure (7) illustrates the impact on correlation for Gender and Age, demonstrating the effectiveness of tDB in reducing correlations for continuous sensitive variables. Similar trends are observed, with correlations decreasing by more than 50% compared to the initial baseline results in both categories. The results from the fairML functions showed mixed trends: applying tDB reduced correlation with respect to Age but increased it for Gender.

In terms of accuracy, Figure (8) shows a modest increase in the misclassification rate for conventional machine learning algorithms compared to tDB. Logistic Regression and K-Nearest Neighbors exhibit relative increases of less than 10% and 25%, respectively, while XGBoost and Neural Networks show increases exceeding 25% for mid-to-high $k$ values. Notably, accuracy remains unchanged when comparing fairML with tDB.

The Dutch Census dataset, collected by the Dutch Central Bureau for Statistics in 2001, is used to predict whether an individual holds a prestigious occupation or not (binary response variable), with “gender” as the sensitive attribute. Note: Zafar’s Logistic Regression encountered errors in training and was excluded from our analysis. The following graphs illustrate the impact of tDB across several models.

Empirical results from the Dutch Census dataset demonstrate significant improvements in fairness with a controlled accuracy loss on classification tasks. The dataset includes 10 categorical predictors, and tDB applies one-hot encoding to convert these into dummy variables, increasing the dimensionality to 61 predictors. This example highlights tDB’s effectiveness on high-dimensional, sparse datasets. Thus, this examples illustrates the effectiveness of tDB on high-dimensional, sparse datasets. In terms of fairness, Figure (9) shows considerable reductions in correlations when applying tDB to both traditional and fair machine learning models. For mid-to-high range of $k$ values, the correlation reduction exceeds 50% compared to the baseline results. Regarding accuracy, Figure (10) hows that tDB leads to a 25% increase in the misclassification rate at mid-to-high range $k$ values for traditional machine learning models, with similar trends observed in comparisons between fairML functions and tDB.

Insights from the COMPAS, Iranian Churn, and Dutch Census datasets highlight the impact of tDB on the fairness-utility tradeoff in classification settings. The consistent reduction in correlation between sensitive variables and predictions across different models underscores the effectiveness of tDB in enhancing fairness, albeit with some utility tradeoff. Overall, these findings suggest that our method shows strong potential for improving fairness in both regression and classification scenarios.