Fair Mixup: Fairness via Interpolation

1 INTRODUCTION . Fairness has increasingly received attention in machine learning , with the aim of mitigating unjustified bias in learned models . Various statistical metrics were proposed to measure the disparities of model outputs and performance when conditioned on sensitive attributes such as gender or race . Equipped with these metrics , one can formulate constrained optimization problems to impose fairness as a constraint . Nevertheless , these constraints do not necessarily generalize since they are data-dependent , i.e they are estimated from finite samples . In particular , models that minimize the disparities on training sets do not necessarily achieve the same fairness metric on testing sets ( Cotter et al. , 2019 ) . Conventionally , regularization is required to improve the generalization ability of a model ( Zhang et al. , 2016 ) . On one hand , explicit regularization such as weight decay and dropout constrain the model capacity . On the other hand , implicit regularization such as data augmentation enlarge the support of the training distribution via prior knowledge ( Hernández-Garcı́a & König , 2018 ) . In this work , we propose a data augmentation strategy for optimizing group fairness constraints such as demographic parity ( DP ) and equalized odds ( EO ) ( Barocas et al. , 2019 ) . Given two sensitive groups such as male and female , instead of directly restricting the disparity , we propose to regularize the model on interpolated distributions between them . Those augmented distributions form a path connecting the two sensitive groups . Figure 1 provides an illustrative example of the idea . The path simulates how the distribution transitions from one group to another via interpolation . Ideally , if the model is invariant to the sensitive attribute , the expected prediction of the model along the path should have a smooth behavior . Therefore , we propose a regularization that favors smooth transitions along the path , which provides a stronger prior on the model class . We adopt mixup ( Zhang et al. , 2018b ) , a powerful data augmentation strategy , to construct the interpolated samples . Owing to mixup ’ s simple form , the smoothness regularization we introduce has a closed form expression that can be easily optimized . One disadvantage of mixup is that the interpolated samples might not lie on the natural data manifold . Verma et al . ( 2019 ) propose Manifold Mixup , which generate the mixup samples in a latent space . Previous works ( Bojanowski et al. , 2018 ; Berthelot et al. , 2018 ) have shown that interpolations between a pair of latent features correspond to semantically meaningful , smooth interpolation in the input space . By constructing the path in the latent space , we can better capture the semantic changes while traveling between the sensitive groups and hence result in a better fairness regularizer that we coin fair mixup . Empirically , fair ∗Work done during an internship at IBM Research AI mixup improves the generalizability for both DP and EO on tabular , computer- vision , and natural language benchmarks . Theoretically , we prove for a particular case that fair mixup corresponds to a Mahalanobis metric in the feature space in which we perform the classification . This metric ensures group fairness of the model , and involves the Jacobian of the feature map as we travel along the path . In short , this work makes the following contributions : • We develop fair mixup , a data augmentation strategy that improves the generalization of group fairness metrics ; • We provide a theoretical analysis to deepen our understanding of the proposed method ; • We evaluate our approach via experiments on tabular , vision , and language benchmarks ; 2 RELATED WORK . Machine Learning Fairness To mitigate unjustified bias in machine learning systems , various fairness definitions have been proposed . The definitions can usually be classified into individual fairness or group fairness . A system that is individually fair will treat similar users similarly , where the similarity between individuals can be obtained via prior knowledge or metric learning ( Dwork et al. , 2012 ; Yurochkin et al. , 2019 ) . Group fairness metrics measure the statistical parity between subgroups defined by the sensitive attributes such as gender or race ( Zemel et al. , 2013 ; Louizos et al. , 2015 ; Hardt et al. , 2016 ) . While fairness can be achieved via pre- or post-processing , optimizing fair metrics at training time can lead to the highest utility ( Barocas et al. , 2019 ) . For instance , Woodworth et al . ( 2017 ) impose independence via regularizing the covariance between predictions and sensitive attributes . Zafar et al . ( 2017 ) regularize decision boundaries of convex margin-based classifier to minimize the disparaty between groups . Zhang et al . ( 2018a ) mitigate the bias via minimizing an adversary ’ s ability to predict sensitive attributes from predictions . Nevertheless , these constraints are data-dependent , even though the constraints are satisfied during training , the model may behave differently at evaluation time . Agarwal et al . ( 2018 ) analyze the generalization error of fair classifiers obtained via two-player games . To improve the generalizability , Cotter et al . ( 2019 ) inherit the two-player setting while training each player on two separated datasets . In spite of the analytical solutions and theoretical guarantees , game-theoretic approaches could be hard to scale for complex model classes . In contrast , our proposed fair mixup , is a general data augmentation strategy for optimizing the