Original: Learning Fair Representations through Uniformly Distributed Sensitive Attributes
Revised: Learning Fair Representations through Uniformly Distributed Sensitive Attributes

Original: Anonymous Author
Revised: Anonymous Author

Original: Anonymous Institution
Revised: Anonymous Institution

Original: Abstract—Machine Learning  models trained on biased data can reproduce and even amplify these biases.
Revised: Abstract—Machine Learning (ML) models trained on biased data can reproduce and even amplify these biases.

Original: These models are deployed to make decisions that can affect people’s lives,thus ensuring their fairness is of critical importance.
Revised: Since such models are deployed to make decisions that can affect people’s lives, ensuring their fairness is critical.

Original: To mitigate unfairness, one approach  is to map the input data into a less- biased new space by means of learning fair representations.
Revised: On approach to mitigate possible unfairness of ML models is to map the input data into a less-biased new space by means of training the model on fair representations.

Original: Several methods based on adversarial learning have been proposed to learn fair representation by fooling an adversary in predicting the sensitive attribute (gender, race, etc.).
Revised: Several methods based on adversarial learning have been proposed to learn fair representation by fooling an adversary in predicting the sensitive attribute (e.g., gender or race).

Original: However, adversarial-based learning can be too difficult to optimize in practice, besides, it penalizes the utility of the representation.
Revised: However, adversarial-based learning can be too difficult to optimize in practice; besides, it penalizes the utility of the representation.

Original: In this research effort we learn bias-free representations from the input data by inducing a uniform distribution over the sensitive attributes in the latent space.
Revised: Hence, in this research effort we train bias-free representations from the input data by inducing a uniform distribution over the sensitive attributes in the latent space.

Original: In particular, we propose a probabilistic framework that learns these representations by enforcing the correct reconstruction of the original data, plus the prediction of the attributes of interest while eliminating the possibility of predicting the sensitive ones.
Revised: In particular, we propose a probabilistic framework that learns these representations by enforcing the correct reconstruction of the original data, plus the prediction of the attributes of interest while eliminating the possibility of predicting the sensitive ones.

Original: Our method leverages the inability of Deep Neural Networks (DNNs) to generalize when trained on a noisy label space to regularize the latent space by using a network head that predicts a noisy version of the sensitive attributes in order to increase the uncertainty of their predictions at test time.
Revised: Our method leverages the inability of Deep Neural Networks (DNNs) to generalize when trained on a noisy label space to regularize the latent space. We use a network head that predicts a noisy version of the sensitive attributes in order to increase the uncertainty of their predictions at test time.

Original: Our experiments in two datasets demonstrates that the proposed model  improves fairness while maintaining the prediction accuracy of downstream tasks.
Revised: Our experiments in two datasets demonstrates that the proposed model significantly improves fairness while maintaining the prediction accuracy of downstream tasks.

Original: I. Introduction
Revised: I. Introduction

Original: The growing deployment of Machine Learning (ML) systems in business and government has shown that these systems can learn, perpetuate and even amplify biases that we are trying to fight in the real world (Mehrabi et al., 2019).
Revised: The growing deployment of Machine Learning (ML) systems in business and government has shown that these systems can learn, perpetuate, and even amplify biases that that society is combating in the real world. (Mehrabi et al., 2019).

Original: The decisions provided by ML models can have a profound effect on the course of our lives: getting a job, getting out of jail, admission to a particular school, etc.
Revised: The decisions provided by ML models can have a profound effect on the course of people’s lives e g., hiring, detention, college admissions

Original: It is, therefore, crucial to ensure that these decisions are not flawed or discriminatory.
Revised: Therefore, it is crucial to ensure that such decisions are sound and fair.

Original: Fairness in ML has received a lot attention during the past years.
Revised: Fairness in ML has received a lot attention during the past years.

Original: Many fairness definitions  have been proposed to quantify the unfairness ofa ML models.
Revised: A number of definitions of fairness have been proposed to quantify the unfairness of ML models.

Original: These fairness notions can be categorized into three main categories.
Revised: These fairness definitions can be categorized into three main categories.

Original: Group fairness (Hardt et al., 2016; Verma and Rubin, 2018; Dwork et al., 2012) evaluates the model’s performance against different groups, based on a given sensitive attribute (gender, race, age, etc.) it requires the groups tobe treated equally by the model.
Revised: Group fairness (Hardt et al., 2016; Verma and Rubin, 2018; Dwork et al., 2012) evaluates the model’s performance within different groups; based on a given sensitive attribute (e.g., gender, race, age), it requires the model to treated groups equally.

Original: Individual fairness (Dwork et al., 2012) requires that similar individuals with respect to given task should receive a similar outcome.
Revised: Individual fairness (Dwork et al., 2012) requires that similar individuals with respect to given task should receive a similar outcome.

Original: And the counterfactual fairness (Kusner et al., 2017) requires that the decisions provided by the model be the same if the sensitive attribute were changed (e.g., in the case of a loan approval system, what the outcome would have been if a female applicant had been born  male).
Revised: counterfactual fairness (Kusner et al., 2017) requires that the decisions provided by the model remain the same if the sensitive attribute were changed (e.g., in the case of a loan approval system, what the outcome would have been if a female applicant had been born a male).

Original: This paper focuses on achieving groupbased fairness notion such as statistical parity, equalize odds and equal opportunity, with an emphasis on statistical parity.
Revised: Our study focuses on achieving group-based fairness notion such as statistical parity, equalize odds, and equal opportunity, with an emphasis on statistical parity.

Original: One way to mitigate unfairness consists on mapping the input data into a less-biased new space; by means of learning fair representations.
Revised: One way to mitigate unfairness is to map the input data into a less-biased new space by means of learning fair representations.

Original: The new representation of the data is, therefore, likely to produce more fair outcomes when used for any downstream tasks such as classification.
Revised: The new representation of the data is therefore likely to produce more fair outcomes when used for any downstream tasks such as classification.

Original: Several approaches based on adversarial learning have been proposed to learn a fair representation (Edwards and
Revised: Several approaches based on adversarial learning have been proposed to learn a fair representation (Edwards and

Original: Storkey, 2015; Madras et al., 2018; Kenfack et al., 2021;
Revised: Storkey, 2015; Madras et al., 2018; Kenfack et al., 2021;

Original: Creager et al., 2019) by removing all dependencies to the sensitive attributes from the data while preserving as much of the other informationof that individual as possible.
Revised: Creager et al., 2019) by removing all dependencies on the sensitive attributes from the data while preserving as much of the other information as possible.

Original: In the adversarial training, an adversary is trained to predict the sensitive attribute from the latent space yielded by a generator, while the generator is trained to fool the adversary in predicting the sensitive attribute.
Revised: In the adversarial training, an adversary is trained to predict the sensitive attribute from the latent space yielded by a generator, while the generator is trained to fool the adversary in predicting the sensitive attribute.

Original: However, such an adversarial setup requires that any adversary cannot predict the protected attribute, and can be too difficult to optimize in practice (Moyer et al., 2018; Edwards and Storkey, 2015; Feng et al., 2019).
Revised: However, such an adversarial setup requires that any adversary cannot predict the protected attribute, and can be too difficult to optimize in practice (Moyer et al., 2018; Edwards and Storkey, 2015; Feng et al., 2019).

Original: Moreover, these approaches always lead to a degradation of the predictor accuracy, in particular, Moyer et al. showed that adversarial training is unnecessary and sometimes counter-productive.
Revised: Moreover, these approaches always lead to a degradation of the predictor accuracy, in particular, Moyer et al. showed that adversarial training is unnecessary and sometimes counter-productive.

Original: In addition, reducing the tradeoff between fairness and accuracy is an active research question within the fair ML community, i.e., to provide models with high accuracy with as little bias as possible.
Revised: In addition, reducing the tradeoff between fairness and accuracy is an active research question within the fair ML community, which essentially implies the task to provide models with high accuracy with as little bias as possible.

Original: This work addresses these issues by proposing an alternative and straightforward approach for learning fair representations without an adversaryand while maintaining high utility on downstream tasks.
Revised: Our study addresses these issues by proposing an alternative and straightforward approach for learning fair representations without an adversary while maintaining high utility on downstream tasks.

Original: In this paper, we introduce a new regularizing objective function that yields a representation that maintains the utility of predictions and mitigates bias comparably better than thecurrent state-of-the-art approaches.
Revised: We introduce a new regularizing objective function that yields a representation that maintains the utility of predictions and mitigates bias comparably better than the state-of-the-art approaches.

Original: In essence, our proposed objective is not just to make the prediction of the sensitive attribute impossible to predict by an adversary, but instead to increase the uncertainty of predicting the sensitive attribute from the latent space, i.e., make it almost uniform (close to 0 . 5 ).
Revised: In essence, our proposed objective is not only to make the prediction of the sensitive attribute impossible to predict by an adversary, but instead to increase the uncertainty of predicting the sensitive attribute from the latent space, i.e., make it almost uniform (close to 0 . 5 ).

