Simple and Effective Specialized Representations for Fair Classifiers
Authors: Alberto Sinigaglia
Davide Sartor
Marina Ceccon
Gian Antonio Susto
Venue: NeurIPS 2025 poster ()
Keywords: Fairness
Invariant Representations
Characteristic Function
Neural Networks
Primary area: social_and_economic_aspects_of_machine_learning
TL;DR: This paper proposes a stable and efficient fair classification method using characteristic function distance to minimize sensitive information in representations.

Abstract: Fair classification is a critical challenge that has gained increasing importance due to international regulations and its growing use in high-stakes decision-making settings.
Existing methods often rely on adversarial learning or distribution matching across sensitive groups; however, adversarial learning can be unstable, and distribution matching can be computationally intensive.
To address these limitations, we propose a novel approach based on the characteristic function distance. Our method ensures that the learned representation contains minimal sensitive information while maintaining high effectiveness for downstream tasks. 
By utilizing characteristic functions, we achieve a more stable and efficient solution compared to traditional methods.
Additionally, we introduce a simple relaxation of the objective function that guarantees fairness in common classification models with no performance degradation. 
Experimental results on benchmark datasets demonstrate that our approach consistently matches or achieves better fairness and predictive accuracy than existing methods. 
Moreover, our method maintains robustness and computational efficiency, making it a practical solution for real-world applications.

Simple and Effective Specialized Representations for Fair Classifiers
Alberto Sinigaglia"
Davide Sartor"
Human Inspired Technology Research Center University of Padua
Department of Information Engineering
University of Padua
alberto. a.berto.sinigagll. Sanigaalli@@hhd unipd.it
davide waaetrr..atrr.... 40phd. unipd. it
Marina Ceccon
Gian Antonio Susto
Department Information Engineering
Department of Information Engineering
University of Padua
University of Padua
marina. ceccon@phd. unipd.
gianantonio. susto@unipd.
Abstract
Fair classification a critical challenge that has gained increasing importance due to international regulations and its growing in high-stakes decision-making set- tings. Existing methods often rely on adversarial learning or distribution matching across sensitive groups; however, adversarial learning can be unstable, and distri- bution matching can be computationally intensive. To address these limitations, propose novel approach based the characteristic function distance. Our method ensures that the learned representation contains minimal sensitive infor- mation while maintaining high effectiveness for downstream tasks. By utilizing characteristic functions, we achieve a more stable and efficient solution compared traditional methods. Additionally, jective function that guarantees fairness in common classification models with no performance degradation. Experimental results benchmark datasets demonstrate that our approach consistently matches or achieves better fairness and predictive accuracy than existing methods. Moreover, our method maintains robustness and computational efficiency, making it practical solution for real- wal-world applications.
introduce simple relaxation of the ob-
Introduction
Algorithmic fairness has become central concern deploying automated decision pocision-maaing systems, especially in high-stakes domains like hiring, lending, criminal justice, and healthcare 4 44 [4] 21].
The growing reliance on these systems has raised concerns about their potential reinforce or In response, large body of research has focused on detecting, analyzing, and mitigating bias throughout the algorithmic pipeline
amplify societal biases 3 15 48
44
55 36
Numerous fairness definitions and metrics have been proposed, reflecting wide range of normative and technical perspectives [18. 27 [6]. A common criterion statistical independence, which requires that model predictions be independent of sensitive attributes like ethnicity or gender 42 B
simple strategy for enforcing this criterion is to exclude the sensitive attribute from the model's input features, an approach commonly referred to as Fairness through Unawareness [10]. However, this method often ineffective practice, as sensitive information may still be indirectly captured through other correlated features, so-called proxy variables [39]. Removing all proxies typically results in significant degradation of performance, as task-relevant information is also discarded [10]
Thhse authors contributed equally.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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CFD,, (p (P vian))
ho(z)
52(2)
2
YU
Figure 1: Overview of the proposed approach. Each sample is associated with sensitive attribute and a target label Y. The conditional distribution I' xis is mapped to new distribution Pzis. which encouraged resemble a Gaussian Distribution via the Characteristic Function Distance (CFD). The encoded representation Z minimizes A(P 22)))
while retaining task-relevant information.
To better balance fairness and predictive accuracy, many approaches aim to learn fair representations by projecting input data into a latent space that obscures sensitive information while preserving task-relevant structure [55 35 37 2]. Once optimized, these representations can replace the original data during both training and testing, enabling fair decision maaiin in scenarios where sensitive attributes must not be explicitly used, such determining loan approvals hiring decisions. These approaches typically fall into two categories, which we refer to as specialized and general. Specialized representations are tailored to specific task, filtering sensitive and irrelevant features for improved performance. General representations, in contrast, are task agnosticc and aim for broad applicability across multiple tasks while being insensitive to protected attributes, but are harder to learn and may compromise predictive accuracy. Returning to the loan approval example, specialized representations learned in that context would be suitable only for predicting creditworthiness. If reused for another task, e.g. default risk or customer lifetime value, they might lack task-relevant information that was discarded during the fairness optimization. General representations, instead, preserve information that remains broadly useful across prediction tasks while still mitigating sensitivity to protected attributes.
A popular class of methods uses Variational Autoencoders (VAEs) to learn fair, specialized represen- tations 32 37. In VAE- based approaches, faimness is optimized indirectly, either via a variational bound or Maximum Mean Discrepancy (MMD), which misalign with true faimess goals and yield suboptimal results 2
Other methods rely Adversarial Networks that ensure specialized fair representations through a minimax game between a predictor and a fairness discriminator [19 28 Such approaches are often unstable and sensitive to hyperparameters, making the hainess-performance trade-off more challenging to manage (20). Additionally, fairness guarantees are tied to the specific adversary used during training and may fail if a stronger or different adversary is applied later 37 52
26 45)
Fair Normalizing Flows (FNF) offer representations that are robust to any adversary, positioning itself as a state-of-the state-of-the-art alternative to adversarial methods. Nonetheless, it has key limitations. It requires sensitive attribute information at inference, conflicting with privacy goals. It also trains separate model for each sensitive group, which is computationally expensive, especially with multi-class attributes or large datasets. Crucially, while FNF aims to learn general representations, the inclusion of a classification loss can bias the model toward task-specific features. This may introduce group-specific information, potentially weakening faimeess guarantees through representation bias.
There is an inherent trade-off between general and specialized representations: while general represen- tations are broadly useful and transferable, achieving high predictive performance often requires some degree of specialization. Even models like FNF, which explicitly aim for generalization, ultimately rely on task task-specific loss functions that guide representations toward specialization to optimize accu- racy. This reliance suggests that specialization is not merely a byproduct but a necessary condition for strong predictive performance. Recognizing this, it becomes clear that embracing specialization is not only pragmatic but essential. Adversarial approaches naturally embody this perspective by tailoring representations to specific prediction tasks. However, they come with significant limitations,
most notably, weaker guarantees around fairness and robustness [ 37 45). This underscores the

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need for alternative methods that, like adversarial techniques, pursue specialization, but do so through more transparent and controllable means, offering stronger fairesss guarantees result.
To address these challenges, we propose a novel framework that embraces specialization in repre- sentations while rigorously enforcing fairness, without relying adversarial training. In addition to stronger guarantees, our method is significantly simpler and more lightweight than existing ap- proaches. This further justifies the aim to achieve specialized representations, as the advantages of transferability offered by general representations might not outweigh the benefit of higher prediction accuracy in settings where re-training for a different task is computationally lightweight. Our key contributions are summarized as follows:
We introduce a new approach based distribution matching via characteristic functions. This formulation avoids reliance on auxiliary components such as variational autoencoders, adversarial networks, or normalizing flows.
We derive a simplified version of the proposed framework tailored to classification, which allows for formal guarantees the sensitive information accessible downstream classifiers.
We conduct extensive experiments demonstrating that our method effectively removes sensitive information while matching SotA accuracy and delivering significantly fairer representations.
In summary, our approach offers a principled alternative for learning fair, specialized representations. It scamlessly combines simplicity, training stability, and robust faimess guarantees, all without relying on sensitive attributes during inference. This makes our method both practical and crivacy-conscious, showing promising improvements over the current state-of-the-art (2]
2
Related Work
Variational Autoncoders In the context of fairness through representation learning, prior work has explored the use of Variational Autoencoders (VAEs) disentangle sensitive attributes from learned data representations [32
[1 B1. The encoder extracts task-relevant representations, while the decoder reconstructs the input. Though reconstruction may retain extra information, the optimization encourages representations tailored to prediction. A notable approach is the Variational Fair Autoencoder (VFAF) 32) which extends the standard VAE framework to enforce invariance in the latent space with respect to protected attributes. VFAE achieves this by introducing penalty that explicitly encourages independence between the latent representation and the sensitive attributes. A related method was proposed by Moyer et al. (37], who combine ideas from VAEs and the Variational Information Bottleneck (VIB) to learn representations that are both informative and robust to variations in sensitive inputs.
37
Adversarial Learning Another line of research focuses fairness-aware learning through ad- versarial and information-ther -hhoorrttccccccccccoooooooocctte methods to induce invariant representations [51] 41 46). In adversarial settings, model such as encoder or predictor is trained in opposition to an adversary whose goal is to recover sensitive attributes from the learned representation. This adversarial pressure encourages the model to generate latent features that do not reveal sensitive information, thereby promoting invariance with respect to protected variables 19 22]. Xie et al. [51] proposed a general adversarial framework for learning representations that are invariant to arbitrary nuisance attributes. Their method formulates the learning process as three-player minimax game involving an encoder, task-specific predictor, and a discriminator that attempts to infer the nuisance attribute. Madras al. (35) introduced learning Adversarially Fair and Transferable Representations (I (LAFTR), which incorporates adversarial objectives aligned with specific faimess definitions, to learn representations that remain fair even when deployed by downstream classifiers without explicit fairness constraints. Roy and Bodetti 4T] extended this adversarial paradigm by proposing MaxEnt-ARe ARL, which maxi- mizes the entropy of the adversary's 's prediction of the sensitive attribute rather than minimizing its accuracy. This approach improves privacy, as it offers the practical benefit of not requiring access to sensitive labels during encoder training. Jaiswal et al. (28 introduced Adversarial Forgetting, a framework that decouples the learning of rich representations from the selective forgetting of sensitive or nuisance information via dedicated forget-gate mechanism. Building on adversarial fairness frameworks, FR-Train 40 incorporates mutual information-ban formulation and adds a second adversary to enhance robustness to poisoned data.
3

