Reductions

On a high level, the reduction algorithms within Fairlearn enable unfairness mitigation for an arbitrary machine learning model with respect to user-provided fairness constraints. All of the constraints currently supported by reduction algorithms are group-fairness constraints. For more information on the supported fairness constraints refer to Fairness constraints for binary classification and Fairness constraints for regression.

Note

The choice of a fairness metric and fairness constraints is a crucial step in the AI development and deployment, and choosing an unsuitable constraint can lead to more harms. For a broader discussion of fairness as a sociotechnical challenge and how to view Fairlearn in this context refer to Fairness in Machine Learning.

The reductions approach for classification seeks to reduce binary classification subject to fairness constraints to a sequence of weighted classification problems (see [1]), and similarly for regression (see [2]). As a result, the reduction algorithms in Fairlearn only require a wrapper access to any “base” learning algorithm. By this we mean that the “base” algorithm only needs to implement fit and predict methods, as any standard scikit-learn estimator, but it does not need to have any knowledge of the desired fairness constraints or sensitive features.

From an API perspective this looks as follows in all situations

>>> reduction = Reduction(base_estimator, constraints, **kwargs)  
>>> reduction.fit(X_train, y_train, sensitive_features=sensitive_features)  
>>> reduction.predict(X_test)  

Fairlearn doesn’t impose restrictions on the referenced base_estimator other than the existence of fit and predict methods. At the moment, the base_estimator’s fit method also needs to provide a sample_weight argument which the reductions techniques use to reweight samples. In the future Fairlearn will provide functionality to handle this even without a sample_weight argument.

Before looking more into reduction algorithms, this section reviews the supported fairness constraints. All of them are expressed as objects inheriting from the base class Moment. Moment’s main purpose is to calculate the constraint violation of a current set of predictions through its gamma function as well as to provide signed_weights that are used to relabel and reweight samples.

Fairness constraints for binary classification

All supported fairness constraints for binary classification inherit from UtilityParity. They are based on some underlying metric called utility, which can be evaluated on individual data points and is averaged over various groups of data points to form the utility parity constraint of the form

\[\text{utility}_{a,e} = \text{utility}_e \quad \forall a, e\]

where \(a\) is a sensitive feature value and \(e\) is an event identifier. Each data point has only one value of a sensitive feature, and belongs to at most one event. In many examples, there is only a single event \(*\), which includes all the data points. Other examples of events include \(Y=0\) and \(Y=1\). The utility parity requires that the mean utility within each event equals the mean utility of each group whose sensitive feature is \(a\) within that event.

The class UtilityParity implements constraints that allow some amount of violation of the utility parity constraints, where the maximum allowed violation is specified either as a difference or a ratio.

The difference-based relaxation starts out by representing the utility parity constraints as pairs of inequalities

\[\begin{split}\text{utility}_{a,e} - \text{utility}_{e} \leq 0 \quad \forall a, e\\ -\text{utility}_{a,e} + \text{utility}_{e} \leq 0 \quad \forall a, e\end{split}\]

and then replaces zero on the right-hand side with a value specified as difference_bound. The resulting constraints are instantiated as

>>> UtilityParity(difference_bound=0.01)  

Note that satisfying these constraints does not mean that the difference between the groups with the highest and smallest utility in each event is bounded by difference_bound. The value of difference_bound instead bounds the difference between the utility of each group and the overall mean utility within each event. This, however, implies that the difference between groups in each event is at most twice the value of difference_bound.

The ratio-based relaxation relaxes the parity constraint as

\[r \leq \dfrac{\text{utility}_{a,e}}{\text{utility}_e} \leq \dfrac{1}{r} \quad \forall a, e\]

for some value of \(r\) in (0,1]. For example, if \(r=0.9\), this means that within each event \(0.9 \cdot \text{utility}_{a,e} \leq \text{utility}_e\), i.e., the utility for each group needs to be at least 90% of the overall utility for the event, and \(0.9 \cdot \text{utility}_e \leq \text{utility}_{a,e}\), i.e., the overall utility for the event needs to be at least 90% of each group’s utility.