Original: To induce the uniformity over the sensitive attribute in the latent space, we use a neural network that takes the latent code as input, and predict a noisy version of the sensitive attribute, i.e., a defined percentage (e.g., 50%) of individuals are randomly switched between different groups considered.
Revised: To induce the uniformity over the sensitive attribute in the latent space, we use a neural network that takes the latent code as input and predict a noisy version of the sensitive attribute, i.e., a defined percentage (e.g., 50%) of individuals are randomly switched between different groups considered.

Original: The intuition here is that like in a counterfactual world, the target variable (here the sensitive attribute) of few samples is flipped, in a way to confuse the inherent characteristics of each individual, i.e., increase the uncertainty of predicting the sensitive attribute of individuals from the latent space and thus, making them unreliable.
Revised: The intuition here is that similarly to a counterfactual world, the target variable (here the sensitive attribute) of few samples is flipped in a way to confuse the inherent characteristics of each individual, i.e., increase the uncertainty of predicting the sensitive attribute of individuals from the latent space and thus make them unreliable.

Original: Model training on noisy labels has been shown to be a source of uncertainty for DNNs (Song et al., 2022; Bai et al., 2021).
Revised: Model training on noisy labels has been shown to be a source of uncertainty for DNNs (Song et al., 2022; Bai et al., 2021).

Original: Furthermore, neural networks can learn and fit data with noisy or random labels but fail to generalize at testing time (Chen et al., 2019), i.e., although the model achieves small training loss, the predictions on unseen data are almost random (unreliable).
Revised: Furthermore, neural networks can learn and fit data with noisy or random labels but fail to generalize at testing time (Chen et al., 2019), i.e., although the model achieves small training loss, the predictions on unseen data are almost random (unreliable).

Original: By leveraging this weakness of DNNs, our regularizer head trained on a noisy version of the sensitive attribute enforces the latent space tobe provide unreliable predictions (uniform) when used to predict the sensitive attributes.
Revised: By leveraging this weakness of DNNs, our regularizer head trained on a noisy version of the sensitive attribute enforces the latent space to provide unreliable predictions (uniform) when used to predict the sensitive attributes.

Original: We show in the experiments that this approach is effective in enforcing the independence between the learned representation and the sensitive attributes.
Revised: We experimentally showed that our approach is effective for enforcing the independence between the learned representation and the sensitive attributes.

Original: As such, the downstream classification tasks using our representation provided better fairness performance in terms on statistical parity along with a positive impact on equalized odds and equal opportunity.
Revised: As such, the downstream classification tasks using our representation provided better fairness performance in terms of statistical parity along with a positive impact on equalized odds and equal opportunity.

Original: We show that this objective is easy to train, provides comparable fairness performances and does not penalize the utility of the model.
Revised: Based on the obtained results, we claim this objective is easy to train, the approach yields higher fairness performance and does not penalize the utility of the model.

Original: II. Related Work
Revised: II. Related Work

Original: The objective of fair representation learning is to find a function to map the input data into a fair space, i.e., a space where the protected and non-protected attributes are indistinguishable.
Revised: The objective of fair representation learning is to find a function to map the input data into a fair space, i.e., a space where the protected and non-protected attributes are indistinguishable.

Original: Zemel et al.
Revised: Zemel et al.

Original: (2013) were the first to propose learning fair intermediate representations: similar to k -means, their method involves finding k prototypes in the same space as the input data.
Revised: (2013) were the first to propose learning fair intermediate representations: similar to k -means, their method involves finding k prototypes in the same space as the input data.

Original: Each sample is assigned to the closest prototype while adding a constraint in the optimization objective to satisfy fairness and classification performance.
Revised: Each sample is assigned to the closest prototype while adding a constraint in the optimization objective to satisfy fairness and classification performance.

Original: Louizos et al.
Revised: Louizos et al.

Original: (2016) proposed the Variational Fair Autoencoder, an adaptation of the variational autoencoder to learn a mapping function that enforces the independence of the sensitive attribute and the latent space.
Revised: (2016) proposed the Variational Fair Autoencoder, an adaptation of the variational autoencoder to learn a mapping function that enforces the independence of the sensitive attribute and the latent space.

Original: The authors treated the sensitive attribute as nuisance variable and factored it out in the latent space.
Revised: The authors treated the sensitive attribute as nuisance variable and factored it out in the latent space.

Original: As opposed to Louizos et al.
Revised: As opposed to Louizos et al.

Original: ’s (2016) work, our fair mapping function does not only focus on encouraging independence to the sensitive attribute, but also on maintaining the high utility of the representation on downstream tasks such as classification.
Revised: ’s (2016) work, our fair mapping function does not only focus on encouraging independence to the sensitive attribute, but also on maintaining the high utility of the representation on downstream tasks such as classification.

Original: More closely related to our work, Edwards and Storkey (2015)
Revised: More closely related to our work, Edwards and Storkey (2015)

Original: proposed an adversarial approach to enforce statistical parity, which involves three main components: the autoencoder that yields the latent space (generator); the adversary, trained to predict the sensitive attribute from the latent space; and a classifier, trained to maximize the utility of the latent representation with respect to the class label.
Revised: proposed an adversarial approach to enforce statistical parity, which involves three main components: the autoencoder that yields the latent space (generator); the adversary, trained to predict the sensitive attribute from the latent space; and a classifier, trained to maximize the utility of the latent representation with respect to the class label.

Original: The autoencoder and classifier try to fool the adversary in predicting the sensitive attribute from the latent space while the adversary tries to maximize its accuracy of prediction.
Revised: The autoencoder and classifier try to fool the adversary in predicting the sensitive attribute from the latent space while the adversary tries to maximize its accuracy of prediction.

Original: Madras et al.
Revised: Madras et al.

Original: (2018) extended the previous work and proposed new adversarial objectives that yield transferable fair representation and also considered other fairness notions like equal opportunity and equalized odds.
Revised: (2018) extended the previous work and proposed new adversarial objectives that yield transferable fair representation and also considered other fairness notions like equal opportunity and equalized odds.

Original: Kenfack et al.
Revised: Kenfack et al.

Original: (2021) showed that applying the adversary at multiple level of representations tightens the fairness bound of the learned representation.
Revised: (2021) showed that applying the adversary at multiple level of representations tightens the fairness bound of the learned representation.

Original: Feng et al.
Revised: Feng et al.

Original: (2019) used an adversary that minimizes the Wasserstein
Revised: (2019) used an adversary that minimizes the Wasserstein

Original: Distance between the distribution of the protected and non-protected groups.
Revised: Distance between the distribution of the protected and non-protected groups.

Original: (Moyer et al., 2018) learn fair representations by minimizing the mutual information between the latent space and sensitive attribute while maximizing the mutual information between the latent space and loss of the task at hand.
Revised: (Moyer et al., 2018) learn fair representations by minimizing the mutual information between the latent space and sensitive attribute while maximizing the mutual information between the latent space and loss of the task at hand.

Original: These previous works mainly focus on improving fairness and have shown a significant drop in the accuracy of predictions compared to the unconstrained models.
Revised: These previous works mainly focus on improving fairness and have shown a significant drop in the accuracy of predictions compared to the unconstrained models.

Original: Our goal is to reduce the fairness/accuracy tradeoff of the learned representation, i.e., maintain a high accuracy of prediction on downstream tasks while improving fairness.
Revised: Our goal is to reduce the fairness/accuracy tradeoff of the learned representation, i.e., maintain a high accuracy of prediction on downstream tasks while improving fairness.

Original: We propose a generalized probabilistic model for fair representation learning and a relaxation of the adversary objective that turns out to be easier to train and provides a better fairness/accuracy tradeoff than previous work.
Revised: We propose a generalized probabilistic model for fair representation learning and a relaxation of the adversary objective that turns out to be easier to train and provides a better fairness/accuracy tradeoff than previous work.

Original: As demonstrated by Dutta et al.
Revised: As demonstrated by Dutta et al.

Original: (2020), it is possible to find a data distribution where optimal fairness and accuracy are achieved simultaneously.
Revised: (2020), it is possible to find a data distribution where optimal fairness and accuracy are achieved simultaneously.

Original: III. Our Method
Revised: III. Our Method

Original: In order to learn fair representations, we propose a probabilistic hierarchical model that learns representations such that it is (ideally) impossible to use them for predicting the sensitive attributes.
Revised: In order to learn fair representations, we propose a probabilistic hierarchical model that learns representations such that it is (ideally) impossible to use them for predicting the sensitive attributes.

Original: Thus, achieving fairness w.r.t.
Revised: Thus, achieving fairness w.r.t.

Original: these attributes.
Revised: these attributes.