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Other Approaches Recognizing the limitations of adversarial frameworks, particularly their insta- researchers have explored alternative paradigms for learning fair representations. Jiang et al. 12 propose a theoretically grounded approach that enforces demographic parity by minimizing the Wasserstein-1 distance between model outputs across different sensitive groups. Tucker and Shah 47 present Concept Subspace Networks (CSNs), prototype-based architecture that unifies fair and hierarchical classífication within a single model. Ultimately, building on these ideas, Balunović et al. [2] introduce Fair Normalizing Flows (FNF), framework that provides provable fairness guarantees against any downstream adversary. FNF repre- sents the current state- ofate-o--hhe-n art in terms of fair representations. It employs separate normalizing flow encoders for each sensitive group and ensures fairesss by minimizing the statistical distance between the resulting latent distributions. This formulation enables exact likelihood computation in the latent space, allowing for theoretical upper bounds on unfairness for any downstream classifier, a property not commonly achieved in existing fair representation learning methods. Furthermore, Balunović et al. [2) show how adversarial learning inherently causes a false sense of fairness [22] 52 20 26]. Indeed, they show how multiple methods based such techniques share tendency to break once the learned representations are tested on more powerful families of classifiers. Though adversarial learning is the only approach offering specialized representations, such evidence reinforces the need to develop novel directions to achieve them without the instabilities of adversarial learning.
bility and lack of formal guarantees [20] 26 22
3 Background
Let X E Idd denote a feature vector, Y E {0. 1} a binary label, and S E ($1..... 8n ) a sensitive attribute, taken from some joint distribution P2, x.... y. The most common scenario in fairness-critical applications is binary sensitive attribute S E {0, 1]. Traditional classification algorithms fit a classifier fo F + {0, 1} to predict the task label Y from X. The sensitive attribute S is often statistically correlated with both the feature vector X and the target label Y. raising fairness concerns. A prevalent approach in sfate-of-hhe-aee fairness-ave aware methods involves constructing fair representations Z from X prior predicting Y.
Statistical Distance and Adversarial Evaluation Since strict faimess criteria often entail accuracy trade-offs due to correlation between the sensitive and the task label Y. fair representations typically result in either performance reductions or leakage of the sensitive information in the classification. Consequently, we adopt the concept of e-fairness, where e represents the statistical distance between the learned representations Pzis: This enables the application of bounds from Madras et al. (35) relating statistical distances to several fairness metrics (see appendix [.4]. Specifically, they consider the Total Variation (TV) distance, defined for distributions Po and P as:
/ x
A(Po, Pi)
[Po(x)) (P)
Balunović et al. (2) estimate the statistical distance directly. However, this can be extremely hard in the general case, and inaccuracy in the estimation might lead to a poor estimation of fairness. Given that the proposed method does not always impose specific distributional assumptions on Z, we uniformly adopt adversarial evaluation to quantify fairness throughout the whole paper. Indeed, for an optimal adversarial classifier f. the statistical distance between conditional distributions can be precisely expressed as:
1
sup max P(Y = o(X)) 2 (1
Paaeo,,P2isel Pz(==1))-
G 9Eg
Adversarial Evaluation should not be confused with Adversarial Learning. Adversarial Learning learns latent fair representation Z from X optimizing eq. (2) directly via min-max training, which can lead to training instability [22 37 20]. Adversarial frameworks are typically robust only to the class of functions used during training. Adversarial Evaluation simply assesses the fairness of a fixed representation Z by quantifying how accurately classifier (often an Multi Layer Perceptron (MLP)) can predict the sensitive attribute S from Z.
Logistic Regression Logistic Regression (LRR is a popular classification algorithm used to model the probability of a binary outcome Y E {0, as function of set of predictors X e Rd. The

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model is defined by the logistic function og : R + (0, 1);
1
xa(x) 1 e P|o) Σ.-1 P(o)
where the coefficients 3 E Rd-1 are the model parameters, with B(0g denoting the bias term. The model output (I) is used to learn P(Y = X z) via maximum likelihood estimation. To achieve this, the objective function to be minimized is given by the negative lieeliilhhood::
C (3) -xx (Y (oog(as(X)) + (1 Y) log(1 ax()))
A key advantage of LR over other families of classifiers, such MLP, is the convexity of its objective function. Since this loss function is convex, has a global optimum that can be provably reached. This property not only guarantees convergence but also enables the use of second-order optimization techniques such as Newton-Raphson or its variant, Iteratively Reweighted Least Squares. These methods leverage the Hessian of the loss function to achieve quadratic convergence, significantly accelerating the optimization compared to first-order methods like gradient descent.
4 Fair Representations matching Characteristic Function
The concept of Characteristic Function has been extensively studied for a long time in the statis- tical testing literature [33, but interest in its ap- plication to ML has only recently grown [I] As discussed in section + incentivize fair repre- sentation, we need to match the different condi- tional distributions Pzis. thus minimizing their relative statistical distance. To achieve this, we propose the addition of a differentiable penalty term based on the Characteristic Function Dis- tance. We refer to the overall approach as Fair- ness matching Characteristic Function (FmCF).
Algorithm 1 FmCF loss term
1: Input: encoder hg, predictor fu. batch
( Mis St) ~ Pxy..)
2: = < ha(x)
3: ú fo(=)
4: for all E S do
5: for all j {l...., ......,. k} do
6: Sample tj ~ Pr
7: sox(l)) e |
8: (2)a(tz) n. E ell)))2))
9: end for
end for
Characteristic Function Distance Given random variable X E Rd, an alternative way to describe the distribution of X is the Characteristic Function (CF) 9x IR" + C. Denoting by Fx the probability measure of X, the characteristic function y is defined as:
k
11: Ler
Σ .ES
- (t))) (
h
1=1
12: L Lolū.v) v) LcF
(x((1== = Ex (e =
(5)
JRERS
The CF closely related to the notion of the Fourier Transform, and inherits several important properties. It always exists, it's bounded [(x(()) < 1, but most importantly, there is one-to-one correspondence between probability measures and CFs [33). That is, given two random variables X and Y. with probability measures Fx and Py then it holds that:
x PY
Px= Py.
(6)
The CF can be used define a distance metric between probability measures M Given two random variables X E Rdd Y E Idd with probability measures Px and IPy, the CFD is defined as:
CFDè, (Px,Py) Er [1xx(T) (T))
The value of CFD7, depends on the weighting kernel Pr. When supp(Pr) Rd, then the CFD is strict, the sense that CFD;, (Pxx, Py) = 3 Px Py Common choices for Pr that ensure this property are the Normal and Laplace distributions.
5

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4u-
2 p -0.50
0.00
68
66
00
20-28
80
P
PXX
Optimal Logistic Regression
Figure 2: The closer B[X 0J is to E[X y = 1]. the less predictive power the feature has, causing
the LR coefficient approach
Derivation of Penalty Term Instead of matching Pzis each other, each I Pzis is matched independently to common target distribution. This simplifies and stabilizes the training procedure.
In this work, standard Normal distribution the target, although in principle, any distribution could be used. The CF of the standard Normal distribution PoNr is given by:
1is1 (( dx
( 'ZER (2x)
The CFD between I22s and the target distribution can be estimated using Monte Carlo sampling. Given a batch i.i.d. samples 21...., 2n taken from P2is and .1,...,4y taken from the weighting kernel P the CFD can be approximated as:
CFD (Pzis,
Σ - 2Z st,(t
where (t) = E (ttees denotes the empirical CF. The total penalty term is obtained by summing the CFD estimate across all different sensitive groups S
C
(or(tt)) - (t)ss(()))
(10)
SES
3=1
Comparison to Adversarial Learning and FNF Similarly to FmCF, Adversarial Learning offers highly specialized representations; however, they have been shown to provide only an illusory sense of fairness [37 52 45), primarily due to the inherent characteristics of the underlying optimization problem. In contrast, the proposed approach seeks to learn representations that are strongly predictive of Y while minimizing information about S, without the need for adversarial losses.
Compared FNF 2 FmCF offers multiple advantages compared to Normalizing Flows (NF). Firstly, NFs require carefully designed architectures. To efficiently compute the Jacobian determinant, only specific layer structures are suitable. Additionally, to ensure invertibility, the dimensionality of the input vector X must be preserved throughout the transformation. This constraint makes NFs computationally expensive for high-dimensional data. In contrast, for FmCF, any function approximator can serve as encoder. This architectural flexibility enables the learned representation to focus solely on the information pertinent to downstream task (e.g.. classification), Moreover, FmCF eliminates the need for sensitive information during deployment. Since the classifier internally debiases X, it does not require access to the sensitive attribute S at test time. On the other hand, FNF not only necessitates access to the sensitive attribute but also requires training separate Normalizing Flow for each sensitive class, significantly increasing training costs. This distinction particularly important for minimizing potential discrimination and addressing concerns related to the explicit collection of sensitive data.

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Fair Classification matching Sufficient Statistics
The method introduced in section d provides versatile framework for learning specialized representations Z from X independent from S for a general task. However, in practice, most
Algorithm 2 FmSS Training
1: Input: encoder he, linear regression Ss, batch
B { yd, 84) ~ Px,,s))
faimess-aware scenarios involve classification
2: t hl(z)
[50 [54]. By simplifying the approach, tailor-
3: y + fn(z)
ing it to fair classification, we can provide prov-
4: for all S do
able post-hoe fairness guarantees. Furthermore,
5: o² t Var Z S= s]
we can also make it computationally cheaper, relaxing the need for Monte Carlo estimate of the CF.
6: M, E S =
7: end for
8: LKI.
Σ 10 + n log o
Connection to Distribution Moments The penalty term introduced in section 4 involves sampling multiple points / ~ Pr from the weighting kernel where the empirical CF is evaluated at. The number of points needed for reliable estimate of the CFD grows rapidly in the number of dimensions [
9: -+ Lo( LKL
In this section, propose alternative to reduce the number of points needed for the evaluation. Consider the multidimensional Maclaurin expansion of the CF:
Px
8
(x(() ~
(1I)
00
1=
R=0 S.t.
Assuming that xx is analytic, matching the derivatives evaluated at = 0 would be sufficient to ensure faireess, without the need for sampling from Pr. There exist statistical tools to estimate the full series (appendix A.3): however, they might lead to strong instabilities in training.
A pragmatic approach is to truncate the expansion to some order N. This objective does not guarantee that the two distributions are exactly equal; however, we will show that truncating to N sufficient to ensure fair classification.
In fact, there is direct connection between the derivatives of the CF and the moments of distribution.
Theorem 5.1. The n-derivative of the CF evaluated at = 0 is related to the n-th moment of the distribution:
0 0( .... 0t [k.]]
=
Ex [ [ky] ... X(K.]]
(12)
Similarly, the n-th empirical moment is equivalent to the n-th derivative of the empirical CF.
Simplified Penalty for Classification A generic MLP with M layers used for classification can be hor),, and final layer of LR g Therefore, by applying a fairness penalty to 2, we can restrict the analysis to the family LR classifiers.
interpreted as M 1 layers of encoder
There exists an optimal condition under which LR is provably fair. In particular, if the first moment [ S] is the same for all E S, then such representation has no predictive power for LR (i.e. = 0) in predicting S.
Theorem 5.2. For a representation 7 E R.d and a binary sensitive attribute (0, 1), the optimal LR classifier og. with S as target invariant to Zμ when EEzs [Zpi] = 1Ez [Zn]]
Theorem 5.2 shows that, building an intermediate representation Z with Ezis [Z] = Ez [Z] is sufficient ensure that LR models do use information about the sensitive attribute S.
However, it possible to derive an even stronger guarantee, analyzing what happens for Ez [Zp (S] Ez [ In particular, theorem 5.3 highlights that the mean and variance of Pzis are enough to fully characterize the predictive power of feature.
7