The two ratio constraints can be rewritten as

\[\begin{split}- \text{utility}_{a,e} + r \cdot \text{utility}_e \leq 0 \quad \forall a, e \\ r \cdot \text{utility}_{a,e} - \text{utility}_e \leq 0 \quad \forall a, e\end{split}\]

When instantiating the ratio constraints, we use ratio_bound for \(r\), and also allow further relaxation by replacing the zeros on the right hand side by some non-negative ratio_bound_slack. The resulting instantiation looks as

>>> UtilityParity(ratio_bound=0.9, ratio_bound_slack=0.01)  

Similarly to the difference constraints, the ratio constraints do not directly bound the ratio between the pairs of groups, but such a bound is implied.

Note

It is not possible to specify both difference_bound and ratio_bound for the same constraint object.

Demographic Parity

A binary classifier \(h(X)\) satisfies demographic parity if

\[\P[h(X) = 1 \given A = a] = \P[h(X) = 1] \quad \forall a\]

In other words, the selection rate or percentage of samples with label 1 should be equal across all groups. Implicitly this means the percentage with label 0 is equal as well. In this case, the utility function is equal to \(h(X)\) and there is only a single event \(*\).

In the example below group "a" has a selection rate of 60%, "b" has a selection rate of 20%. The overall selection rate is 40%, so "a" is 0.2 above the overall selection rate, and "b" is 0.2 below. Invoking the method gamma shows the values of the left-hand sides of the constraints described in Fairness constraints for binary classification, which is independent of the provided difference_bound. Note that the left-hand sides corresponding to different values of sign are just negatives of each other. The value of y_true is in this example irrelevant to the calculations, because the underlying utility in demographic parity, selection rate, does not consider performance relative to the true labels, but rather proportions in the predicted labels.

Note

When providing DemographicParity to mitigation algorithms, only use the constructor and the mitigation algorithm itself then invokes load_data. The example below uses load_data to illustrate how DemographicParity instantiates inequalities from Fairness constraints for binary classification.

>>> from fairlearn.reductions import DemographicParity
>>> from fairlearn.metrics import MetricFrame, selection_rate
>>> import numpy as np
>>> import pandas as pd
>>> dp = DemographicParity(difference_bound=0.01)
>>> X                  = np.array([[0], [1], [2], [3], [4], [5], [6], [7], [8], [9]])
>>> y_true             = np.array([ 1 ,  1 ,  1 ,  1 ,  0,   0 ,  0 ,  0 ,  0 ,  0 ])
>>> y_pred             = np.array([ 1 ,  1 ,  1 ,  1 ,  0,   0 ,  0 ,  0 ,  0 ,  0 ])
>>> sensitive_features = np.array(["a", "b", "a", "a", "b", "a", "b", "b", "a", "b"])
>>> selection_rate_summary = MetricFrame(metrics=selection_rate,
...                                      y_true=y_true,
...                                      y_pred=y_pred,
...                                      sensitive_features=pd.Series(sensitive_features, name="SF 0"))
>>> selection_rate_summary.overall.item()
    0.4
>>> selection_rate_summary.by_group
SF 0
a    0.6
b    0.2
Name: selection_rate, dtype: float64
>>> dp.load_data(X, y_true, sensitive_features=sensitive_features)
>>> dp.gamma(lambda X: y_pred)
sign  event  group_id
+     all    a           0.2
             b          -0.2
-     all    a          -0.2
             b           0.2
dtype: float64

The ratio constraints for the demographic parity with ratio_bound \(r\) (and ratio_bound_slack=0) take form

\[r \leq \dfrac{\P[h(X) = 1 \given A = a]}{\P[h(X) = 1]} \leq \dfrac{1}{r} \quad \forall a\]

Revisiting the same example as above we get

>>> dp = DemographicParity(ratio_bound=0.9, ratio_bound_slack=0.01)
>>> dp.load_data(X, y_pred, sensitive_features=sensitive_features)
>>> dp.gamma(lambda X: y_pred)
sign  event  group_id
+     all    a           0.14
             b          -0.22
-     all    a          -0.24
             b           0.16
dtype: float64