Original: First, we detail the probabilistic model for a single level of hierarchy.
Revised: First, we detail the probabilistic model for a single level of hierarchy.

Original: Then, we discuss how to expand this model to several levels within a hierarchical model.
Revised: Then, we discuss how to expand this model to several levels within a hierarchical model.

Original: A. Probabilistic Model
Revised: A. Probabilistic Model

Original: Let the user data be the triplet ( x, s, y ) where there is nonsensitive information x and sensitive one s .
Revised: Let the user data be the triplet ( x, s, y ) where there is nonsensitive information x and sensitive one s .

Original: Our task is to predict the dependent variable y .
Revised: Our task is to predict the dependent variable y .

Original: We introduce a latent variable z that models the user information while taking care of obfuscating the sensitive information s .
Revised: We introduce a latent variable z that models the user information while taking care of obfuscating the sensitive information s .

Original: We assume a generative model (see Fig. 1)
Revised: We assume a generative model (see Fig. 1)

Original: [Equation]
Revised: [Equation]

Original: and we are interested in maximizing its log-probability, i.e., max log p ( x, s, y ) .
Revised: and we are interested in maximizing its log-probability, i.e., max log p ( x, s, y ) .

Original: We assume the inference model to be q ( x, s, z ) =
Revised: We assume the inference model to be q ( x, s, z ) =

Original: q ( z | x, s ) q ( x, s ) (see Fig. 1).
Revised: q ( z | x, s ) q ( x, s ) (see Fig. 1).

Original: By using a z x y s
Revised: By using a z x y s

Original: [Figure]
Revised: [Figure]

Original: z
Revised: z

Original: [Figure]
Revised: [Figure]

Original: variational inference approach, we obtain the evidence lower bound of the log likelihood, that is, log p
Revised: variational inference approach, we obtain the evidence lower bound of the log likelihood, that is, log p

Original: [Equation]
Revised: [Equation]

Original: x, s, y
Revised: x, s, y

Original: [Equation]
Revised: [Equation]

Original: This model tells us that we need to maximize the likelihood of the class prediction, p ( y | z ) , as well as the sensitive data prediction, p ( s | z ) .
Revised: This model tells us that we need to maximize the likelihood of the class prediction, p ( y | z ) , as well as the sensitive data prediction, p ( s | z ) .

Original: Simultaneously, we need to guarantee the generation of the original data, p ( x | z ) .
Revised: Simultaneously, we need to guarantee the generation of the original data, p ( x | z ) .

Original: Moreover, we need to minimize the Kullback-Leibler divergence of the prior and our proposed encoder, E q ( z | x,s ) log q ( z | x, s ) /p
Revised: Moreover, we need to minimize the Kullback-Leibler divergence of the prior and our proposed encoder, E q ( z | x,s ) log q ( z | x, s ) /p

Original: ( z ) .
Revised: ( z ) .

Original: For our objective, we simplify this model.
Revised: For our objective, we simplify this model.

Original: First, we approximate the expectation over the latent q ( z | x ) by using sampling.
Revised: First, we approximate the expectation over the latent q ( z | x ) by using sampling.

Original: And in its simplest form, we use a single sample.
Revised: And in its simplest form, we use a single sample.

Original: Moreover, we do not work with the divergence, and use implicit distributions instead.
Revised: Moreover, we do not work with the divergence, and use implicit distributions instead.

Original: Our simplified optimization problem is
Revised: Our simplified optimization problem is

Original: [Equation]
Revised: [Equation]

Original: where the sampling z ∼ q ( z | x ) is replaced with a deterministic function that substitutes the distribution (implemented through a neural network).
Revised: where the sampling z ∼ q ( z | x ) is replaced with a deterministic function that substitutes the distribution (implemented through a neural network).

Original: However, this problem yields unfair representations since it maximizes the log-likelihood of the sensitive attribute as well.
Revised: However, this problem yields unfair representations since it maximizes the log-likelihood of the sensitive attribute as well.

Original: To solve this problem, we propose to make the sensitive distribution p ( s | x ) close to a uniform distribution instead.
Revised: To solve this problem, we propose to make the sensitive distribution p ( s | x ) close to a uniform distribution instead.

Original: The idea is that, instead of maximizing the prediction of the sensitive attribute, we will make the prediction unreliable (thus, close to a uniform distribution).
Revised: The idea is that, instead of maximizing the prediction of the sensitive attribute, we will make the prediction unreliable (thus, close to a uniform distribution).

Original: For example, if the sensitive attribute is binary, instead of maximizing or minimizing the likelihood (which will make it predictable), we try to maintain the likelihood close to 0 .
Revised: For example, if the sensitive attribute is binary, instead of maximizing or minimizing the likelihood (which will make it predictable), we try to maintain the likelihood close to 0 .

Original: For our implementation, we approximate the loglikelihood log p ( s | z ) with the cross-entropy
Revised: For our implementation, we approximate the loglikelihood log p ( s | z ) with the cross-entropy

Original: [Equation]
Revised: [Equation]

Original: where ˆ s is our prediction, and rather than the ground truth we use a stochastic process, s r , that randomly
Revised: where ˆ s is our prediction, and rather than the ground truth we use a stochastic process, s r , that randomly

Original: Algorithm
Revised: Algorithm

Original: Require: t ∈ [0 , 1] a flipping probability (default 0 . 5 ), N training epochs, K levels, and Θ = { θ i } K − 1 i =0 set of initialized network parameters.
Revised: Require: t ∈ [0 , 1] a flipping probability (default 0 . 5 ), N training epochs, K levels, and Θ = { θ i } K − 1 i =0 set of initialized network parameters.

Original: for n ← 1 ,N do X,Y,S ← random mini-batch from dataset
Revised: for n ← 1 ,N do X,Y,S ← random mini-batch from dataset

Original: Z 1 ← [ XS ] for i ← 1 ,K do (cid:46)
Revised: Z 1 ← [ XS ] for i ← 1 ,K do (cid:46)

Original: Iterate over levels Z i +1 ← Enc i
Revised: Iterate over levels Z i +1 ← Enc i

Original: ( Z i )
Revised: ( Z i )

Original: (cid:46)q ( z i +1 |
Revised: (cid:46)q ( z i +1 |

Original: z i )
Revised: z i )

Original: ˆ Z i ← Dec i ( Z i +1 )
Revised: ˆ Z i ← Dec i ( Z i +1 )

Original: (cid:46)p ( z i |
Revised: (cid:46)p ( z i |

Original: z i +1 ) end for ˆ Y ←
Revised: z i +1 ) end for ˆ Y ←

Original: Pred y
Revised: Pred y

Original: ( Z K ) (cid:46)p ( y | z K )
Revised: ( Z K ) (cid:46)p ( y | z K )

Original: ˆ S ←
Revised: ˆ S ←

Original: Pred s ( Z K ) (cid:46)p ( s | z K ) S r ← flip ( S,t ) (cid:46)
Revised: Pred s ( Z K ) (cid:46)p ( s | z K ) S r ← flip ( S,t ) (cid:46)

Original: Randomly flip S with prob.
Revised: Randomly flip S with prob.

Original: t L x ← MSE ( ˆ Z K ,Z K ) (cid:46) Reconstruction loss L y ← CE (ˆ Y ,Y ) (cid:46)
Revised: t L x ← MSE ( ˆ Z K ,Z K ) (cid:46) Reconstruction loss L y ← CE (ˆ Y ,Y ) (cid:46)

Original: Prediction loss L s ← CE (ˆ S,S r ) (cid:46)
Revised: Prediction loss L s ← CE (ˆ S,S r ) (cid:46)

Original: Sensitive attribute loss L = α L x + β L y + λ L s Θ ← Θ − α ∇ Θ L (cid:46) Update network param.
Revised: Sensitive attribute loss L = α L x + β L y + λ L s Θ ← Θ − α ∇ Θ L (cid:46) Update network param.

Original: end for flips between the possible classes for s .
Revised: end for flips between the possible classes for s .

Original: This last part is the key transformation that obfuscates the sensitive attribute.
Revised: This last part is the key transformation that obfuscates the sensitive attribute.

Original: The other losses approximate the other two log-likelihoods.
Revised: The other losses approximate the other two log-likelihoods.

Original: Namely, they are a mean-squared error minimization between the reconstruction ˆ x and the data x
Revised: Namely, they are a mean-squared error minimization between the reconstruction ˆ x and the data x

Original: [Equation]
Revised: [Equation]

Original: and the cross-entropy for the downstream task
Revised: and the cross-entropy for the downstream task

Original: [Equation]
Revised: [Equation]

Original: Thus, our optimization problem (3) becomes the minimization of
Revised: Thus, our optimization problem (3) becomes the minimization of

Original: [Equation]
Revised: [Equation]

Original: where α , β and λ are weights associated to each loss.
Revised: where α , β and λ are weights associated to each loss.