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Theorem 5.3. The predictive power of a feature 2 goes to zero the more its conditional expectation is close to the marginal one. That is, given Q
E21s
2nn
B2
E2 (4
lim S1 = 0 2-+0
(13)
One empirical example of the consequences of theorem 5.3 is shown in fig. where Gaussian and
mixture of Gaussians are used to model TP szs and E Z/s gradually converge the same value, while the variance is preserved.
Similarly to section 4 we adopt Gaussian target distribution for F2is This is partieularly compelling choice from the lens of the maximum entropy principle
The fairness penalty term can be obtained from the Kullback-Leible (KL) divergence between standard Gaussian (o 1) and ((, o²), which is given by
1 (o² + x² log o2)
KL(P, σ)
Accordingly, the total penalty term can be obtained as
SS
Σ
|ar[ S
+ E = s] - log (Var | =+]][l-
(14)
#ES
where Var[Z S) and [Z denote the empirical moments.
Fairness Guarantees This approach fundamentally departs from popular methods in the literature, which primarily focus on minimizing A(Pz)s=o- Pz]s=1), by permitting the sensitive attribute $ to be encoded within the learned representation. Crucially, it ensures that S is embedded in manner that renders it provably inaccessible to the classifier being trained. Leveraging the convexity of LR, can precisely quantify the e-fairness of given representation thus establishing that logistic classifier Tg (I) can extract more than an amount of information regarding the sensitive attribute s. This constitutes a substantially stronger guarantee than adversarial evaluation, which lacks provable tightness and robustness.
Experimental Evaluation
Sensitive
Tassk
FNF offers the strongest theoretical and em- pirical guarantees, making it the current state of the art in fair representation learning. Therefore, primarily evaluate the performance of our proposed methods on the same suite of bench- marks, while also considering additional meth- ods and datasets for a more comprehensive anal- ysis. In all tested datasets, both proposed ap- proaches match or surpass the performances of state-of-the-a approaches in terms of accuracy, while providing fairer representations. Surpris- ingly. even though the approach presented in section[]] guarantees against the worst-case linear attacker, it still holds good performances against adversarial evaluations using deep neu- ral networks. Other results on different setups can be found in appendix A.5] where we addi- tionally report more Pareto plots using different hyperparameters for both FmSS (section 5 and FmCF (section
AdBIT
Gerrman
S
$=1
Y=
Y-1
Figure 3: Latent distributions of TPzi and PzY obtained using FmCF
When constrained by fixed first and second moments, the Gaussian distribution uniquely maximizes entropy.

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0.50
0.80
0.75
0.75
0.70
0.65
0.60
0.60
0.6
0.7
0.8
0.50
0.52
0.54
0.56
0.58
0.60
Adversarial Accuracy
Adversarial Accuracy
Compas
(Ours)
Figure 4: Pareto front between FmCF and FNF comparing task accuracy and fairness (95% confidence intervals from 10 seeds).
Adult Dataset This dataset is the most fun- damental benchmark in fairness, as almost
Table Comparison on Adult dataset.
all methods in literature evaluated it
Adv. Accuracy
28 35] It requires predicting the income of person without considering their gender
Model
Acc.
E g
9 4 g
14 For this dataset, we use the same setup as approach both on classi- fier in the same family as the classifier used for the task, and from family of much more expressive ones, thus deeper and wider. This allows for a much tighter evaluation of the sta- tistical distance A (P 2 s- P =))-1). However, appendix AA.5] we compare other baselines that only offer guarantees on the trained family of functions. For the proposed methods in table J E the class of L.R classifiers, while g are deep MLPs (see appendix A.7 for details). By construction, FmCF should be evaluated for g g g. while FmSS for g E
Random
75.20
50.00
50,00
2. by evaluating
Forgrtttg
28
85.99
66.68
74.50
MaxEnt-ARL.
41
84.80
69.47
85.18
LAFTR 33
86.09
72.05
84.58
FNF 2
84.43
N/A
59.56
FmCF (ours)
85.01
54.92
56.64
FmSS (ours)
85.02
51.70
57.47
German dataset The German dataset has been explored in the literature under different settings. In particular, (37) considers gender as the sensitive attribute, with the task of predicting income based on set of personal features. To ensure a fair comparison, we evaluate our approach against several existing methods using the same exper- imental setup as in [37], as presented in table 2 The proposed method outperforms the other approaches both in terms accuracy and fairness, showcasing the effec- tiveness of the novel penalties. Further comparison under sensitive attributes is presented appendix A.3
Table 2: Comparison on German dataset.
Model
Acc.
Adv. Acc.
Random
69.00
70.00
VFAE
332
72.00
71.70
CIAFL 32
69.50
81.10
IRwAL 37
71.00
69.80
FmCF (ours)
74.60
69.00
FmSS (ours)
74.20
69.00
Comparison with FNF 2 Since 2 achieves the best
balance between fairness and accuracy, in section 6 we compare proposed method from section against all datasets used their evaluation. These include four distinct datasets: Crime and Health, which contain continuous features that NFs can handle directly, and Compas and Adult, which include discrete features requiring separate handling by Normalizing Flows.
The plots demonstrate that our proposed method consistently outperforms matches 2 with the added advantage of not requiring the sensitive attribute during evaluation. Additionally, in appendix A.5] we report the Pareto fronts obtained by varying the latent dimension size, highlighting the impact on performance.
Visualization of Learned Representations order to have better sense of what the CF approach presented in section 4 is learning, visualize the learned representation in fig.

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Indeed, it can be seen that for both datasets, the distribution Pzis, which is the distribution of interest evaluate the fairness of representation. very close being a Gaussian. Thus, the CF is effectively enforcing the similarity to such distribution. Instead, P Pzly, which instead is the distribution of interest learn the downstream task does not resemble a Gaussian distribution, and in particular, while P Z|s=0 Ri s= = = & =1
7 Conclusions
In this work, we introduce novel framework for learning fair specialized representations using CFD, addressing key limitations of existing adversarial and normalizing flow-based approaches. By leveraging CFs, our method ensures stable and efficient fairness by minimizing sensitive information leakage while maintaining high predictive performance. We also present simplified version of the framework that allows for certifiable bounds on the amount of sensitive information used by downstream tasks. Experimental results show that approaches consistently outperform match saate-of-the-eee methods in terms of the fairness- fairness-accuracy trade-off across a range of benchmark datasets. Unlike many existing approaches, at inference time, making it more practical for real-world applications. Our approach opens a new avenue for research in fair representation learning by offering principled, non-adversarial alternative to existing methods. However, despite its simplicity and effectiveness, the proposed framework has certain limitations. Similar to many fairness-oriented algorithms. it encounters challenges in high-dimensional settings, where the CFD tends to lose effectiveness. Moreover, the moment-basee approach is inherently tailored to classification tasks, leaving opportunities for extending such approaches to regression contexts. These limitations underscore the need for further research to improve the scalability and robustness of our method in such scenarios. Additionally, current fairness benchmarks largely focus on low-dimensional datasets, limiting the ability to comprehensively evaluate and compare methods in more realistic and complex settings. Developing more sophisticated. high-dimensional fairness datasets would greatly enhance the evaluation of approaches like ours and drive further innovation in the field.
method does not require access to sensitive attributes
Broader Impact
This research utilizes the Adult, COMPAS, German, Crime, and Health datasets, each of which is extensively acknowledged as a benchmark in the domain machine learning fairness. Our study conducted with stringent adherence to ethical standards and commitment to transparency. By meticulously employing responsible methodologies, we strive to make substantive contributions to the advancement of Al ethics. This work underscores dedication to fostering fairness and promoting socially responsible practices within the broader machine learning community.
Acknowledgements
We thank the anonymous reviewers for their insightful comments and suggestions, which helped improve the clarity and quality of this work. We additionally thank Alessandro Fabris and Davide Dalle Pezze, for the valuable early-stage feedback and discussions on this paper.
References
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A
Appendix
This appendix provides additional theoretical derivations, algorithmic details, and extended empirical results to support the main paper. It is organized follows:
Appendix A.1
Complete derivations for Theorems L.1 5.2 and 5.3
Appendix
Analysis of MMD, its relation characteristic funtetioo distances, and links to
adversarial representation learning.
Appendix A.3
estimator for unbiased 1opendix/A.4 Formal definitions and commentary on Demographic Parity, Equalized Odds, and other group Saroup-fauresss criteria.
Discussion of full noment-matchiin the series truncation, and practical considerations.
Appendix A.5 dimensions, and robustness checks.
Extended comparisons against baselines, Pareto-front ablations over latent
Detailed descriptions of all benchmark datasets, preprocessing pipelines, and protected attribute encoding.
Appendix A.6
Appendix A.7 configuration.
Implementation specifics, optimizer settings, hyperparameters, and hardware
A.1
Proofs of Theorems
Proof. of theorem
Taking the derivative inside the expectation
0m 4x
O [())
(15)
01 p. @t...))
Ttpa).... ..... @t
gmel(Xit)
Ex
(16)
ailk.1 p1
Ex
j" X(Kaa ... X Xin.je
(17)
Evaluating both sides at 0 concludes the proof:
Proof. of theorem 5.2
Defining 2[o] = 1, the derivative of the logistic function can be expressed as
(18)
The derivative of the log-likelihood function in eq. can be expressed
(B)
E Ex, (Y log(as(X)) + (1 Y)log( xx(())
(19)
(x,(X) - (1 Y) (1 - (-o 0311 --Ex,y Y xx(X) 0Bg
dos(X)
(20)
=-Ex,y [Yz (1 -a(())) - -11 Y) 2 [3(x)]
(21)
=-Ex,y [mn] (Y -g((x))
(22)
At the optimum fr, the gradient of the log-likelihood is zero. Considering the partial derivative with respect 310]
(LL)
-Ex, ( -3-- (X)]
(23)
03,0
--Ey 1 Ex 10 (X)
(24)
14

---

showing that By [Y] = Ex lag- (X)].
Considering the partial derivative with respect to S/ for + 0
0 (B*) -Ex,y [x (Y 0z [X]]]
(25)
1811
= =Ex ( 03- (X) By [Y Exjy [xa]]
(26)
Substituting Ex loo- (X)) By [Y] and the hypotesis Exty [(ii]] Ex [1aa]
ACLR
Ex [X]] og. og. (X)] Ey Y Ex [
(27)
03
Ex [X og. (X)] - Ex la (X) Ex [11]]
(28)
Cov (3 03 (X))
(29)
Unless [[s] is constant, the condition Cov (xi Øg. (X)) = is obtained for S/a 0, making
0g (X) independent of Xij--
Proof. theorem 5.3
Following the same steps of the proof for theorem 5.2 we can show that Ex [X)] Ey [A] and
Ex [X 0-- (X)] Ey [Y Exjr [[]]]-
Therefore, the correlation Corr (X1i a. (X)) can be expressed as:
Corr (X1j: (X)) = Ex [(x
Ex
(xe))
(o (X) Ex [o (X)I)
(30)
Ex
X7 [1]
Ex
o3 (X)
Ex go (X)
-Ex
[X
Ex (0 (X)
(31)
Ex
[7]]
Ex
[o3-
Using the triangle inequality Ex (x)( Ex [(4] a-- (X)] Ey [Y Exix [x]]]:
Ex [o)(X)] and substituting xx(X) By [Y] and
Ex [Xioo (X)]
Ex
X 11
Ex
[x3-(X)]
Corr (X (x)) s
(32)
Ex
X2 [1
Ex
[0 (X)
Ey Y Ex
[X
[1]]
Ex
[
By [Y]
(33)
Ex
X2
Ey [Y]
Ey YY (Exx
[xp]
Ex [xx])
(34)
Ex
x
Ey
[Y]
|xxx
[(]]
Ex
xx
Assuming Ω
can bound the correlation with
Ex
x2
Corr (X18300- (X)) S Ve.
(35)
Showing that lima-p Corr (X11:0s-(X)) limg-+ [7) 0
When X13 is not constant, this also implies
15