Following the expressions for the left-hand sides of the constraints, we obtain

\[\begin{split}r \cdot \text{utility}_{a,*} - \text{utility}_* = 0.9 \times 0.6 - 0.4 = 0.14 \\ r \cdot \text{utility}_{b,*} - \text{utility}_* = 0.9 \times 0.2 - 0.4 = -0.22 \\ - \text{utility}_{a,*} + r \cdot \text{utility}_* = - 0.6 + 0.9 \times 0.4 = -0.24 \\ - \text{utility}_{b,*} + r \cdot \text{utility}_* = - 0.2 + 0.9 \times 0.4 = 0.16 \\\end{split}\]

True Positive Rate Parity and False Positive Rate Parity

A binary classifier \(h(X)\) satisfies true positive rate parity if

\[\P[h(X) = 1 \given A = a, Y = 1] = \P[h(X) = 1 \given Y = 1] \quad \forall a\]

and false positive rate parity if

\[\P[h(X) = 1 \given A = a, Y = 0] = \P[h(X) = 1 \given Y = 0] \quad \forall a\]

In first case, we only have one event \(Y=1\) and ignore the samples with \(Y=0\), and in the second case vice versa. Refer to Equalized Odds for the fairness constraint type that simultaneously enforce both true positive rate parity and false positive rate parity by considering both events \(Y=0\) and \(Y=1\).

In practice this can be used in a difference-based relaxation as follows:

>>> from fairlearn.reductions import TruePositiveRateParity
>>> from fairlearn.metrics import true_positive_rate
>>> import numpy as np
>>> tprp = TruePositiveRateParity(difference_bound=0.01)
>>> X                  = np.array([[0], [1], [2], [3], [4], [5], [6], [7], [8], [9]])
>>> y_true             = np.array([ 1 ,  1 ,  1 ,  1 ,  1,   1 ,  1 ,  0 ,  0 ,  0 ])
>>> y_pred             = np.array([ 1 ,  1 ,  1 ,  1 ,  0,   0 ,  0 ,  1 ,  0 ,  0 ])
>>> sensitive_features = np.array(["a", "b", "a", "a", "b", "a", "b", "b", "a", "b"])
>>> tpr_summary = MetricFrame(metrics=true_positive_rate,
...                           y_true=y_true,
...                           y_pred=y_pred,
...                           sensitive_features=sensitive_features)
>>> tpr_summary.overall.item()
0.5714285714285714
>>> tpr_summary.by_group
sensitive_feature_0
a    0.75...
b    0.33...
Name: true_positive_rate, dtype: float64
>>> tprp.load_data(X, y_true, sensitive_features=sensitive_features)
>>> tprp.gamma(lambda X: y_pred)
sign  event    group_id
+     label=1  a           0.1785...
               b          -0.2380...
-     label=1  a          -0.1785...
               b           0.2380...
dtype: float64

Note

When providing TruePositiveRateParity or FalsePositiveRateParity to mitigation algorithms, only use the constructor. The mitigation algorithm itself then invokes load_data. The example uses load_data to illustrate how TruePositiveRateParity instantiates inequalities from Fairness constraints for binary classification.

Alternatively, a ratio-based relaxation is also available:

>>> tprp = TruePositiveRateParity(ratio_bound=0.9, ratio_bound_slack=0.01)
>>> tprp.load_data(X, y_true, sensitive_features=sensitive_features)
>>> tprp.gamma(lambda X: y_pred)
sign  event    group_id
+     label=1  a           0.1035...
               b          -0.2714...
-     label=1  a          -0.2357...
               b           0.1809...
dtype: float64

Equalized Odds

A binary classifier \(h(X)\) satisfies equalized odds if it satisfies both true positive rate parity and false positive rate parity, i.e.,

\[\P[h(X) = 1 \given A = a, Y = y] = \P[h(X) = 1 \given Y = y] \quad \forall a, y\]

The constraints represent the union of constraints for true positive rate and false positive rate.