Original: B. Hierarchical Representations
Revised: B. Hierarchical Representations

Original: To produce even fairer representations, we explore a hierarchical structure of K levels of representations.
Revised: To produce even fairer representations, we explore a hierarchical structure of K levels of representations.

Original: The overall idea is to construct representations following the probabilistic model presented before (see Section III-A).
Revised: The overall idea is to construct representations following the probabilistic model presented before (see Section III-A).

Original: The i th level posses a latent variable z
Revised: The i th level posses a latent variable z

Original: i that follows
Revised: i that follows

Original: [Equation]
Revised: [Equation]

Original: as in our original joint (1), such that z i +1 is the latent representation for the z i variable.
Revised: as in our original joint (1), such that z i +1 is the latent representation for the z i variable.

Original: In this way, we can construct a set of K joints and learn them simultaneously by constructing a cascade of encoders and decoders (following Fig. 2).
Revised: In this way, we can construct a set of K joints and learn them simultaneously by constructing a cascade of encoders and decoders (following Fig. 2).

Original: We regularize them with our loss function (7) on the highest level K .
Revised: We regularize them with our loss function (7) on the highest level K .

Original: In our implementation, without loss of generality, the first level is the input data, i.e., z 1 =
Revised: In our implementation, without loss of generality, the first level is the input data, i.e., z 1 =

Original: [ x s ] .
Revised: [ x s ] .

Original: C. How the uniformity of the sensitive attribute implies fairness
Revised: C. How the uniformity of the sensitive attribute implies fairness

Original: Our method aims to enforce statistical parity on any downstream tasks that uses the representation learned by the encoder part of our method, i.e., a mapping of the input z
Revised: Our method aims to enforce statistical parity on any downstream tasks that uses the representation learned by the encoder part of our method, i.e., a mapping of the input z

Original: i q (cid:0) z i +1 |
Revised: i q (cid:0) z i +1 |

Original: z i (cid:1)
Revised: z i (cid:1)

Original: z i + p (cid:0) z i |
Revised: z i + p (cid:0) z i |

Original: z i +1 (cid:1)
Revised: z i +1 (cid:1)

Original: ˆ z i p (cid:0) y |
Revised: ˆ z i p (cid:0) y |

Original: z i +1 (cid:1)
Revised: z i +1 (cid:1)

Original: y p (cid:0) s |
Revised: y p (cid:0) s |

Original: z i +1 (cid:1) s q (cid:0)
Revised: z i +1 (cid:1) s q (cid:0)

Original: z i |
Revised: z i |

Original: z
Revised: z

Original: i − 1 (cid:1)
Revised: i − 1 (cid:1)

Original: z i −z i p (cid:0)
Revised: z i −z i p (cid:0)

Original: z
Revised: z

Original: i − 1 | z i (cid:1)
Revised: i − 1 | z i (cid:1)

Original: ˆ z i −q (cid:0)
Revised: ˆ z i −q (cid:0)

Original: z
Revised: z

Original: i +

Revised: i +

Original: |
Revised: |

Original: z i +1 (cid:1)
Revised: z i +1 (cid:1)

Original: z i +
Revised: z i +

Original: z
Revised: z

Original: i + p (cid:0) z
Revised: i + p (cid:0) z

Original: i +1 |
Revised: i +1 |

Original: z i +2 (cid:1)
Revised: z i +2 (cid:1)

Original: ˆ z i + ... ...
Revised: ˆ z i + ... ...

Original: [Figure]
Revised: [Figure]

Original: data into a fairer space.
Revised: data into a fairer space.

Original: We recall that statistical parity promotes the independence between the sensitive attribute ( S ) and the classifier outcome ( ˆ Y ), i.e., ˆ Y ⊥ S .
Revised: We recall that statistical parity promotes the independence between the sensitive attribute ( S ) and the classifier outcome ( ˆ Y ), i.e., ˆ Y ⊥ S .

Original: Given the input data X , the encoder part of our model maps it into the latent space Z .
Revised: Given the input data X , the encoder part of our model maps it into the latent space Z .

Original: We regularize the mapping by enforcing the uniformity of predicting S given Z , we do this by optimizing the regularizer network over a noisy version of S , in which the sensitive attribute of some individual is flipped.
Revised: We regularize the mapping by enforcing the uniformity of predicting S given Z , we do this by optimizing the regularizer network over a noisy version of S , in which the sensitive attribute of some individual is flipped.

Original: Our experiments shows in final representation, individuals are equally likely to be assigned a given group label.
Revised: Our experiments shows in final representation, individuals are equally likely to be assigned a given group label.

Original: That is, p ( S | Z ) ≈ 0 .
Revised: That is, p ( S | Z ) ≈ 0 .

Original: Moreover, Avigad (2013) showed how randomness in a sequence of binary numbers can induce uniform distribution.
Revised: Moreover, Avigad (2013) showed how randomness in a sequence of binary numbers can induce uniform distribution.

Original: When S is binary, enforcing the uniformity implies that p ( S = 1 | Z ) =
Revised: When S is binary, enforcing the uniformity implies that p ( S = 1 | Z ) =

Original: p ( S = 

Revised: p ( S =

Original: | Z )
Revised: | Z )

Original: ≈ 0 .
Revised: ≈ 0 .

Original: 5 , therefore S ⊥ Z .
Revised: 5 , therefore S ⊥ Z .

Original: Any classifier f trained from Z will also be independent to S , i.e.,
Revised: Any classifier f trained from Z will also be independent to S , i.e.,

Original: [Equation]
Revised: [Equation]

Original: , thus, achieving statistical parity.
Revised: , thus, achieving statistical parity.

Original: To enforce the prediction of sensitive attribute from learned representation to be uniform (unreliable), our method leverages the weakness of DNNs to generalize when trained on noisy label Song et al.
Revised: To enforce the prediction of sensitive attribute from learned representation to be uniform (unreliable), our method leverages the weakness of DNNs to generalize when trained on noisy label Song et al.

Original: In particular,
Revised: In particular,

Original: Zhang et al.
Revised: Zhang et al.

Original: (2021) show that deep neural networks can easily fit data with partially or completely noisy labels.
Revised: (2021) show that deep neural networks can easily fit data with partially or completely noisy labels.

Original: In the case of partially noisy labels, the network still receives a signal from clean samples but also learns noise.
Revised: In the case of partially noisy labels, the network still receives a signal from clean samples but also learns noise.

Original: As a result, the model does not generalize well on testing data, i.e., small training error but large error on the test set.
Revised: As a result, the model does not generalize well on testing data, i.e., small training error but large error on the test set.

Original: As we want the latent variable to not exhibit information about the sensitive attributes, we regularize the latent space with a network head that predicts a noisy version of the sensitive attributes.
Revised: As we want the latent variable to not exhibit information about the sensitive attributes, we regularize the latent space with a network head that predicts a noisy version of the sensitive attributes.

Original: As such, the latent will degenerate w.r.t.
Revised: As such, the latent will degenerate w.r.t.

Original: sensitive attribute and at test time the prediction of the sensitive attribute from the latent space will be unreliable, i.e., almost uniform.
Revised: sensitive attribute and at test time the prediction of the sensitive attribute from the latent space will be unreliable, i.e., almost uniform.

Original: Assuming that the conditional probability of the sensitive attribute distributes as a Bernoulli with probability u ( z ) , p ( s | z )
Revised: Assuming that the conditional probability of the sensitive attribute distributes as a Bernoulli with probability u ( z ) , p ( s | z )

Original: =
Revised: =

Original: B ( u ( z )) —hereinafter, we refer to u ( z ) as u for brevity.
Revised: B ( u ( z )) —hereinafter, we refer to u ( z ) as u for brevity.

Original: Given that distribution of flipping the ground truth is a Bernoulli distribution B ( r ) , we obtain that our flipping process (described in Section III-A) is a Bernoulli
Revised: Given that distribution of flipping the ground truth is a Bernoulli distribution B ( r ) , we obtain that our flipping process (described in Section III-A) is a Bernoulli

Original: B ( u − 2 ur + r ) (see the proof in the Appendix A).
Revised: B ( u − 2 ur + r ) (see the proof in the Appendix A).

Original: We show, by Theorem 1, that our cross-entropy loss (4) is an upper bound of the Kullback-Leibler (KL-) between the conditional p ( s | z ) and a uniform distribution (when
Revised: We show, by Theorem 1, that our cross-entropy loss (4) is an upper bound of the Kullback-Leibler (KL-) between the conditional p ( s | z ) and a uniform distribution (when

Original: B ( r = 0 . 5) ).
Revised: B ( r = 0 . 5) ).

Original: Thus, by minimizing this loss (4), we make the conditional distribution to be the uniform distribution.
Revised: Thus, by minimizing this loss (4), we make the conditional distribution to be the uniform distribution.