---

A.2 Connections to Maximum Mean Discrepancy
The Maximum Mean Diserepancy (MMD) is popular distance measure between distributions. Given a Reproducing Kernel Hilber Space (RKHS) H. it is defined as follows
MMDx (P Py)
sup I
E [f(x) - E [)())
(36)
Connections from Characteristic Function can be expressed in terms of the CFs (8.
For random variables E R., Y E R., the MMD
MMD (Ixx / [ox(t) (t)(
(())
(37)
where the weighting function (e(t) the inverse Fourier Transform of the kernel of H. When w is a probability density function, then the integral can be interpreted as an expectation, and
MMD# (Px, PPy) CFD,, (Pxx Py)
(38)
Connections from Sufficient Statistics The formulation presented in section 5 shares some similarities with Maximum Mean Discrepancy (MMD) used by VFAE. Indeed, MMD states that given two distributions, they are equal as long as the maximum distance between their expected value under transformation is zero.
Thus, if MMDH IPy which tries to find f by means of a min-max optimization. However, as shown by (2), Adversarial Learning fails as finding such f extremely hard, due the fact that the two distributions are constantly changing based the current f. Therefore, if MMDx (Px, Py) * 0. no provable guarantee on the fairness can be given. Instead, if f in the family of LR, theorem 5.3 shows that the second moment bounds the predictive power of such representation.
PPy
then Px = Py. This indeed exploited by Adversarial Learning,
This is weaker condition than the one that (2) Adversarial Learning, CVAE, uses learn, by matching the distribution. Indeed, Pz|s= = Pz = Z S = E|Z S = 0] but not the other way around.
Yet, given LR has convex loss function, the optimal adversarial classifier can be provably trained to convergence with second-order optimizer, thus giving guarantees the amount of sensitive S that can at most be used in the classification.
A.3 Matching All Moments
The approaches presented in sections and 5 can be reconducted to matching the moment generating function, or directly the moments a distribution. Indeed, matching all moments of two distributions with bounded support ensures that they equal almost everywhere. Even though it is trivial to force the support of Z to be bounded, directly applying moment matching to the distributions IP P21 might needed characterize a distribution, which is computationally
be impractical, as all moments unfeasible. Furthermore, truncating to the k-th moment leads to a biased estimation, and, in the context of this work, potential leakage of the sensitive attribute S.
One powerful tool for unbiased estimation of the complete infinite series of moments the Russian Roulette estimator. Consider infinite series of the form Y >i= Y, The Russian Roulette estimator randomly chooses truncation point N and calculates the partial sum up to N, while adjusting the weight of the computed sum ensure that the estimator remains unbiased:
Y, 2(N
(39)
While the Russian Roulette estimator is unbiased, it can result in high variance if the truncation probabilities are not chosen carefully. Therefore, balancing computational efficiency and variance is essential in practical applications.
Instead, the proposed approaches presented in sections and use a more principled and computa- tionally stable way of matching two distributions.
16

---

A.4
Additional Fairness Metrics
Although various fairness criteria have been proposed, balancing the fairness-aceuracy trade-off remains challenging.
Demographic Parity (DP) is widely used group fairness criterion that requires the outcome of a decision-making process to be independent of a sensitive attribute such as race, gender, or age 18 13 Let hg denote a representation function that maps inputs X to latent space Z he (X), and let Sa denote a classifier applied to this representation, producing predictions Y = (ho (X)). Then, DP requires the probability of receiving a favorable outcome (e.g., Ý = 1) to be the same across all demographic groups É S:
P(Y = 1 =) P) 1) E.
(40)
This states that the positive outcome rate for any group & should be equal to the overall positive outcome rate in the population. Since demographic parity is independent of the ground truth labels, is especially salient in contexts where reliable ground truth information is hard to obtain and a positive outcome is desirable, including employment, credit, and criminal justice [16 23
To quantify deviations from this ideal, we measure the Demographic Parity Difference (DPD). In binary group settings (e.g., involving ad advantaged group and disadvantaged group d), it is defined as:
DPD P(Ý = a)
S d).
(41)
In real-world scenarios, sensitive attributes are often multi-class (e.g., race with more than two categories). In these cases, generalizations for measuring DPD violation include:
The Maximum Pairwise Difference, capturing the largest disparity in positive outcome rates between any two groups:
P P(( = =s,))
DPD = max
(42)
The Average Absolute Pairwise Difference, computing the average disparity across all pairs:
DPD
EE s,ES
P(Ý = I 55 s.) P(Ý 1 S = (j)
(43)
S
Since achieving perfect demographic parity (where all the above differences are zero) can be imprac- tical, the objective is often relaxed to minimizing the Demographic Parity Distance:
A DP
Σ P( = S = s) P(Ý 1) SES
(44)
While demographic parity is often defined in terms of these outcome rate differences, it can also be understood more generally terms of statistical distance between group-conditions distributions of model predictions or representations [35]. Let Zo and Z₁ denote the distributions of these representations conditioned on different sensitive attribute values 0 and respectively. Then, for any measurable test function fa applied to the model outputs, one can define the test discrepancy as:
A DP h4) [fzz] Ez, (Z)]]]
(45)
The statistical distance (or total variation distance) between Zo and Z1 the supremum of this discrepancy over all measurable test functions μ::
Zo. Z1) = sup |E [u( E22 [2,()(]
(46)
It follows that for any fe.
A o ho) < (7,, 71), and A DDP (fo h2) 0 if and only if fe(ho(X)) 1 S, i.e., demographic parity holds.
(47)
17

---

In contrast to Demographic Parity, the Equal Opportunity (EO) criterion incorporates information about the target variable [27] Specifically, EO requires that the true positive rate (TPR) be equal across all demographic groups. Formally:
P(Ý 5 s, Y =1 1) = Y 1) ss E S.
(48)
This fairess criterion is especially salient in domains where false negatives are particularly harmful and reasonably accurate ground truth labels available, such in healthcare, criminal justice, and risk assessment 7 43 38). The corresponding EO distance is defined as:
RO (Jo o ho) = [ 5 =, 0, = 1] Ex (2) S = 1]
(49)
which can be upper bounded by the objective value of an optimal adversary g" trained distinguish between Z Y 1 for different groups (35)
An alternative fairness criterion is Equalized Odds (EOd) (77), which extends Equal Opportunity by requiring that prediction outcomes be conditionally independent of the sensitive attribute given the true label. Formally, EOd demands that for all S:
P( = 1 Y = = P(Ý = 11 A
(50)
and
P(Ý = Y = = - = V
(51)
These constraints ensure that both the true positive rate (TPR) and false positive rate (FPR) are equal across all groups [56] 25). This criterion is particularly applicable in decision-making contexts where reliable ground truth labels available and both false negatives and false positives incur significant societal or personal costs 12 533. The EOd distance is defined as:
BOd (fe o he)
= (Z) S = Y, 1| - Ez]ff(Z) S = 1, = 1]]l
+ +| S 0 0] [Z][((Z) 1. O]l-
(52)
A.5 Additional empirical evaluations and ablations
In table Ba we compare our proposed approaches to other baselines on the Adult dataset, with the
same setup in 33.. Given that distinction is made the family of functions used for the evaluation, we report the Adv. Accuracy of FmCF and FmSS approaches using a deep and wide where compare out methods the
MLP. The same holds true for the results reported in table 3b
same setup from 47
(b) Comparison on Adult dataset using the setup from 47
(a) Comparison on Adult dataset using the setup
from 37
Model
Acc.
Adv. Acc.
Model
Acc.
Adv. Acc.
Random
70.0
81.0
Random
75.2
67.5
CSN
47
73.1
81.3
VFAE 32
84.2
88.2
CIAFL [5
73.6
81.1
CIAFL [1
83.1
88.8
VFAE 32
72.8
81.2
IRwAL 37
84.2
77.6
FRTrain
40
72.7
80.9
WassDB
29
72.8
81.1
FmCF (ours)
85.0
67.8
FmSS (ours)
85.0
67.9
FmCF (ours)
74.1
81.1
FmSS (ours)
74.2
81.1
In figs. to 8 we report the tradeoffs between adversarial balanced accuracy and task accuracy. We evaluate FmCF section Jagainst an adversarial MLP as reported in appendix A.77 Instead, we evaluate FmSS section 5 both against an MLP and against a LR. In particular, for all images, in black reported the performance of Fair Normalizing Flows [2]. while in blue reported the performance using I dimension, in orange 2 dimensions, and in green 3 dimensions. We observe that the lower the dimension, the easier it is for the penalties to be effective, giving surprising performances for extremely low-dimensional latent spaces. Notably, such characteristic is not limiting for the
18

---

downstream task, as even in the nowest-dimension setting, both approaches outperform all the baselines.
3.00
0.81
T0.ck Accuracy
0.00
0.78
2.08
1.00
1.55
0.72
2.8
2.0
05
28
10
Adversarial Accuracy (MLP)
Adversarial Accuracy (MLP)
Adverserial Accuraty (Linear)
Figure 5: Pareto varying latent dimension on Health dataset.
2.00
376
Thick Accurory
348
3.00
0.64
3.6
0.64
0.54
3.64
0.52
0.53
2.40
0.53
3.58
0.55
335
6.52
6.51
5,40
6.14
6.5
6.51
0.56
5.6
Adversarial Accursty (MLP)
Adversatiel Accuracy (MLP)
Adversarial Accursty (Linear)
Figure 6: Pareto varying latent dimension Compas dataset.
2.0
006
Th Accurcyy
2.75
0.72
0.9
3.00
0.55
035
Advarsorial Accuracy
Adversarial
(Linear)
Figure 7: Pareto varying latent dimension on Crime dataset.
2.24
0.54
Tock Accurary
2.00
2.52
0.0
3.75
5.85
089
076
1.90
032 0.34
0.30
0.50
0.54
0.0
8.52
0.44
830
0.00
3.52
0.64
3.2
056
034 0.0
Adversarial Accuracy (MLP)
Adversanial Accuracy (MLP)
Adverserial Accuracy (Linear)
Figure Pareto varying latent dimension on Adult dataset.
Furthermore, we evaluated the effects of batch size on the reliability of the CFD estimation. confirmed by appendix A.5 larger batches yield more stable and reliable CFD estimates. The variance of the estimates decreases consistently with increasing batch size, aligning with the expected theoretical scaling of (
19