>>> from fairlearn.reductions import EqualizedOdds
>>> eo = EqualizedOdds(difference_bound=0.01)
>>> eo.load_data(X, y_true, sensitive_features=sensitive_features)
>>> eo.gamma(lambda X: y_pred)
sign  event    group_id
+     label=0  a          -0.3333...
               b           0.1666...
      label=1  a           0.1785...
               b          -0.2380...
-     label=0  a           0.3333...
               b          -0.1666...
      label=1  a          -0.1785...
               b           0.2380...
dtype: float64

Error Rate Parity

The error rate parity requires that the error rates should be the same across all groups. For a classifier \(h(X)\) this means that

\[\P[h(X) \ne Y \given A = a] = \P[h(X) \ne Y] \quad \forall a\]

In this case, the utility is equal to 1 if \(h(X)\ne Y\) and equal to 0 if \(h(X)=Y\), and so large value of utility here actually correspond to poor outcomes. The difference-based relaxation specifies that the error rate of any given group should not deviate from the overall error rate by more than the value of difference_bound.

>>> from fairlearn.reductions import ErrorRateParity
>>> from sklearn.metrics import accuracy_score
>>> accuracy_summary = MetricFrame(metrics=accuracy_score,
...                                y_true=y_true,
...                                y_pred=y_pred,
...                                sensitive_features=sensitive_features)
>>> accuracy_summary.overall.item()
0.6
>>> accuracy_summary.by_group
sensitive_feature_0
a    0.8
b    0.4
Name: accuracy_score, dtype: float64
>>> erp = ErrorRateParity(difference_bound=0.01)
>>> erp.load_data(X, y_true, sensitive_features=sensitive_features)
>>> erp.gamma(lambda X: y_pred)
sign  event  group_id
+     all    a          -0.2
             b           0.2
-     all    a           0.2
             b          -0.2
dtype: float64

Note

When providing ErrorRateParity to mitigation algorithms, only use the constructor. The mitigation algorithm itself then invokes load_data. The example uses load_data to illustrate how ErrorRateParity instantiates inequalities from Fairness constraints for binary classification.

Alternatively, error rate parity can be relaxed via ratio constraints as

\[r \leq \dfrac{\P[h(X) \ne Y \given A = a]}{\P[h(X) \ne Y]} \leq \dfrac{1}{r} \quad \forall a\]

with a ratio_bound \(r\). The usage is identical with other constraints:

>>> from fairlearn.reductions import ErrorRateParity
>>> erp = ErrorRateParity(ratio_bound=0.9, ratio_bound_slack=0.01)
>>> erp.load_data(X, y_true, sensitive_features=sensitive_features)
>>> erp.gamma(lambda X: y_pred)
sign  event  group_id
+     all    a          -0.22
             b           0.14
-     all    a           0.16
             b          -0.24
dtype: float64

Control features

The above examples of Moment (Demographic Parity, True and False Positive Rate Parity, Equalized Odds and Error Rate Parity) all support the concept of control features when applying their fairness constraints. A control feature stratifies the dataset, and applies the fairness constraint within each stratum, but not between strata. One case this might be useful is a loan scenario, where we might want to apply a mitigation for the sensitive features while controlling for some other feature(s). This should be done with caution, since the control features may have a correlation with the sensitive features due to historical biases. In the loan scenario, we might choose to control for income level, on the grounds that higher income individuals are more likely to be able to repay a loan. However, due to historical bias, there is a correlation between the income level of individuals and their race and gender.

Control features modify the above equations. Consider a control feature value, drawn from a set of valid values (that is, \(c \in \mathcal{C}\)). The equation given above for Demographic Parity will become:

\[P[h(X) = 1 | A = a, C = c] = P[h(X) = 1 | C = c] \; \forall a, c\]

The other constraints acquire similar modifications.

Fairness constraints for multiclass classification

Reductions approaches do not support multiclass classification yet at this point. If this is an important scenario for you please let us know!