fairness constraints , which is easily compatible with any dataset modality or model class . Data Augmentation and Regularization Data augmentation expands the training data with examples generated via prior knowledge , which can be seen as an implicit regularization ( Zhang et al. , 2016 ; Hernández-Garcı́a & König , 2018 ) where the prior is specified as virtual examples . Zhang et al . ( 2018b ) proposes mixup , which generate augmented samples via convex combinations of pairs of examples . In particular , given two examples zi , zj ∈ Rd where z could include both input and label , mixup constructs virtual samples as tzi + ( 1 − t ) zj for t ∈ [ 0 , 1 ] . State-of-the-art results are obtained via training on mixup samples in different modalities . Verma et al . ( 2019 ) introduces manifold mixup and shows that performing mixup in a latent space further improves the generalization . While previous works focus on general learning scenarios , we show that regularizing models on mixup samples can lead to group fairness and improve generalization . 3 GROUP FAIRNESS . Without loss of generality , we consider the standard fair binary classification setup where we obtain inputs X ∈ X ⊂ Rd , labels Y ∈ Y = { 0 , 1 } , sensitive attribute A ∈ { 0 , 1 } , and prediction score Ŷ ∈ [ 0 , 1 ] from model f : Rd → [ 0 , 1 ] . We will focus on demographic parity ( DP ) and equalized odds ( EO ) in this work , while our approach also encompasses other fairness metrics ( detailed discussion in section 5 ) . DP requires the predictions Ŷ to be independent of the sensitive attribute A , that is , P ( Ŷ |A = 0 ) = P ( Ŷ |A = 1 ) . However , DP ignores the possible correlations between Y and A and could rule out the perfect predictor if Y 6⊥⊥ A. EO overcomes the limit of DP by conditioning on the label Y . In particular , EO requires Ŷ and A to be conditionally independent with respect to Y , that is , P ( Ŷ |A = 1 , Y = y ) = P ( Ŷ |A = 0 , Y = y ) for y ∈ { 0 , 1 } . Given the difficulty of optimizing the independency constraints , Madras et al . ( 2018 ) propose the following relaxed metrics : ∆DP ( f ) = |Ex∼P0f ( x ) − Ex∼P1f ( x ) | ∆EO ( f ) = ∑ y∈ { 0,1 } ∣∣∣Ex∼Py0 f ( x ) − Ex∼Py1 f ( x ) ∣∣∣ where we define Pa = P ( ·|A = a ) and P ya = P ( ·|A = a , Y = y ) , a , y ∈ { 0 , 1 } . We denote the joint distribution of X and Y by P . Similar metrics have also been used in Agarwal et al . ( 2018 ) , Wei et al . ( 2019 ) , and Taskesen et al . ( 2020 ) . One can formulate a penalized optimization problem to regularize the fairness measurement , for instance , ( Gap Regularization ) : min f E ( x , y ) ∼P [ ` ( f ( x ) , y ) ] + λ∆DP ( f ) , ( 1 ) where ` is the classification loss . In spite of its simplicity , our experiments show that small training values of ∆DP ( f ) do not necessarily generalize well at evaluation time ( See section 6 ) . To improve the generalizability , we introduce a data augmentation strategy via a dynamic form of group fairness metrics . 4 DYNAMIC FORMULATION OF FAIRNESS : PATHS BETWEEN GROUPS . For simplicity , we will first consider ∆DP as the fairness metric , and extend our development to ∆EO in section 5 . ∆DP provides a static measurement by quantifying the expected difference at P0 and P1 . In contrast , one can consider a dynamic metric that measures the change of Ŷ while transitioning gradually from P0 to P1 . To convert from the static to the dynamic formulations , we start with a simple Lemma that bridges two groups with an interpolator T ( x0 , x1 , t ) , which generates interpolated samples between x0 and x1 based on step t. Lemma 1 . Let T : X 2 × [ 0 , 1 ] → X be a function continuously differentiable w.r.t . t such that T ( x0 , x1 , 0 ) = x0 and T ( x0 , x1 , 1 ) = x1 . For any differentiable function f , we have ∆DP ( f ) = ∣∣∣∣∣∣∣ ∫ 1 0 d dt ∫ f ( T ( x0 , x1 , t ) ︸ ︷︷ ︸ interpolation ) dP0 ( x0 ) dP1 ( x1 ) dt ∣∣∣∣∣∣∣ = : ∣∣∣∣∫ 1 0 d dt µf ( t ) dt ∣∣∣∣ , ( 2 ) where we define µf ( t ) = Ex0∼P0 , x1∼P1f ( T ( x0 , x1 , t ) ) , the expected output of f with respect to T ( x0 , x1 , t ) . Figure 2 provides an illustrative example of the idea . Lemma 1 relaxes the binary sensitive attribute into a continuous variable t ∈ [ 0 , 1 ] , where µf captures the behavior of f while traveling from group 0 to group 1 along the path constructed with the interpolator T . In particular , given two examples x0 and x1 drawn from each group , T generates interpolated samples that change smoothly with respect to t. For instance , given two racial backgrounds in the dataset , µf simulates how the prediction of f changes while the data of one group smoothly transforms to another . We can then detect whether there are “ unfair ” changes in µf along the path . The dynamic formulation allows us to measure the sensitivity of f with respect to a relaxed continuous sensitive attribute t via the derivative ddtµf ( t ) . Ideally , if f is invariant to the sensitive attribute , ddtµf ( t ) should be small along the path from t = 0 to 1 . Importantly , a small ∆DP does not imply | ddtµf ( t ) | is small for t ∈ [ 0 , 1 ] since the derivative could fluctuate as it can be seen in Figure 2 .

Summary: This paper proposed a data augmentation method for training a classifier which is intended to have predictive parity between two identified groups. It is based on the “mixup” idea – samples from the two groups are interpolated between, and the smoothness of this path is encouraged. The authors recommend doing the interpolation in latent space. An optimal solution in a constrained setting is derived, and experiments show the empirical success of the model at this task on 3 datasets.