---

Table Standard deviation of FmCF Penalty estimator,
Batch Size
Adult
German
Compas
Health
8
0.1150
0.1252
0.0924
0.1004
16
0.0596
0.0854
0.0361
0.0422
32
0.0164
0,0172
0,0129
0.0121
64
0,0063
0,0056
0.0052
0.0054
128
0,0026
0.0024
0,0022
0,0027
256
0.0016
0.0004
0.0010
0.0015
512
0,0007
0,0000
0.0007
0.0009
Given that all benchmarks considered thus far pertain to classification tasks, example demonstrating the applicability of FmCF to alternative problem settings. Specifically, in fig. conditional convolutional autoencoder is employed to learn latent representation of the MNIST handwritten digits dataset. Subsequently, the loss function described in algorithm iis utilized penalize the divergence between Pz and Gaussian distribution. When this condition satisfied, it ensures that Z TT S, indicating that the latent representation Z encodes information about X that is independent of S. Consequently, when the model is prompted to reconstruct an input with a different label, it retains the essential characteristics of the original input while generating a digit consistent with the newly specified label.
present simple
Indeed, the decoder able to generate images preserving characteristics such as rotation, while only relying the fed label for the kind of digit to show.
O
/
2
3
4
5 6
7 8 9
10
2
3
4
5
6
7
8
9
2012
5
8
9
3
0
2
3
4
5
6
8
9
4
O
1
2
3
4
5
6
7
8
9
5
O
2
3
4
5
7
8
9
6
0
2
3
4
8
9
7
O
2
3
5
6
7
8
9
8
5
6
8
9
9
2
3
4
5
6
7
8
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Figure 9: Latent representations Z are extracted from the left-most column and combined with arbitrary target labels generate stylistically consistent images of different classes, shown the right.
For more details on the datasets, please refer to appendix A.6
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A.6 Datasets
We utilize six well-known datasets in our study, sourced from both the UCI Machine Learning Repository and other publicly available resources. These datasets include Adult, Crime, Compas, Law School, Hcalth, and the Statlog (German Credit Data) dataset. Each of these datasets presents distinct challenges and characteristics relevant to faimess and predictive modeling. Below, we briefly introduce each dataset and outline the preprocessing steps employed:
Adul The Adult dataset, also known as the Census Income dataset, originates from the 1994 Census database and is available through the UCI repository [17]. It contains 14 attributes, including age, workclass, education, marital status, occupation, race, and sex. The prediction task is to determine whether an individual's income exceeds $50,000 per year. To facilitate modeling, we discretize continuous features and retain categorical variables related to demographics and employment. Sex is treated as the protected attribute in fairness analyses.
Crime The Communities and Crime dataset combines socio-economic data from the 1990 US Census, law enforcement data from the 1990 LEMAS survey, and crime data from the 1995 FBI UCR. The goal is to predict whether the violent crime rate of community is above below the median. We utilize attributes such as race percentages, income levels, and family structure indicators. Race is designated the protected attribute, derived from the proportions of racial groups within each community.
C The Compas dataset contains data related to criminal history, jail and prison time, demographics, and COMPAS risk scores from Broward County (2012-2013) [5). The prediction task involves forecasting recidivism within two years. Key attributes include age, prior count, and charge degree, with race heing the protected attribute. We preprocess the dataset by discretizing continuous variables and retaining categorical ones, critical for risk prediction.
H The Health dataset was part of the Heritage Health Prize competition on Kaggle and contains medical records of over 55,000 patients. We focus the merged claims, drug count, and lab count attributes while removing personal identifiers to ensure privacy. Age is treated as a protected attribute, divided into binary groups: above and helow 60 years. The primary prediction task to assess the maximum Charlson Comorbidity Index, reflecting the long-term survival prospects of patients with multiple conditions.
Law Schoo ! The Law School dataset consists of data from admissions cycles between 2005 and 2007, covering undergraduate GPA, race, gender, and residency status. To enhance privacy, data has been aggregated where necessary. We consider race as the protected attribute, binarizing it into white and non-white categories. The main task is to predict law school admission outcomes.
100,000 individual applications. Attributes include LSAT scores,
Germar The German Credit Data dataset, sourced from the UCI Machine Learning Reposi- tory 17]. consists of 1,000 records of credit applicants. Each instance is labeled as either "good" "bad" credit risk. The dataset contains 20 features, including age, credit amount, employment status, and personal status, among others. Both categorical and numerical data are present, requiring careful preprocessing. We encode categorical variables using one-hot encoding and normalize numerical features. The protected attribute for fairness evaluation in this dataset is age, segmented into groups representing different age ranges.
However, in (2), the Lawschood dataset is also used to show the effectiveness of FNF. Through the extensive research, did not find any publicly available version of such a dataset, thus our approaches are not evaluated on it.
A.7 Training details
All experiments reported in this paper were implemented using PyTorch. The models were trained on a server equipped with an AMD Ryzen Threadripper PRO 5995WX CPU (64 cores, 128 threads),
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512 GiB of RAM, and three NVIDIA RTX A6000 GPUs. Despite this powerful setup, none of the models utilized more than GiB of VRAM during training.
All reported results are averages over 10 different seeds of the proposed approaches. No extensive hyperparameter tuning was performed for any of the reported results, underscoring the effectiveness of the proposed methods. All MLPs used for adversarial evaluations, encoding, and classification consist of four layers with 64 neurons each. Additionally, larger MLPs for fairness adversarial evaluations, configured with 128 and 256 neurons, were tested but resulted in poorer performance.
The Adam optimizer [30] was employed for all training sessions, with a learning rate of 0.0003. Training was conducted for 100 epochs, incorporating L2 regularization with a weight penalty of 0.0001 to mitigate overfitting. Since accuracy and faimess often involve trade-off, early stopping was not applied. Instead, the model from the final epoch was used for evaluation.
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Reviews:
[{"review_id": "eDj0ozjxtq", "reviewer": "Reviewer_KSAf", "summary": "The paper develops a method for fair representation learning which focuses on \"specialized\" (i.e., task-specific) representations. It aims to avoid the issues with previous approaches such as fair extensions of VAEs, GANs, and normalizing flows. Basically, the approach is to incorporate a penalty term based on characteristic function distance. The method is general, but the case of classification is specifically handled. Some theoretical results are provided, including fairness guarantees in the case of logistic regression. Experiments verify the efficacy of the method.", "questions": "- Does it matter that the P(Z|S) distributions are matched to a common target distribution instead of each other? How does the choice of target distribution affect the results?\n\n- Can you please more fully explain the relationship to Ansari et al. (2020) [1]? In what ways does this work build on that paper, and in what ways is it different?\n\n- How well does the method work with small datasets? Does the characteristic function distance strategy suffer from statistical inefficiency and biased estimation, as suggested by its relation to the method of moments?", "limitations": "A more thorough discussion on the downsides of the characteristic function distance methodology vs. other fair representation learning techniques is warranted. The paper spends a lot of time talking about the downsides of other methodologies, but does not dedicate much time to discussing the downsides of their own one (except briefly in the conclusion).", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 4, "strengths_and_weaknesses": "*Strengths*\n\n - The proposed method is quite novel, particularly in the context of fairness. The use of characteristic function distance is an emerging approach to model fitting which has previously been employed for GANs [1], and has some interesting relationships to maximum mean discrepancy (MMD), as well as the method of moments.\n \n - This work could impact the field by providing a new paradigm for fair representation learning.\n \n - The proposed method is algorithmically simple (even while being technically deep), which is elegant and increases the chance that it will be widely used.\n\n - The technique does not need access to the sensitive attribute during test time, which is useful for privacy.\n \n - This manuscript did a good job of contextualizing the research in terms of competing approaches for fair representation learning.\n\n\n*Weaknesses*\n\n - I would have liked to have seen an exploration of sample size versus estimation/fairness performance, with comparison to baselines. Since the proposed characteristic function distance is closely related to moment matching (Theorem 5.1), it may share some of its downsides such as statistical inefficiency and biased estimation with small sample sizes. (Perhaps the results would be unflattering, which could be why this experiment was not reported!)\n \n - The paper could be improved by adding more discussion on the method's potential practical value for concrete real-world applications.\n \n - This research seems to be heavily inspired by Ansari et al. (2020) [1]. The paper should further discuss its relation to that work, as well as other papers that use characteristic functions for model estimation. As well as giving proper credit to prior work, such a discussion would help contextualize the work within that subarea, and explain the extent to which characteristic function distance is an established and/or emerging estimation principle.\n\n\n*Minor suggestions*\n\n- Fix subscripted \"(\" in Equation 2", "was_revised": true, "final_justification": "I have read the authors' response and the other reviews. The authors provided a thorough response which addressed my questions and the points raised in the review. The scores and opinions of the other reviewers are in alignment with my own. My positive view of the work remains unchanged. Therefore, I have kept my rating the same.", "rebuttal": "We thank the reviewer for the thoughtful and detailed feedback. The points raised regarding statistical efficiency, real-world applicability, and connections to prior work are important and appreciated. Below, we provide clarifications on each and outline how we plan to incorporate these improvements into the final version.\n\nPlease note that, due to the updated NeurIPS guidelines, we are no longer permitted to upload supplementary materials or a revised version of the paper at this stage. We hope that the clarifications provided here sufficiently address your concerns.\n\n----\n\n### Weaknesses\n\n_\"I would have liked to have seen an exploration of sample size versus estimation/fairness performance, with comparison to baselines. Since the proposed characteristic function distance is closely related to moment matching (Theorem 5.1), it may share some of its downsides such as statistical inefficiency and biased estimation with small sample sizes. (Perhaps the results would be unflattering, which could be why this experiment was not reported!)\"_\n\nWe did not explore such a possibility; all reported results have been achieved with minimal changes to the networks and optimizers. In particular, all results are achieved using a fixed batch size of 128 for all datasets. Thus, for each protected class, about 64 samples are used for the estimation, but the exact number does depend on the class imbalance and fluctuates across batches. \nWe do expect the estimation to become unreliable with very low sample sizes. However, we believe that $n>50$ is a reasonable assumption to make in practice.\nRegardless, we agree this is an interesting direction that should have been properly explored. We have run a small-scale evaluation of the accuracy of the CF estimation, varying the batch size, and have confirmed that it exhibits a similar scaling behavior to the estimation of moments (i.e., variance scaling as $O\\left(\\frac{1}{N}\\right)$ etc.). Yet, given the impossibility of uploading additional media during the review process, it is relatively unfeasible to share the details and results properly. We can provide the actual numerical results in a table in a future response (due to character limitations). In any case, we will make sure to address this point in the revised paper if it gets accepted. Thanks for pointing this out.\n\n_\"The paper could be improved by adding more discussion on the method's potential practical value for concrete real-world applications.\"_\n\nThanks for the suggestion. We totally agree with it, though the page limit did not allow us to discuss this aspect further in the main text. However, given that you share our feelings, we will make sure to discuss this aspect further in the camera-ready version, given the additional page allowed for integrating the reviewers' suggestions.\n\n_\"This research seems to be heavily inspired by Ansari et al. (2020) [1]. The paper should further discuss its relation to that work, as well as other papers that use characteristic functions for model estimation. As well as giving proper credit to prior work, such a discussion would help contextualize the work within that subarea, and explain the extent to which characteristic function distance is an established and/or emerging estimation principle.\"_\n\nIndeed, it is. We agree that section 4 does not properly discuss the relation to prior literature and that the paper would greatly benefit from such inclusion. A similar point was also raised by reviewer a3eD. For the camera-ready version, we will expand the section with a discussion on statistical testing based on CF and its use in the field of ML.\n\n### Questions\n\n_\"Does it matter that the P(Z|S) distributions are matched to a common target distribution instead of each other? How does the choice of target distribution affect the results?