Original: Theorem 1.
Revised: Theorem 1.

Original: Let two independent Bernoulli distributions be B ( u ) and B ( r ) , and a third related to the parameters of the previous ones be B ( u − 2 ur + r ) .
Revised: Let two independent Bernoulli distributions be B ( u ) and B ( r ) , and a third related to the parameters of the previous ones be B ( u − 2 ur + r ) .

Original: The cross-entropy between B ( u ) and B ( u − 2 ur + r ) is an upper bound of the Kullback-Leibler divergence between the two original distributions, i.e.,
Revised: The cross-entropy between B ( u ) and B ( u − 2 ur + r ) is an upper bound of the Kullback-Leibler divergence between the two original distributions, i.e.,

Original: [Equation]
Revised: [Equation]

Original: Proof.
Revised: Proof.

Original: See Appendix B.
Revised: See Appendix B.

Original: D. Model Training
Revised: D. Model Training

Original: We present the pseudo-code of our learning process in
Revised: We present the pseudo-code of our learning process in

Original: Algorithm 1.
Revised: Algorithm 1.

Original: We generalize the algorithm for layer-wise training.
Revised: We generalize the algorithm for layer-wise training.

Original: We trained the model’s components simultaneously to minimize the loss function (7).
Revised: We trained the model’s components simultaneously to minimize the loss function (7).

Original: To enforce the uniformity of the sensitive attributes, a key step in the training process is the flipping of the sensitive attribute with a probability t .
Revised: To enforce the uniformity of the sensitive attributes, a key step in the training process is the flipping of the sensitive attribute with a probability t .

Original: In contrast to adversarial training, our training process is more stable and straightforward.
Revised: In contrast to adversarial training, our training process is more stable and straightforward.

Original: In Feng et al.’s (2019) work, the adversary is trained within each minibatch until convergence for one training step of the generator, while Edwards and Storkey’s (2015) alternates the gradient within each minibatch to update the generator or adversary.
Revised: In Feng et al.’s (2019) work, the adversary is trained within each minibatch until convergence for one training step of the generator, while Edwards and Storkey’s (2015) alternates the gradient within each minibatch to update the generator or adversary.

Original: This adversarial setting is hard to optimize in practice and does not provide convergence guarantees.
Revised: This adversarial setting is hard to optimize in practice and does not provide convergence guarantees.

Original: Our method is easier to train and straightforward.
Revised: Our method is easier to train and straightforward.

Original: It is more efficient in terms of accuracy for similar levels of fairness (cf.
Revised: It is more efficient in terms of accuracy for similar levels of fairness (cf.

Original: Section IV).
Revised: Section IV).

Original: G e r m a n d a t a s e t
Revised: G e r m a n d a t a s e t

Original: Original data
Revised: Original data

Original: Female
Revised: Female

Original: [Figure]
Revised: [Figure]

Original: LR
Revised: LR

Original: CR
Revised: CR

Original: LFR-AF
Revised: LFR-AF

Original: LFR
Revised: LFR

Original: LFR-U
Revised: LFR-U

Original: A du l t d a t a s e t
Revised: A du l t d a t a s e t

Original: IV. Experiments
Revised: IV. Experiments

Original: In the following, we present the experimental setup, empirical results, and the comparison of our approach, namely LFR-U, against other fair-representation learning methods.
Revised: In the following, we present the experimental setup, empirical results, and the comparison of our approach, namely LFR-U, against other fair-representation learning methods.

Original: We compare all the models in terms of accuracy, fairness, and their ability to learn representations that cannot predict the sensitive attribute 1 .
Revised: We compare all the models in terms of accuracy, fairness, and their ability to learn representations that cannot predict the sensitive attribute 1 .

Original: A. Experimental setup
Revised: A. Experimental setup

Original: a) Baseline methods: We consider as baseline a simple logistic regression (LR) model trained on the original data with no fairness constraints.
Revised: a) Baseline methods: We consider as baseline a simple logistic regression (LR) model trained on the original data with no fairness constraints.

Original: For the comparison, we consider Correlation Remover (CR) (Bird et al., 2020) which applies a linear transformation to the non-sensitive feature columns in order to remove their correlation with the sensitive feature columns.
Revised: For the comparison, we consider Correlation Remover (CR) (Bird et al., 2020) which applies a linear transformation to the non-sensitive feature columns in order to remove their correlation with the sensitive feature columns.

Original: We also consider adversarial based approaches such as work done by Madras et al.
Revised: We also consider adversarial based approaches such as work done by Madras et al.

Original: (2018); Edwards and Storkey (2015) who trained an adversary to distinguish between samples from the protected and unprotected groups (named LFR), here we consider statistical parity and since we are not interested in the transferability of the learned representation, we used the binary cross entropy loss for the adversary and training process proposed by (Edwards and Storkey, 2015).
Revised: (2018); Edwards and Storkey (2015) who trained an adversary to distinguish between samples from the protected and unprotected groups (named LFR), here we consider statistical parity and since we are not interested in the transferability of the learned representation, we used the binary cross entropy loss for the adversary and training process proposed by (Edwards and Storkey, 2015).

Original: And, the work done by Feng et al.
Revised: And, the work done by Feng et al.

Original: (namely LFRAF) where they trained an adversary to minimize the
Revised: (namely LFRAF) where they trained an adversary to minimize the

Original: Wasserstein distance between the distribution of protected and unprotected groups.
Revised: Wasserstein distance between the distribution of protected and unprotected groups.

Original: We used the implementation provided by the authors.
Revised: We used the implementation provided by the authors.

Original: We also consider a simple
Revised: We also consider a simple

Original: Autoencoder (AE) model that minimizes the loss of reconstructing the input data from the latent space, and then use encoder part to map (compress) the training data logistic regression model.
Revised: Autoencoder (AE) model that minimizes the loss of reconstructing the input data from the latent space, and then use encoder part to map (compress) the training data logistic regression model.

Original: The goal of using an autoencoder is to analyze the effect of the compression over the fairness on downstream task.
Revised: The goal of using an autoencoder is to analyze the effect of the compression over the fairness on downstream task.

Original: b) Data: We experiment in two datasets: German (Jeff et al., 2016) and UCI Adult Income (Asuncion and
Revised: b) Data: We experiment in two datasets: German (Jeff et al., 2016) and UCI Adult Income (Asuncion and

Original: Newman, 2007) datasets.
Revised: Newman, 2007) datasets.

Original: The German dataset comprises 1000 samples of bank account information described with 21 features.
Revised: The German dataset comprises 1000 samples of bank account information described with 21 features.

Original: The target variable y predicts whether an account is good or bad.
Revised: The target variable y predicts whether an account is good or bad.

Original: The adult income dataset contains 48 843 instances of demographic information of American adults, described with 14 features that are a mixture of categorical, ordinal, and numerical data types.
Revised: The adult income dataset contains 48 843 instances of demographic information of American adults, described with 14 features that are a mixture of categorical, ordinal, and numerical data types.

Original: The target indicates whether personal income levels is above or below
Revised: The target indicates whether personal income levels is above or below

Original: USD 50 000 per year.
Revised: USD 50 000 per year.

Original: We use sex (Male or Female) as sensitive attribute in both datasets.
Revised: We use sex (Male or Female) as sensitive attribute in both datasets.

Original: c) Metrics:
Revised: c) Metrics:

Original: In addition to the typical accuracy, we also consider the following fairness metrics used in the literature.
Revised: In addition to the typical accuracy, we also consider the following fairness metrics used in the literature.

Original: Let X
Revised: Let X

Original: = { x i } Ni ; x i ∈ R
Revised: = { x i } Ni ; x i ∈ R

Original: d be our data, S =
Revised: d be our data, S =

Original: { s i } Ni our binary sensitive attribute such that
Revised: { s i } Ni our binary sensitive attribute such that

Original: s i is contained within x i , and η ( · )
Revised: s i is contained within x i , and η ( · )

Original: a classifier that maps a given sample x i to a class label ˆ y i .
Revised: a classifier that maps a given sample x i to a class label ˆ y i .

Original: Statistical parity (Dwork et al., 2012) is a fairness notion that promotes the independence between the positive prediction and sensitive attribute through
Revised: Statistical parity (Dwork et al., 2012) is a fairness notion that promotes the independence between the positive prediction and sensitive attribute through

Original: [Equation]
Revised: [Equation]

Original: We measure statistical parity as
Revised: We measure statistical parity as

Original: [Equation]
Revised: [Equation]

Original: where N s is the number samples with the sensitive attribute set to s .
Revised: where N s is the number samples with the sensitive attribute set to s .

Original: We also consider equalized odds (Hardt et al., 2016;
Revised: We also consider equalized odds (Hardt et al., 2016;

Original: Verma and Rubin, 2018), which promotes the conditional independence between the prediction outcome and the sensitive attribute given the class label.
Revised: Verma and Rubin, 2018), which promotes the conditional independence between the prediction outcome and the sensitive attribute given the class label.