Fairness constraints for regression

The performance objective in the regression scenario is to minimize the loss of our regressor \(h\). The loss can be expressed as SquareLoss or AbsoluteLoss. Both take constructor arguments min_val and max_val that define the value range within which the loss is evaluated. Values outside of the value range get clipped.

>>> from fairlearn.reductions import SquareLoss, AbsoluteLoss, ZeroOneLoss
>>> y_true = [0,   0.3, 1,   0.9]
>>> y_pred = [0.1, 0.2, 0.9, 1.3]
>>> SquareLoss(0, 2).eval(y_true, y_pred)
array([0.01, 0.01, 0.01, 0.16])
>>> # clipping at 1 reduces the error for the fourth entry
>>> SquareLoss(0, 1).eval(y_true, y_pred)
array([0.01, 0.01, 0.01, 0.01])
>>> AbsoluteLoss(0, 2).eval(y_true, y_pred)
array([0.1, 0.1, 0.1, 0.4])
>>> AbsoluteLoss(0, 1).eval(y_true, y_pred)
array([0.1, 0.1, 0.1, 0.1])
>>> # ZeroOneLoss is identical to AbsoluteLoss(0, 1)
>>> ZeroOneLoss().eval(y_true, y_pred)
array([0.1, 0.1, 0.1, 0.1])

When using Fairlearn’s reduction techniques for regression it’s required to specify the type of loss by passing the corresponding loss object when instantiating the object that represents our fairness constraint. The only supported type of constraint at this point is BoundedGroupLoss.

Bounded Group Loss

Bounded group loss requires the loss of each group to be below a user-specified amount \(\zeta\). If \(\zeta\) is chosen reasonably small the losses of all groups are very similar. Formally, a predictor \(h\) satisfies bounded group loss at level \(\zeta\) under a distribution over \((X, A, Y)\) if

\[\E[loss(Y, h(X)) \given A=a] \leq \zeta \quad \forall a\]

In the example below we use BoundedGroupLoss with ZeroOneLoss on two groups "a" and "b". Group "a" has an average loss of \(0.05\), while group "b"’s average loss is \(0.5\).

>>> from fairlearn.reductions import BoundedGroupLoss, ZeroOneLoss
>>> from sklearn.metrics import mean_absolute_error
>>> bgl = BoundedGroupLoss(ZeroOneLoss(), upper_bound=0.1)
>>> X                  = np.array([[0], [1], [2], [3]])
>>> y_true             = np.array([0.3, 0.5, 0.1, 1.0])
>>> y_pred             = np.array([0.3, 0.6, 0.6, 0.5])
>>> sensitive_features = np.array(["a", "a", "b", "b"])
>>> mae_frame = MetricFrame(metrics=mean_absolute_error,
...                         y_true=y_true,
...                         y_pred=y_pred,
...                         sensitive_features=pd.Series(sensitive_features, name="SF 0"))
>>> mae_frame.overall.item()
0.275
>>> mae_frame.by_group
SF 0
a    0.05
b    0.50
Name: mean_absolute_error, dtype: float64
>>> bgl.load_data(X, y_true, sensitive_features=sensitive_features)
>>> bgl.gamma(lambda X: y_pred)
group_id
a    0.05
b    0.50
Name: loss, dtype: float64

Note

In the example above the BoundedGroupLoss object does not use the upper_bound argument. It is only used by reductions techniques during the unfairness mitigation. As a result the constraint violation detected by gamma is identical to the mean absolute error.

References

[1]

Alekh Agarwal, Alina Beygelzimer, Miroslav Dudík, John Langford, and Hanna M. Wallach. A reductions approach to fair classification. In ICML, volume 80 of Proceedings of Machine Learning Research, 60–69. PMLR, 2018. URL: http://proceedings.mlr.press/v80/agarwal18a.html.

[2]

Alekh Agarwal, Miroslav Dudík, and Zhiwei Steven Wu. Fair regression: quantitative definitions and reduction-based algorithms. In ICML, volume 97 of Proceedings of Machine Learning Research, 120–129. PMLR, 2019. URL: http://proceedings.mlr.press/v97/agarwal19d.html.