\"_\n \nHaving a common target offers many advantages:\n- Makes the approach scalable beyond binary sensible attributes. In the case of n classes, the pairwise interactions to consider are $O(n^2)$ compared to $O(n)$ for a fixed target. Moreover, with a fixed target, the loss can be computed independently for each sensible attribute.\n- Reduces the variance of the estimators. Consider the case of matching the mean of the distributions (but a similar discussion holds also for other moments and for $\\phi$).\nWhen estimating $|\\mathbb{E}[X_0]-\\mathbb{E}[X_1]|$ directly, the variance of the sample means compounds on each other. On the other hand, the variance of the estimation of $|\\mathbb{E}[X_0]-R|$ ($R$ being the reference) only depends on the variance of $X_0$.\n- Forces the network output to remain in a numerically stable range. Indeed, matching the distributions to each other is a scale-invariant objective that could lead to exploding outputs. This is not the case for a fixed common target.\nWe did not explore how the choice of target could affect the estimation. The choice of a Normal distribution seems quite natural and unlikely to be restrictive. Its characteristic function is well-behaved and decays rapidly for high frequencies. \nFurthermore, for the specific case of classification, the choice of a target distribution is not needed at all. \n\n_\"Can you please more fully explain the relationship to Ansari et al. (2020) [1]? In what ways does this work build on that paper, and in what ways is it different?\"_\n\nWhile we developed our approach independently of [1] based on the applications of CF in statistical testing [33], our notation and discussion have been directly inspired by [1]. Both works share the core idea of using ECFD as a differentiable 2-sample statistical test, and thus have some core similarity. Yet our approach deviates significantly from [1] in many aspects. \nThe two applications are different, as [1] handles generative models, while our approach focuses on fair representations. \nMoreover, [1] employs adversarial training to optimize the test kernel, which we avoid in order to simplify the approach.\nAs mentioned in the weakness section, we will add a more in-depth discussion on the relation to the literature on the use of CF in machine learning as a whole.\n\n_\"How well does the method work with small datasets? Does the characteristic function distance strategy suffer from statistical inefficiency and biased estimation, as suggested by its relation to the method of moments?\"_\n\nMany of the datasets tested are relatively small. For instance, the German dataset is composed of only 1000 samples, of which only 600 are used for training and the rest for validation and testing. Both approaches achieve comparable or better performances to other baselines, empirically validating their effectiveness.\nFor even smaller datasets, the samples might not be a good representation of the underlying distribution, which could indeed lead to poor performance. However, we believe this failure case is not a limitation of the approach itself.\n\n### Limitations\n\n_\"A more thorough discussion on the downsides of the characteristic function distance methodology vs. other fair representation learning techniques is warranted. The paper spends a lot of time talking about the downsides of other methodologies, but does not dedicate much time to discussing the downsides of their own one (except briefly in the conclusion).\"_\n\nWe believe that the acknowledgement of limitations is a fundamental part of scientific progress. We will make sure to expand the discussion in the camera-ready version, with particular care on the specific points that have been highlighted in the reviews.\n\n----\n\nWe appreciate the reviewer’s comments and suggestions. We will make sure to reflect the necessary clarifications and improvements in the camera-ready version, if the paper is accepted."}, {"review_id": "ELc3kBanc1", "reviewer": "Reviewer_a3eD", "summary": "The area of fair representation learning (FRL) seeks to learn a hidden representation which “looks the same” with respect to multiple different sensitive attributes. That is, for example, we would like to learn embeddings of individuals such that if a downstream user had access to our embedding for a particular input x, they would not be able to determine (with good accuracy) whether the user can predict what sensitive attribute x belongs to. This is what the current paper calls adversarial evaluation.\n\nIn addition, FRL methods may fall into two categories: “specialized” or “generalized”. Specialized methods learn representations useful for a particular task or classification problem, while generalized methods seek to learn representations which may be used by any downstream learner on any task, and should allow for fairness guarantees to be inherited.\n\nFair normalizing flows (FNF) is positioned as a state-of-the-art method which achieves this “generalized” FRL. However, the submitted paper claims FNF has various flaws. For example, it requires training a separate model for each group, and also requires the protected subgroup at inference time.\n\nThe paper instead introduces a FRL method based on characteristic functions. Define $P_Z$ as the marginal distribution over the learned latent space. The proposed method, FmCF, seeks to match the conditional distributions for $P_{Z|S}$ for all sensitive attribute values $S$. Intuitively, if the latent distributions are indistinguishable, then the FRL problem is solved. FmCF does this via matching the characteristic functions of the distributions $P_{Z|S}$. Importantly, characteristic functions and probability distributions map one-to-one. Therefore, if we have efficient methods for matching the characteristic functions, then we can solve FRL (equation 7).\n\nWe would like to regularize loss minimization in order to penalize deviation from equal characteristic functions of $P_{Z|S}$. Instead of penalizing this for all pairs of sensitive attributes, the paper suggests instead penalizing the distance of each characteristic function of $P_{Z|S}$ to the characteristic function of the normal distribution $\\mathcal{N}$. This can be done with a montecarlo approach.\n\nIn Section 5, the paper expands on FmCF by introducing a refinement for classification, FmSS. Instead of matching the entire characteristic functions for $P_{Z|S}$, it turns out that for classification with logistic regression, it suffices to match only the first moments $\\mathbb{E}[Z | S]$ in order to achieve FRL. This is possible due to a fact relating characteristic functions and the moments of the distributions they represent (Theorem 5.1). In fact, a stronger result is obtained for the predictive power of a feature $Z_i$ in obtaining the sensitive attribute $S$ in Theorem 5.3. In particular, matching the mean and variance of $P_{Z|S}$ for all $S$ is sufficient to restrict the predictive power of all features.\n\nFmSS utilizes this result and works by minimizing the KL divergence between the standard normal and the empirical conditional distributions $P_{Z|S}$.\n\nEmpirical experiments are conducted in section 6, where the paper demonstrates that FmCF and FmSS both preserve standard accuracy of the original task (relative to existing FRL methods), and decrease the adversarial accuracy of a predictor attempting to infer the sensitive attributes $S$ from the hidden representations $Z$.", "questions": "1. To estimate the empirical CF in equation (9) of $P_{Z|S}$, must we do rejection sampling? Is this why the cost of monte carlo is so high? Then, does the computational complexity scale with the size of the smallest group?", "limitations": "Some drawbacks of FmCF are discussed, but discussion of computational complexity could be enhanced.", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "I know nothing about fair representation learning and am unfamiliar with most (if not all) of the papers cited by this paper. Hence, my confidence is low.\n\nStrengths\n1. The characteristic function matching approach is novel for FRL (to my knowledge) and seems to really simplify the optimization problem in a convenient manner. The proposed methods FmCF and FmSS also require no architectural adjustments, which seem to be quite prevalent in previous works such as variational fair autoencoders and fair normalizing flows.\n2. The fact that one can only care about matching the mean and variance of the conditional distributions $P_{Z|S}$ for classification is quite surprising and seems to simplify the computation of the regularized loss by quite a bit.\n\nMinor Weaknesses:\n1. Are low latent dimensions like 1,2,3 standard for the FRL literature (line 652)? This seems far lower than one would expect to, for example, learn a sigmoid function on. The final hidden layer of NNs is usually much larger. Is this because of computational constraints? Using such low dimensions for the latent representations to me detracts from strength 2. In particular, if we are in only two or three dimensions, it’s not clear that the higher order moments really have anything interesting to say.\n2. Some of the details in table 1 were unclear. For example, was $\\mathcal{G}$ defined earlier? Also, what I don’t think $\\mathcal{F}$ is defined? I could be missing something, however.\n3. Missing discussion of runtime of FmCF and FmSS vs. standard methods like training fair normalizing flows or VAEs.\n4. It seems like FmCF has a runtime exponential in the dimension of the latent space. However, FmSS seems much more efficient, since we are only matching the mean and variance. \n\nThe paper seems very well motivated and presents what seems like a much simpler approach for achieving FRL, and hence, I believe that it will be a valuable contribution.\n\nOther comments:\n1. As someone outside of FRL, I was a bit confused about the distinction between generalized and specialized FRL methods. I think providing some examples of models or tasks in lines 30-36 could improve clarity.\n2. The paper shows that FmCF extends to settings other than classification in Appendix A.5. To my knowledge, this experiment is not explicitly highlighted in Section 4 or anywhere else in the paper. You should mention it because it is an interesting experiment!\n3. Typo equation (2), parens in subscript.\n4. Consider including a down arrow for adversarial accuracy, to show the reader at a glance that it should be minimized.\n5. No legend for figures 5-8 in appendix, although explained in line 652, could be clearer if you included them in the figure descriptions or as a legend in the plots.\n6. Line 240/241: Maybe remind the reader what the $\\beta$ coefficient is again, as you do in the description of Figure 2.\n7. Line 250-252: How did we move from matching only the mean to matching the mean and variance of $P_{Z|S}$? More explanation may be warranted here if possible.\n8. Relatedly, how did we go from matching the mean of $P_{Z|S}$ (line 239-242) to matching the mean of each feature in Theorem 5.2? A bit more explanation here could help bring some clarity.\n9. Period missing end of line 286.\n10. How does this paper relate to prior work with characteristic functions (e.g., citation [1] from the submission)", "was_revised": true, "final_justification": "The authors responded to each of my minor gripes and questions. I think the main strengths of the paper are the ones I stated:\n\nStrengths\n1. The characteristic function matching approach is novel for FRL (to my knowledge) and seems to really simplify the optimization problem in a convenient manner. The proposed methods FmCF and FmSS also require no architectural adjustments, which seem to be quite prevalent in previous works such as variational fair autoencoders and fair normalizing flows.\n2. The fact that one can only care about matching the mean and variance of the conditional distributions $P_{Z|S}$ for classification is quite surprising and seems to simplify the computation of the regularized loss by quite a bit.\n\nThe experiments are also thorough. My confidence is \"low\" only since I do not work in the fair representation area.", "rebuttal": "We thank the reviewer for the thorough and constructive feedback. Several valuable points were raised regarding dimensionality, theoretical clarity, computational aspects, and presentation. Below, we provide detailed responses and clarifications, and we will ensure these are reflected appropriately in the final version.\n\nPlease note that, due to the updated NeurIPS guidelines, we are no longer permitted to upload supplementary materials or a revised version of the paper at this stage. We hope that the clarifications provided here sufficiently address your concerns.\n\n----\n\n### Weaknesses\n_\"Are low latent dimensions like 1,2,3 standard for the FRL literature (line 652)? This seems far lower than one would expect to, for example, learn a sigmoid function on. The final hidden layer of NNs is usually much larger. Is this because of computational constraints?\"_\n\nFRL aims at learning fair representations, with no restriction on them. Depending on the task, the dimensionality needed for the representations might vary. \nThe higher the dimension, the harder it is to ensure the representation is fair, especially for small datasets.  \nThe most common task is one-class classification, which ideally requires only 1 bit of information. \nIn fact, directly matching the distribution of the logits for each sensible group $S$ (which is a one-dimensional representation) would be sufficient in this case.\nIn practice, we do not want to limit ourselves to modelling the logits directly. Instead, we simply focus on learning a low-dimensional representation that is fair, while also being useful for the task. This approach is much more general and can also be applied to multi-class classification, regression, and so on.  \nPlease consider that the common set of benchmark datasets have very low intrinsic dimension to start with. For instance, FNF [2] employs a heavy feature selection step, retaining only 4-6 of the original features with almost no degradation in task performance.\n\n_\"Using such low dimensions for the latent representations to me detracts from strength 2. In particular, if we are in only two or three dimensions, it’s not clear that the higher order moments really have anything interesting to say.\"_\n\nIf the two distributions we are trying to distinguish are both close to a Gaussian distribution, then indeed higher-order moments are not very informative. \nHowever, for two general distributions, high-order moments can be critical in distinguishing the two. This is true even for the one-dimensional case.  \nFor instance, if the two distributions are $x_0 \\sim N(0, 10)$ and the mixture $x_1 \\sim 0.5 N(-3, 1) + 0.5 N(+3, 1)$, both distributions have the same mean and variance, but are clearly extremely different and have large statistical distance from one another.  \nThis example is shown visually in Figure 2, where we also show that despite the two distributions being clearly separable, having the same first and second moment is sufficient to guarantee a fair classification.\n\n_\"Some of the details in table 1 were unclear. For example, was $mathcal{G}$ defined earlier? Also, what I don’t think $mathcal{F}$ is defined? I could be missing something, however.\"_\n\n$\\mathcal{G}$ is defined in line 299/303, and aims at describing whether we adversarially evaluate our approach on a specific class of functions $\\mathcal{G}$ or not, following the convention used by Fair Normalizing Flows. Regarding $\\mathcal{F}$, it refers to a generic family of classifiers. Thanks for spotting it. Indeed, it was not properly defined. Since the two concepts are indeed related, we will unify the notation in the revised version.\n\n_\"Missing discussion of runtime of FmCF and FmSS vs. standard methods like training fair normalizing flows or VAEs.\"_\n\nIndeed, we agree that we should have discussed this aspect more in depth. In fact, our approach does not require additional parts/networks and is computationally cheaper than other approaches. \nFor the SS approach, estimating mean and variance of a batch of $n$ samples that are $m$-dimensional, requires  $4nm$ FLOPs, which is negligible compared to the forward pass of the network. The same is true for the CF approach, where each frequency can be computed with a similar number of flops.\nUnfortunately, it would be extremely hard to make a fair comparison across all the different methods since many require specific architecture to work. \nIndeed, comparing the number of parameters would shadow the computational requirements, comparing the computational requirements would shadow the hyperparameter sensibility of the approaches, and so on. \nFor this reason, we decided to focus mainly on the fairness/task performance tradeoffs of the different approaches. \n\n_\"It seems like FmCF has a runtime exponential in the dimension of the latent space. However, FmSS seems much more efficient, since we are only matching the mean and variance.``` \n\nThe FmCF runtime is (practically speaking) constant in the dimension, and mainly depends on the number of frequencies sampled from the kernel. \nHowever, the variance of the estimator might grow rapidly with the number of dimensions, making the signal weak.\nIn practice, we did not observe this in the experimental evaluation, and the number of frequencies was kept fixed regardless of the dimension. However, we did not investigate this aspect in depth. \nInstead, as you point out, FmSS is not particularly affected by dimensionality, and this is one of the reasons why we specifically propose such an approach for classification. \n\n### Other comments\n_\"As someone outside of FRL, I was a bit confused about the distinction between generalized and specialized FRL methods. I think providing some examples of models or tasks in lines 30-36 could improve clarity.\"_\n\nThank you for the feedback. We will make sure to add examples and more motivations on why somebody would want one or the other.\n\n_\"The paper shows that FmCF extends to settings other than classification in Appendix A.5. To my knowledge, this experiment is not explicitly highlighted in Section 4 or anywhere else in the paper. You should mention it because it is an interesting experiment!\"_\n\nThank you for the nice comment. Indeed, the CF approach can match any distributions. Thus, it's not necessarily tied to fairness (for example, you can build a sort of VAE like the one shown in the appendix). \nThis also empirically shows and validates that the approach proposed is not tied to classification in any way, but rather that it really suits classification tasks.\nWe decided not to mention this aspect directly in the main text but rather to focus on the core contributions. However, in hindsight, we definitely should have explicitly referenced it. \nWe will include a discussion of the use of CF in related works (as suggested by rev KSAf), and we will make sure to also reference this experiment.\n\n_\"Line 250-252: How did we move from matching only the mean to matching the mean and variance of $P(Z|S)$ More explanation may be warranted here if possible.\"_\n\nThe statement is meant to introduce he theorem that follows in line 253. We recognize that the phrasing is misleading, and we will clarify this.\n\n_\"Relatedly, how did we go from matching the mean of $P(Z|S)$ (line 239-242) to matching the mean of each feature in Theorem 5.2? A bit more explanation here could help bring some clarity.\"_\n\nThe two formulations are equivalent. We used the component wise notation in the proof, however we recognize that the theorem statement would be more clear if expressed in vector notation. We will fix this in the revised version.\n\n_\"How does this paper relate to prior work with characteristic functions (e.g., citation [1] from the submission)\"_\n\nTo our knowledge, this is the first application of CF to fair representation learning. However, there are several approaches that employ CF for different applications.\nAs mentioned before, rev KSAf also pointed out that the paper would benefit from a broader discussion on the connections between this work and other approaches based on the CF. Thus, we will include a proper discussion at the beginning of section 4.\nSpecifically in relation to [1], both works share the core idea of using ECFD as a differentiable 2-sample statistical test, yet our approach deviates significantly from [1]. The two applications are different, as [1] handles generative models, while our approach focuses on fair representations. \nMoreover, [1] employs adversarial training to optimize the test kernel, which we avoid in order to simplify the approach.\n\n\nRegarding the rest of the comments, thanks, we will address them in the camera ready version if the paper gets accepted.\n\n### Questions \n\n_\"To estimate the empirical CF in equation (9) of $P(Z|S)$...\"_\n\nThe Monte Carlo estimate is relatively inexpensive to compute. It involves a fixed number of operations for each datapoint, which is negligible compared to the forward pass of the network.\nMoreover, we do not necessarily need rejection sampling in order to have balanced batch. In fact, the term in (9) for each of the groups can be computed independently and possibly with a different number of data samples. In all experiments, we used a fixed batch size of 128 and computed the penalty term for each group separately, even for unbalanced batches. \n\n### Limitations:\n\n_\"Some drawbacks of FmCF are discussed, but discussion of computational complexity could be enhanced.\"_\n\nYou are right, we strongly believe that defining clear limitation are a key part of the scientific process, and if you feel that more attention should be given to them, we will make sure to add more informations in the camera ready about the limitations of the CF approach if the paper gets accepted (NeurIPS does not allow for resubmission)\n\n----\n\nWe appreciate the reviewer’s comments and suggestions. We will make sure to reflect the necessary clarifications and improvements in the camera-ready version, if the paper is accepted."}, {"review_id": "UGTINxwVWC", "reviewer": "Reviewer_HsWD", "summary": "This paper introduces a fair representation approach. It minimizes the characteristic function distance of the different demographic groups' representation and a Gaussian distribution in the latent space. To keep the prediction power, a cross-entropy loss is used. The author also shows some theoretical insights into their methods. In the experiments section, it compares with Fair Normalizing flow methods, which is also a fair guarantee method. The results show that the method proposed by the author achieved a good tradeoff between fairness and utility.", "questions": "1. Why do the authors select a Gaussian distribution as the reference? While I understand it is discussed in the context of stability training, I am still unclear why aligning with different demographic groups would be considered a suboptimal choice\n\nI don't have further questions. Please see the weakness section for my major concerns.", "limitations": "Yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 2, "originality": 3, "strengths_and_weaknesses": "**Stengths**\n\n(1) This paper is well-presented; it introduces the prerequisites in detail, and the notation is neatly arranged.\n \n(2) The algorithm is clear and easy to reproduce. I personally think this algorithm is correct.\n\n(3) It is indeed more flexible than FNF, as it can be used as any backbone model, instead of just flow flow-based model. \n\n**Weakness**\n\n(1) The major concern I have is the tradeoff between utility and fairness. As in Algorithm 1 line 11, different demographic groups' representations are aligned with the same distribution (Gaussian in the text). Although a cross-entropy loss is added to guide the learning procedure, the fairness constraint may overwhelmingly dominate the learning procedure, resulting in less effective performance of the classification task. \n\n(2)  There exist many fair representation learning methods. For example, MMD-B Fair [1] uses MMD as the statistical independence criterion to learn a fair representation. I am wondering why the author's proposed method is more needed than existing fair representation works. A rigorous experiment comparison would help to illustrate that.\n\n(3) Datasets are narrowed in tabular datasets, it narrows the scope of this work.\n\n\n[1]Deka, Namrata, and Danica J. Sutherland. \"Mmd-b-fair: Learning fair representations with statistical testing.\" International Conference on Artificial Intelligence and Statistics. PMLR, 2023.", "was_revised": true, "final_justification": "After the rebuttal, most of the concerns are well addressed. All in all, I think this paper provides more flexibility than using a flow-based model to learn a fair representation.", "rebuttal": "We thank the reviewer for the thoughtful comments and valuable suggestions. The concerns raised regarding fairness-utility tradeoffs, comparative baselines, and dataset scope are appreciated and help us better position the work. We address each point below and will incorporate the necessary clarifications in the final version.\n\nPlease note that, due to the updated NeurIPS guidelines, we are no longer permitted to upload supplementary materials or a revised version of the paper at this stage. We hope that the clarifications provided here sufficiently address your concerns.\n\n----\n\n### Weaknesses \n\n_\"The major concern I have is the tradeoff between utility and fairness. As in Algorithm 1 line 11, different demographic groups' representations are aligned with the same distribution (Gaussian in the text). Although a cross-entropy loss is added to guide the learning procedure, the fairness constraint may overwhelmingly dominate the learning procedure, resulting in less effective performance of the classification task.\"_\n\nThanks for pointing it out. Indeed, the fairness-accuracy tradeoff is a very well-known effect.  \nFrom an Information theoretical standpoint, it can be easily proven by observing that $H(Y|X, S) \\le H(Y|X)$.   \nTherefore, removing information must (theoretically) increase the uncertainty in the prediction, worsening the classification accuracy. \nIn our approach, the hyperparameter $\\alpha$ can be used to control the weight given to the fairness objective, effectively controlling how much information on the sensitive attribute is removed.\nIndeed, this tradeoff can be observed in Figure 3, where, even though the phenomenon you are describing indeed happens, it appears to be much more gentle compared to FNF [2].\n\n_\"There exist many fair representation learning methods. For example, MMD-B Fair [1] uses MMD as the statistical independence criterion to learn a fair representation. I am wondering why the author's proposed method is more needed than existing fair representation works. A rigorous experiment comparison would help to illustrate that.\"_\n\nThanks for bringing this paper to our attention. Even though we tried to have an exhaustive literature review, some works might have been overlooked or missed. \nThis specific work seems to focus on a statistical test standpoint. Indeed, it tries to minimize the probability that a statistical test (MMD) finds a difference in the distributions.  \nIf on one side MMD is indeed connected to the CFD, such a connection seems to be a consequence of the common objective of fair representation learning. In fact, even Adversarial Learning can be interpreted as an MMD minimization approach. \nThe two approaches differ in many ways. For instance, MMD-B-fair requires splitting into blocks in order to have an efficient approach to estimate MMD, where the CF approach can be straightforwardly applied. \nAs suggested by reviewers a3eD and KSAf, we will include a more detailed discussion of related literature in the camera-ready version. We will make sure to also properly address this work, as we believe that it would be a valid addition to the discussion.\n\n_\"Datasets are narrowed in tabular datasets, it narrows the scope of this work.\"_\n\nWe agree that testing the approach mainly on tabular data narrows the scope of the paper. However, we would like to highlight that the datasets used are the default benchmarks to assess fair representations. \nThough being the default, we agree with you that might be interesting to test it on other domains.   \nFor instance, a small example about image generation is shown in the appendix.\nBeing only a proof of concept and somewhat out of scope of the paper, we opted to put it in the appendix rather than in the main text.\nAs reviewer a3eD also pointed out, we should have referenced this experiment explicitly in the main text, as it shows that the proposed approach is in no way restricted to tabular data nor to classification.\nIn the revised version, we will better highlight this aspect as we believe it is a major strength of the approach that might otherwise be easily overlooked.\n\n### Questions\n\n_\"Why do the authors select a Gaussian distribution as the reference? While I understand it is discussed in the context of stability training, I am still unclear why aligning with different demographic groups would be considered a suboptimal choice\"_\n\nIndeed, using a Gaussian distribution is not necessarily the optimal choice, but it is a rather natural one.  \nFor instance, the Gaussian distribution has a very simple closed-form characteristic function that is well-behaved and rapidly decays to zero for high frequencies.\nIn practice, if the encoder network is expressive enough, the specific target distribution chosen should not have a major influence on the result.  \nOn the other hand, aligning with different demographic groups directly has some significant drawbacks. Since there is no closed-form expression for the characteristic function, both terms in Eq. 9 would need to be estimated. This significantly increases the variance of the loss term, making it noisier.\nMoreover, when matching different groups to one another, there is no unique optimal solution. This makes the training unstable and might also lead to numerical instabilities.   \nOverall, having a common target solves multiple critical issues, while introducing minimal downsides (if any).\n\n\n----\n\n\nWe appreciate the reviewer’s comments and suggestions. We will make sure to reflect the necessary clarifications and improvements in the camera-ready version, if the paper is accepted."}, {"review_id": "v8JJ6rluCv", "reviewer": "Reviewer_HZGi", "summary": "This paper proposes a representation learning method that overcomes the unstable issues of adversarial learning and computationally intensive issue of distribution matching. The method first uses CFD to align group-wise representations to a common distribution, and then proposes a more efficient moment-matching approximation that matches low-order moments (N = 2) to guarantee that downstream classifiers cannot access sensitive information.", "questions": "- How sensitive is the method to the choice of the shared target distribution?\n- Have the authors considered whether aligning group-wise representations to a learned barycenter or a task-specific distribution might be more effective than enforcing a shared target distribution, such as the Gaussian used in this paper?\n- Line 230: “…however, we show that truncating to N=2 is sufficient to ensure fair classification,” but I cannot find the evidence to support this claim. Can the authors specify and justify it?", "limitations": "Yes, the authors acknowledge the limitation regarding high-dimensional settings in the conclusion section. However, I respectfully disagree with the claim that \"current fairness benchmarks largely focus on low-dimensional datasets, limiting the ability to comprehensively evaluate and compare methods in more realistic and complex settings.\" In the domain of computer vision, for instance, there exist several well-established image datasets for fairness evaluation, such as CelebA, FairFace, and LFW.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n- The proposed method replaces the need to match full distributions (via monte carlo CFD estimation) with a tractable alternative by truncating the characteristic function expansion to a low order (e.g., N = 2), which leads to more stable and efficient training compared to monte carlo based methods.\n- The method uses CFD to align group-wise representations at the distributional level, offering a clearer and more principled formulation of fairness compared to adversarial learning or MMD-based approaches.\n- The visualization of learned representations in Figure 4 is very intuitive to capture the effectiveness of the proposed method.\n\nWeakness:\n- The proposed method aligns representations of each group to a shared target distribution, and in this paper, uses the standard normal distribution (likely for computational simplicity and methodological clarity). However, consider a situation where the target distribution is not well-aligned with any group (e.g., when the data resemble a mixture of Gaussians) and enforcing all groups to align with the shared target may actually distort them unnecessarily.\n- One major limitation of the paper lies in its limited experimental scope. The method is only evaluated on two small tabular datasets, which are low-dimensional and structurally simple. In such cases, aligning low-order moments (N=2) may already suffice for fair representation learning. However, it remains unclear whether the proposed approach can generalize to more complex scenarios, such as high-dimensional, nonlinear, or multi-modal tasks, where group-wise distributional differences are more intricate. This limits the method’s relevance to current practice and raises concerns about its applicability in more realistic, large-scale, or multi-modal learning settings.", "was_revised": true, "final_justification": "The authors have addressed all of my questions and concerns, and I was already inclined to accept the paper from the beginning. In particular, I appreciate the authors’ detailed response regarding the major limitation I raised about the scope of the experimental evaluation. They clarified that although the main focus was on the Adult and German datasets, they aimed to ensure fairness in comparison by restricting evaluations to datasets where shared training configurations were accessible. Additionally, they mentioned a generative task to demonstrate the broader applicability of their CF approach to latent representations, although they deliberately left it out of the main paper due to its divergence from the main classification focus. I find this reasoning acceptable and appreciate the authors’ efforts to maintain fairness and clarity in the empirical study. Taking all of this into account, I have raised my score accordingly.", "rebuttal": "We thank the reviewer for the detailed and constructive feedback. Below, we address each point raised, clarifying theoretical aspects and experimental choices, and acknowledging areas that can be improved in the final version. \n\nPlease note that, due to the updated NeurIPS guidelines, we are no longer permitted to upload supplementary materials or a revised version of the paper at this stage. We hope that the clarifications provided here sufficiently address your concerns.\n\n----\n### Weaknesses\n\n_\"The proposed method aligns representations of each group to a shared target distribution, and in this paper, uses the standard normal distribution (likely for computational simplicity and methodological clarity). However, consider a situation where the target distribution is not well-aligned with any group (e.g., when the data resemble a mixture of Gaussians) and enforcing all groups to align with the shared target may actually distort them unnecessarily.\"_\n\nThe proposed method based on CF aims to make two changes of variable. First $X$, the original sample, is mapped to $Z$, a fair representation, and then $Z$ is mapped to $Y$, the classification. An MLP can provably perform any arbitrary change of variable (thus is able to map $X$ to $Z$, a Gaussian distribution, with no problem) thanks to the universal approximation theorem. \nIndeed, this is the core idea behind every fair representation learning algorithm (VAE maps to a Gaussian, FNF to an estimated distribution, Adversarial learning to two distributions that should be indistinguishable). \nNonetheless, in order to have guarantees about the expressivity, we just need to ensure that $H(Z) \\ge H(Y)$, thus that it has enough capacity to “be able to carry the relevant information for the classification”. \nSince $Z$ is continuous (and no noise is injected) and $Y$ is discrete, we know this is possible. To conclude, though mapping to a distribution might sound tedious for certain cases, an MLP is more than capable of doing so. \nHowever, if you feel that this discussion (ie, that mapping to a Gaussian is not “theoretically limiting”) needs more attention, let us know and we will make sure to add more details to the camera-ready version of the paper if it gets accepted.\n\n_\"One major limitation of the paper lies in its limited experimental scope. The method is only evaluated on two small tabular datasets, which are low-dimensional and structurally simple. In such cases, aligning low-order moments (N=2) may already suffice for fair representation learning. However, it remains unclear whether the proposed approach can generalize to more complex scenarios, such as high-dimensional, nonlinear, or multi-modal tasks, where group-wise distributional differences are more intricate. This limits the method’s relevance to current practice and raises concerns about its applicability in more realistic, large-scale, or multi-modal learning settings.\"_\n\nIndeed, a thorough evaluation has been carried out on the Adult and German datasets, as they are the benchmarks most commonly used in the literature. \nHowever, the empirical evaluation is not limited to this two datasets. For instance, Figure 3 also shows results for CRIME, COMPAS, and HEALTH datasets in comparison to FNF [2].  \nThe reason why other methods were not evaluated on such datasets was to be as fair as possible to the compared methods. Indeed, all the other baselines considered for this study performed their evaluation only on German and Adult, and shared the code and training configuration (network, optimizer, and so on) only for such datasets.\nInstead, FNF also tested their beyond GERMAN and ADULT. Thus, we were able to do a fair comparison with such work.  \nOverall, we tried to make our experimental evaluation as broad and comprehensive as any of the baselines we evaluated against.  \nFurthermore, while we acknowledge that it might be a simple toy scenario, we felt that showing that the proposed approach is valid on different modalities was important too. Thus, we included a simple generative task in the appendix. \nThis showcases the effectiveness of the CF approach for modelling the distribution of latent representations learned by autoencoders.\nYet, given that the main focus of the paper was to deal with fair classification, we thought that it would have been a bit off-topic to put it in the main text.\n\n### Question\n\n_\"How sensitive is the method to the choice of the shared target distribution?\"_\n\nGiven that we wanted an effective and simple approach to learn fair representations, we tried to do as minimal hyperparameter tuning as possible. Therefore, we did not test other reference distributions beyond the Gaussian one. \nIdeally, if we consider that an MLP can perform an arbitrary change of variable, we can easily show that we should be invariant to the reference distribution. \nIn fact, the network can learn the map $X \\rightarrow Z \\rightarrow N(0,1) \\rightarrow Y$, where $Z$ is modelled with a different distribution. \n\n_\"Have the authors considered whether aligning group-wise representations to a learned barycenter or a task-specific distribution might be more effective than enforcing a shared target distribution, such as the Gaussian used in this paper?\"_\n\nWe did not explore this direction, but the idea of learning the reference distribution might be an interesting line of work, as it would allow the MLP to choose the distribution that is “the simplest”. \nWe will make sure to include this option in the future works in the camera-ready version if the paper gets accepted.\n\n_\"Line 230: “…however, we show that truncating to N=2 is sufficient to ensure fair classification,” but I cannot find the evidence to support this claim. Can the authors specify and justify it?\"_\n\nThe result showing that we only need 2 moments is the conjunction of Theorems 5.2 and 5.3. Theorem 5.2 shows that as long as the first moment is exact, such a feature has no predictive power for any LogRegression. Theorem 5.3 shows that if the first moment is an epsilon away between the two sensitive groups, the predictive power of such a feature is bound by its variance. To conclude, if we have bounded variance and we push the means towards each other, we can ensure fairness. \nLine 230 is meant to anticipate these results, but we realize that this could be expressed much more clearly. Thus, we will make sure to revise it in the camera-ready version, if the paper gets accepted.\n\n### Limitations\n\n_\"Yes, the authors acknowledge the limitation regarding high-dimensional settings in the conclusion section. However, I respectfully disagree with the claim that \"current fairness benchmarks largely focus on low-dimensional datasets, limiting the ability to comprehensively evaluate and compare methods in more realistic and complex settings.\" In the domain of computer vision, for instance, there exist several well-established image datasets for fairness evaluation, such as CelebA, FairFace, and LFW.\"_\n\nWe thank the reviewer for the honest comment and for bringing such benchmarks to our attention. Given the presence of such benchmarks, we agree that the claim might be a bit of a stretch. What we meant is that none of the other considered baselines evaluate their approach on such datasets.\nWe can include in the appendix an evaluation of the proposed approaches (FmCF, FmSS) on such datasets, in order to highlight that the input modality is not a point of friction.\nIn any case, we will make sure to properly address this aspect in the revised paper for the camera-ready version if it gets accepted.\n\n\n\n### Formatting Concerns\n\n_\"Equation (2), the left parenthesis in $p(y=f(x))$ seems to have a formatting issue._  \nThanks for spotting the typo\n\n_\"The positions of Algorithm 1 and Algorithm 2 can be adjusted appropriately, especially to ensure that the top and bottom margins are aligned with the surrounding text._  \nThanks, we will indeed try to make it that way for the camera-ready if the paper gets accepted\n\n_\"Lines 242–267: The formatting appears somewhat chaotic, maybe consider reordering or restructuring them in a cleaner and more organized way._  \nThanks, we will do our best to improve it according to the additions requested during the rebuttal.\n\n----\n\nWe appreciate the reviewer’s comments and suggestions. We will make sure to reflect the necessary clarifications and improvements in the camera-ready version, if the paper is accepted."}]

Author rebuttal:

Meta-review: This paper tackles the important challenge of fair classification and introduces a novel characteristic function distance–based approach that addresses the instability of adversarial methods and the inefficiency of distribution matching. The proposed method is both stable and computationally efficient, with a relaxed objective that ensures fairness without sacrificing accuracy. Strong experimental results across benchmarks demonstrate consistent improvements in both fairness and predictive performance, highlighting the method’s robustness and practicality for real-world deployment.

Good clarity, contribution, and all positive reviews. This paper is recommended to be accepted.
Decision: Accept (poster)