Original: That is
Revised: That is

Original: [Equation]
Revised: [Equation]

Original: for y ∈ { 0 , 1 } .
Revised: for y ∈ { 0 , 1 } .

Original: We measure it as ∆ EOD = (cid:88) y ∈{ 0 , 1 } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88)
Revised: We measure it as ∆ EOD = (cid:88) y ∈{ 0 , 1 } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88)

Original: i : s i =1 ,y i =
Revised: i : s i =1 ,y i =

Original: y η
Revised: y η

Original: ( x i )
Revised: ( x i )

Original: N ( y )− (cid:88)
Revised: N ( y )− (cid:88)

Original: i : s i =0 ,y i =
Revised: i : s i =0 ,y i =

Original: y η
Revised: y η

Original: ( x i )
Revised: ( x i )

Original: N ( y )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (13) where N ( y ) s is the number samples with the sensitive attribute set to s and the class label set to y .
Revised: N ( y )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (13) where N ( y ) s is the number samples with the sensitive attribute set to s and the class label set to y .

Original: Equal opportunity is similar to equalized odds (13) (Hardt et al., 2016; Verma and Rubin, 2018).
Revised: Equal opportunity is similar to equalized odds (13) (Hardt et al., 2016; Verma and Rubin, 2018).

Original: However, it only considers the case where y = 1 .
Revised: However, it only considers the case where y = 1 .

Original: We refer to it as ∆ EOP .
Revised: We refer to it as ∆ EOP .

Original: d) Model: All the components of models are defined as multi-layer perceptron (MLP).
Revised: d) Model: All the components of models are defined as multi-layer perceptron (MLP).

Original: The autoencoder part is defined with model with two levels of representation (layers).
Revised: The autoencoder part is defined with model with two levels of representation (layers).

Original: The first hidden layer is 15 units ( z 1 ), while the second one (i.e., latent space) is 10 and 8 units ( z 2 ) for the Adult and German datasets, respectively.
Revised: The first hidden layer is 15 units ( z 1 ), while the second one (i.e., latent space) is 10 and 8 units ( z 2 ) for the Adult and German datasets, respectively.

Original: Classifiers and regularizer networks are defined as single-layer MPL with the number of input units equal to the dimension of the latent space ( z 2 ).
Revised: Classifiers and regularizer networks are defined as single-layer MPL with the number of input units equal to the dimension of the latent space ( z 2 ).

Original: LFR-AF, AE, and LFR models are built with the same autoencoder architecture.
Revised: LFR-AF, AE, and LFR models are built with the same autoencoder architecture.

Original: For a fairer comparison, we enforced uniformity only at highest hierarchy ( z 2 ).
Revised: For a fairer comparison, we enforced uniformity only at highest hierarchy ( z 2 ).

Original: We use Adam optimizer (Kingma and Ba, 2014) with step size of 0 .
Revised: We use Adam optimizer (Kingma and Ba, 2014) with step size of 0 .

Original: 001 for 1000 steps and a batch size of 64 .
Revised: 001 for 1000 steps and a batch size of 64 .

Original: We, then, train a logistic regression model on the leaned representations to predict the class labels or the sensitive attribute to assess the level of dependency between the given representation and the learned representation.
Revised: We, then, train a logistic regression model on the leaned representations to predict the class labels or the sensitive attribute to assess the level of dependency between the given representation and the learned representation.

Original: We split the dataset into train and test sets.
Revised: We split the dataset into train and test sets.

Original: We use the training set to learn the representations.
Revised: We use the training set to learn the representations.

Original: We map the test set into the learned space, and used it to train the logistic regressor using 10-fold cross-validation.
Revised: We map the test set into the learned space, and used it to train the logistic regressor using 10-fold cross-validation.

Original: We run the experiment seven times and average the results shown in Fig.
Revised: We run the experiment seven times and average the results shown in Fig.

Original: 5 and 6.
Revised: 5 and 6.

Original: Note that this evaluation regressor is different from the classifier, p ( y | z ) , used at training time to regularize the model—cf.
Revised: Note that this evaluation regressor is different from the classifier, p ( y | z ) , used at training time to regularize the model—cf.

Original: L y (6).
Revised: L y (6).

Original: B. Learned Representations
Revised: B. Learned Representations

Original: To examine how the groups are distributed in the latent space, we learned representations using the different methods considered, and projected them into two dimensions using PCA.
Revised: To examine how the groups are distributed in the latent space, we learned representations using the different methods considered, and projected them into two dimensions using PCA.

Original: Figure 4 showcases the distribution of groups (Male and Female) in the latent space.
Revised: Figure 4 showcases the distribution of groups (Male and Female) in the latent space.

Original: We trained representation using our approach without inducing the uniformity of the sensitive attribute in the latent space, i.e., without the noisy prediction of the sensitive.
Revised: We trained representation using our approach without inducing the uniformity of the sensitive attribute in the latent space, i.e., without the noisy prediction of the sensitive.

Original: Therefore, the learned representation (fig 4 second column–LR) is enforced to discriminate between group label and the class label.
Revised: Therefore, the learned representation (fig 4 second column–LR) is enforced to discriminate between group label and the class label.

Original: As a result, the second column of the figure 4 shows that without the uniformity, groups are linearly separable in the latent space.
Revised: As a result, the second column of the figure 4 shows that without the uniformity, groups are linearly separable in the latent space.

Original: In contrast, we enforce the uniformity in latent space, the learned representations are entangled and therefor hard to distinguish (or predict).
Revised: In contrast, we enforce the uniformity in latent space, the learned representations are entangled and therefor hard to distinguish (or predict).

Original: [Figure]
Revised: [Figure]

Original: German dataset
Revised: German dataset

Original: [Figure]
Revised: [Figure]

Original: ∆ DP ∆
Revised: ∆ DP ∆

Original: ∆
Revised: ∆

Original: Figure 4 also shows that similarly to other fair representation learning methods, our methods is able entangle different groups in the latent space.
Revised: Figure 4 also shows that similarly to other fair representation learning methods, our methods is able entangle different groups in the latent space.

Original: This suggest that our method of making the prediction of the sensitive attribute unreliable by enforcing its uniformity is as good as defeating an adversary (LFR and LFR-U).
Revised: This suggest that our method of making the prediction of the sensitive attribute unreliable by enforcing its uniformity is as good as defeating an adversary (LFR and LFR-U).

Original: The advantage of our approach for obtaining this representation is that it is easier to train and does not suffer from the instability of the adversarial approach, which requires to defeat an adversary and can sometimes be counterproductive.
Revised: The advantage of our approach for obtaining this representation is that it is easier to train and does not suffer from the instability of the adversarial approach, which requires to defeat an adversary and can sometimes be counterproductive.

Original: C. Fairness Analysis
Revised: C. Fairness Analysis

Original: Figure 5 presents the fairness performances of logistic regressions models trained with data representations of each particular method.
Revised: Figure 5 presents the fairness performances of logistic regressions models trained with data representations of each particular method.

Original: Overall, the representation yielded by our method (LFR-U) outperforms the baseline models in term of accuracy and provides better performance in terms of statistical parity, equalized odds and equal opportunity compared on the considered datasets.
Revised: Overall, the representation yielded by our method (LFR-U) outperforms the baseline models in term of accuracy and provides better performance in terms of statistical parity, equalized odds and equal opportunity compared on the considered datasets.

Original: Whilst our method is designed to enforce fairness in terms of statistical parity by inducing uniformity of the sensitive attribute, our empirical results show that this has a positive impact on other fairness notions (equalized odds and equal opportunity).
Revised: Whilst our method is designed to enforce fairness in terms of statistical parity by inducing uniformity of the sensitive attribute, our empirical results show that this has a positive impact on other fairness notions (equalized odds and equal opportunity).

Original: While other methods learned representations that decreased the accuracy compared to the baseline methods, ours impacted it less (Fig. 4).
Revised: While other methods learned representations that decreased the accuracy compared to the baseline methods, ours impacted it less (Fig. 4).

Original: The goal of the adversarial approach for fair representation learning is to completely remove the dependence to the sensitive attribute by defeating an adversary.
Revised: The goal of the adversarial approach for fair representation learning is to completely remove the dependence to the sensitive attribute by defeating an adversary.

Original: However, this optimization process does not guarantee that the trained adversary is the optimal one, and, that there is no other adversary that can predict the sensitive attribute from the learned representation.
Revised: However, this optimization process does not guarantee that the trained adversary is the optimal one, and, that there is no other adversary that can predict the sensitive attribute from the learned representation.

Original: In contrast, we enforce the uniformity of the sensitive attribute in the latent space by leveraging a noisy prediction of the sensitive attribute from the latent space, such that each data point has the same probability ( 0 . 5 ) to be assigned to a given group.
Revised: In contrast, we enforce the uniformity of the sensitive attribute in the latent space by leveraging a noisy prediction of the sensitive attribute from the latent space, such that each data point has the same probability ( 0 . 5 ) to be assigned to a given group.

Original: Thus, instead of focusing on learning optimal features that fool a classifier, we focus on obtaining features that are equally likely to produce every class in the sensitive attribute.
Revised: Thus, instead of focusing on learning optimal features that fool a classifier, we focus on obtaining features that are equally likely to produce every class in the sensitive attribute.

Original: The results (Fig. 5) also show that in an unsupervised setting, a simple autoencoder (AE) reduces the dependence on the sensitive attribute in the latent space and thus improves fairness.
Revised: The results (Fig. 5) also show that in an unsupervised setting, a simple autoencoder (AE) reduces the dependence on the sensitive attribute in the latent space and thus improves fairness.

Original: This shows that sensitive information can be cancelled out by compression, which can be useful in the case where sensitive attributes are not observed.
Revised: This shows that sensitive information can be cancelled out by compression, which can be useful in the case where sensitive attributes are not observed.

Original: D. Sensitive Attribute Obfuscation
Revised: D. Sensitive Attribute Obfuscation

Original: We analyze the amount of information of the sensitive attribute that is still encoded in the latent space after the learning by training a classifier to predict it from the learned representation.
Revised: We analyze the amount of information of the sensitive attribute that is still encoded in the latent space after the learning by training a classifier to predict it from the learned representation.

Original: We expect (ideally) the accuracy of this classifier to be very low, i.e., the dependence to the sensitive attribute has being removed from the latent space.
Revised: We expect (ideally) the accuracy of this classifier to be very low, i.e., the dependence to the sensitive attribute has being removed from the latent space.

Original: A classifier trained on the original data is quite accurate in predicting the sensitive attribute s .
Revised: A classifier trained on the original data is quite accurate in predicting the sensitive attribute s .

Original: As shown in the figure 6, a logistic regression model trained to predict the sensitive attribute in the original dataset achieves more than 83 % and 75 % of accuracy in the Adult and German datasets respectively.
Revised: As shown in the figure 6, a logistic regression model trained to predict the sensitive attribute in the original dataset achieves more than 83 % and 75 % of accuracy in the Adult and German datasets respectively.

Original: While our method and adversarial based methods provide less than 68 % accuracy.
Revised: While our method and adversarial based methods provide less than 68 % accuracy.

Original: This shows that all approaches try to obscure information about s in the latent space, while ours also maintains better utility.
Revised: This shows that all approaches try to obscure information about s in the latent space, while ours also maintains better utility.

Original: From privacy perspectives, a logistic regression model can be seen as a weak adversary, i.e., a stronger adversary such as a neural network can archive better performance on predicting the sensitive attributes from the learned representation.
Revised: From privacy perspectives, a logistic regression model can be seen as a weak adversary, i.e., a stronger adversary such as a neural network can archive better performance on predicting the sensitive attributes from the learned representation.

Original: However, although the objectives (fairness and privacy) converge, the primary goal is not to make the latent space robust against an adversary wanting to reconstruct the sensible attributes.
Revised: However, although the objectives (fairness and privacy) converge, the primary goal is not to make the latent space robust against an adversary wanting to reconstruct the sensible attributes.

Original: Thus, the use of logistic regression in this case is not to evaluate the robustness against an adversary but to evaluate the level of obfuscation of the sensitive attribute, which is necessary for fairness.
Revised: Thus, the use of logistic regression in this case is not to evaluate the robustness against an adversary but to evaluate the level of obfuscation of the sensitive attribute, which is necessary for fairness.

Original: We also test how much information about the sensitive attribute is retained in the representation learned by our method using a stronger adversary,
Revised: We also test how much information about the sensitive attribute is retained in the representation learned by our method using a stronger adversary,

Original: i
Revised: i

Original: [Table]
Revised: [Table]

Original: e., a neural networks.
Revised: e., a neural networks.

Original: [Table] [Table]
Revised: [Table] [Table]

Original: The adversary is a two layer MLP with 25 and 15 units in the first and the second hidden layers respectively.
Revised: The adversary is a two layer MLP with 25 and 15 units in the first and the second hidden layers respectively.

Original: The network is trained seven times and the accuracy is averaged across runs.
Revised: The network is trained seven times and the accuracy is averaged across runs.

Original: The optimizer and the parameters used are the same as in the previous experiment.
Revised: The optimizer and the parameters used are the same as in the previous experiment.

Original: Table I shows the accuracy of predicting the sensitive attribute and class label using the original representation and the representation provided by our methods in Adult and German datasets.
Revised: Table I shows the accuracy of predicting the sensitive attribute and class label using the original representation and the representation provided by our methods in Adult and German datasets.

Original: These results show that even a stronger adversary such as a neural network is not accurate in reconstructing the sensitive attribute from the representation regularized with our method.
Revised: These results show that even a stronger adversary such as a neural network is not accurate in reconstructing the sensitive attribute from the representation regularized with our method.

Original: For instance, on Adult dataset, the MLP model achieves 84.5% accuracy in predicting the sensitive attribute vs. 68.6% with our representation, while achieving 82.6% in predicting the class label compared to 85% with the original representation.
Revised: For instance, on Adult dataset, the MLP model achieves 84.5% accuracy in predicting the sensitive attribute vs. 68.6% with our representation, while achieving 82.6% in predicting the class label compared to 85% with the original representation.

Original: Note that the representation used to train the adversary was optimized using both the classifier and the regularize heads, i.e., β = 1 and λ =(Equation 7).
Revised: Note that the representation used to train the adversary was optimized using both the classifier and the regularize heads, i.e., β = 1 and λ =(Equation 7).

Original: Showing that the representation preserved the useful information for target downstream task while obfuscating information about sensitive attributes.
Revised: Showing that the representation preserved the useful information for target downstream task while obfuscating information about sensitive attributes.

Original: E. Hierarchical Regularization
Revised: E. Hierarchical Regularization

Original: We analyze the effect of hierarchically regularizing our proposed model instead of only regularizing the highest level.
Revised: We analyze the effect of hierarchically regularizing our proposed model instead of only regularizing the highest level.

Original: To evaluate the effect of this full regularization, we train (using the Adult dataset) our method (LFR-U) enforcing uniformity at he highest level ( z 2 ) and at both levels ( z 1 and z 2 ).
Revised: To evaluate the effect of this full regularization, we train (using the Adult dataset) our method (LFR-U) enforcing uniformity at he highest level ( z 2 ) and at both levels ( z 1 and z 2 ).

Original: We report the results in Table II.
Revised: We report the results in Table II.

Original: They show that regularizing at multiple levels further improve fairness w.r.t.
Revised: They show that regularizing at multiple levels further improve fairness w.r.t.

Original: equalized odds and equal opportunity.
Revised: equalized odds and equal opportunity.

Original: We highlight that existing models only regularize a single level.
Revised: We highlight that existing models only regularize a single level.

Original: Thus, all our other experiments are with a single level regularization for a fair comparison.
Revised: Thus, all our other experiments are with a single level regularization for a fair comparison.

Original: For this set of
Revised: For this set of

Original: LR
Revised: LR

Original: CR LFR-AF AE
Revised: CR LFR-AF AE

Original: LFR
Revised: LFR

Original: LFR-U 0 .
Revised: LFR-U 0 .

Original: A cc u r a c y ( % ) y prediction ( ↑ )
Revised: A cc u r a c y ( % ) y prediction ( ↑ )

Original: LR
Revised: LR

Original: CR LFR-AF AE
Revised: CR LFR-AF AE

Original: LFR
Revised: LFR

Original: LFR-U 0 .
Revised: LFR-U 0 .

Original: A cc u r a c y ( % ) s prediction ( ↓ )
Revised: A cc u r a c y ( % ) s prediction ( ↓ )

Original: German
Revised: German

Original: Adult experiments, we trained the representation with higher weight ( β = 1 ) on classifier head without reconstruction ( λ = 0 ), which lead to better accuracy on predicting the class label and lower fairness.
Revised: Adult experiments, we trained the representation with higher weight ( β = 1 ) on classifier head without reconstruction ( λ = 0 ), which lead to better accuracy on predicting the class label and lower fairness.

Original: F. Ablation Study of Uniformity
Revised: F. Ablation Study of Uniformity

Original: We analyze how the flipping probability t affects the learned representation.
Revised: We analyze how the flipping probability t affects the learned representation.

Original: The probability t is a hyperparameter that can be tuned to improve the accuracy or fairness performance of the learned representations.
Revised: The probability t is a hyperparameter that can be tuned to improve the accuracy or fairness performance of the learned representations.

Original: The best choice of the flipping ratio depends on how balanced are the different groups in the dataset.
Revised: The best choice of the flipping ratio depends on how balanced are the different groups in the dataset.

Original: When groups are not well balanced in the dataset, samples from well represented group will have higher probability to be flipped compared to samples from underrepresented groups, which might restore the group balance during the training.
Revised: When groups are not well balanced in the dataset, samples from well represented group will have higher probability to be flipped compared to samples from underrepresented groups, which might restore the group balance during the training.

Original: In previous experiments, we used t = 0 .
Revised: In previous experiments, we used t = 0 .

Original: Figures 8 andshow accuracy and fairness performance across a range of different values of t for German and Adult dataset respectively.
Revised: Figures 8 andshow accuracy and fairness performance across a range of different values of t for German and Adult dataset respectively.

Original: [Figure]
Revised: [Figure]

Original: A cc u r a c y ( % ) 0 .
Revised: A cc u r a c y ( % ) 0 .

Original: Probability of flipping data
Revised: Probability of flipping data

Original: F a i r n e ss ( % ) ∆ DP ∆ EOD ∆ EOP
Revised: F a i r n e ss ( % ) ∆ DP ∆ EOD ∆ EOP

Original: [Figure]
Revised: [Figure]

Original: A cc u r a c y ( % ) 0 .
Revised: A cc u r a c y ( % ) 0 .

Original: Probability of flipping data
Revised: Probability of flipping data

Original: [Figure]
Revised: [Figure]

Original: i r n e ss
Revised: i r n e ss

Original: [Figure]
Revised: [Figure]

Original: When we set t = 0 , it is equivalent to use the original dataset.
Revised: When we set t = 0 , it is equivalent to use the original dataset.

Original: Thus, the model learns to separate different groups in the latent space.
Revised: Thus, the model learns to separate different groups in the latent space.

Original: Therefore, the non-uniformity of the sensitive attribute is enforced.
Revised: Therefore, the non-uniformity of the sensitive attribute is enforced.

Original: Similarly, when t = 1 , all the sensitive attributes are flipped for the samples; and, again, the non-uniformity is enforced but in a reversed way.
Revised: Similarly, when t = 1 , all the sensitive attributes are flipped for the samples; and, again, the non-uniformity is enforced but in a reversed way.

Original: Figure 7 shows in Adult dataset, the value of t that provides the best fairness performances is 0 . 5 , the figure shows that the level of unfairness increases as the value t moves away from 0.5.
Revised: Figure 7 shows in Adult dataset, the value of t that provides the best fairness performances is 0 . 5 , the figure shows that the level of unfairness increases as the value t moves away from 0.5.

Original: Similarly for the German dataset, figure 8 shows that the value of t that provides better fairness-accuracy trade-off is between 0 . 5 and 0 . 6 .
Revised: Similarly for the German dataset, figure 8 shows that the value of t that provides better fairness-accuracy trade-off is between 0 . 5 and 0 . 6 .

Original: V. Conclusion and Future Work
Revised: V. Conclusion and Future Work

Original: In this work, we introduced the idea that making a model’s learned representations unreliable w.r.t.
Revised: In this work, we introduced the idea that making a model’s learned representations unreliable w.r.t.

Original: the sensitive attributes enforces fairness.
Revised: the sensitive attributes enforces fairness.

Original: Our experiments showed that even while using full data samples (i.e., including the sensitive attribute) for training, we could obfuscate the sensitive attributes while maintaining the prediction accuracy of the related tasks.
Revised: Our experiments showed that even while using full data samples (i.e., including the sensitive attribute) for training, we could obfuscate the sensitive attributes while maintaining the prediction accuracy of the related tasks.

Original: Moreover, we showed using experiments on two datasets that demonstrated the advantages of our proposal w.r.t.
Revised: Moreover, we showed using experiments on two datasets that demonstrated the advantages of our proposal w.r.t.

Original: the accuracy and fairness metrics.
Revised: the accuracy and fairness metrics.

Original: Interestingly, our method is designed to improve fairness in terms of statistical parity, our results showed that other fairness notions such equalized odds and equal opportunities get positively impacted.
Revised: Interestingly, our method is designed to improve fairness in terms of statistical parity, our results showed that other fairness notions such equalized odds and equal opportunities get positively impacted.

Original: In a nutshell, our proposal is to enforce uniformity over a prediction head for the sensitive attributes, and use it as a regularizer for the learned representations.
Revised: In a nutshell, our proposal is to enforce uniformity over a prediction head for the sensitive attributes, and use it as a regularizer for the learned representations.

Original: Moreover, we explore a hierarchical model of representations that helps in enforcing the fairness while providing a better prediction performance on class labels.
Revised: Moreover, we explore a hierarchical model of representations that helps in enforcing the fairness while providing a better prediction performance on class labels.

Original: However, we observe a drop in accuracy compared to model trained without fairness constrains.
Revised: However, we observe a drop in accuracy compared to model trained without fairness constrains.

Original: Although this drop is less than other fair representation learning techniques, reducing the tradeoff between fairness and accuracy remains an important issue within the fair
Revised: Although this drop is less than other fair representation learning techniques, reducing the tradeoff between fairness and accuracy remains an important issue within the fair

Original: ML community.
Revised: ML community.

Original: Finally, it would also be interesting to study theoretical guarantees for the representation learned in Algorithm 1, i.e., given the participants in a decisionmaking process, what fairness guarantees the proposed method can provide.
Revised: Finally, it would also be interesting to study theoretical guarantees for the representation learned in Algorithm 1, i.e., given the participants in a decisionmaking process, what fairness guarantees the proposed method can provide.

Original: Appendix A Flipping process
Revised: Appendix A Flipping process

Original: The flipping process (cf.
Revised: The flipping process (cf.

Original: Section III-A) takes a Bernoulli over the sensitive attribute, B ( u ) , and flips it using another independent Bernoulli with probability r , B ( r ) .
Revised: Section III-A) takes a Bernoulli over the sensitive attribute, B ( u ) , and flips it using another independent Bernoulli with probability r , B ( r ) .

Original: Given s ∼ B ( u ) , f ∼ B ( r ) , we compute the resulting sensitive value, s r , from their joint p ( s, f ) =
Revised: Given s ∼ B ( u ) , f ∼ B ( r ) , we compute the resulting sensitive value, s r , from their joint p ( s, f ) =

Original: p ( s ) p ( f ) , where there are four possibilities:
Revised: p ( s ) p ( f ) , where there are four possibilities:

Original: [Equation]
Revised: [Equation]

Original: It follows that, p
Revised: It follows that, p

Original: [Equation]
Revised: [Equation]

Original: 18d) and, similarly we find
Revised: 18d) and, similarly we find

Original: [Equation]
Revised: [Equation]

Original: 19b
Revised: 19b

Original: [Equation]
Revised: [Equation]

Original: Consequently, the randomized sample p ( s r ) ∼ B ( u − 2 ur + r ) .
Revised: Consequently, the randomized sample p ( s r ) ∼ B ( u − 2 ur + r ) .

Original: Appendix B Proof of Theorem
Revised: Appendix B Proof of Theorem

Original: (Restated) Theorem 1: Let two independent Bernoulli distributions be B ( u ) and B ( r ) , and a third related to the parameters of the previous ones be B ( u − 2 ur + r ) .
Revised: (Restated) Theorem 1: Let two independent Bernoulli distributions be B ( u ) and B ( r ) , and a third related to the parameters of the previous ones be B ( u − 2 ur + r ) .

Original: The cross-entropy between B ( u ) and B ( u − 2 ur + r ) is an upper bound of the Kullback-Leibler divergence between the two original distributions, i.e.,
Revised: The cross-entropy between B ( u ) and B ( u − 2 ur + r ) is an upper bound of the Kullback-Leibler divergence between the two original distributions, i.e.,

Original: [Equation]
Revised: [Equation]

Original: Proof.
Revised: Proof.

Original: The KL divergence between the Bernoullis is
Revised: The KL divergence between the Bernoullis is

Original: [Equation]
Revised: [Equation]

Original: and similarly
Revised: and similarly

Original: KL
Revised: KL

Original: [Equation]
Revised: [Equation]

Original: By manipulating the former KL (21)
Revised: By manipulating the former KL (21)

Original: [Equation]
Revised: [Equation]

Original: we obtain
Revised: we obtain

Original: [Equation]
Revised: [Equation]

Original: where C
Revised: where C

Original: [Equation]
Revised: [Equation]

Original: u
Revised: u

Original: [Equation]
Revised: [Equation]

Original: ) is a function of u and r .
Revised: ) is a function of u and r .

Original: Then
Revised: Then

Original: [Equation]
Revised: [Equation]

Original: we can expand the LHS
Revised: we can expand the LHS

Original: [Equation]
Revised: [Equation]

Original: and we can bound it by
Revised: and we can bound it by

Original: [Equation]
Revised: [Equation]