Computational-Unidentifiability in Representation for Fair Downstream Tasks
Authors: Taeuk Jang
Xiaoqian Wang
Venue: Submitted to ICLR 2023 (2023)
Keywords: fairness
representation learning

Abstract: Deep representation learning methods are highlighted as they outperform classical algorithms in various downstream tasks, such as classification, clustering, generative models, etc. Due to their success and impact on the real world, fairness concern is rising with noticeable attention. However, the focus of the fairness problem was limited to a certain downstream task, mostly classification, and few were studied from the perspective of representation itself. We claim that the fairness problems to various downstream tasks originated from the input feature space, i.e., the learned representation space. While several studies explored fair representation for the classification task, the fair representation learning method for unsupervised learning is not actively discussed. To fill this gap, we define a new notion of fairness, computational-unidentifiability, which suggests the fairness of the representation as the distributional independence of the sensitive groups. We demonstrate motivating problems that achieving computationally-unidentifiable representation is critical for fair downstream tasks. Moreover, we propose a novel fairness metric, Fair Fréchet distance (FFD), to quantify the computational-unidentifiability and address the limitation of a well-known fairness metric for unsupervised learning, i.e., balance. The proposed metric is efficient in computation and preserves theoretical properties. We empirically validate the effectiveness of the computationally-unidentifiable representations in various downstream tasks.

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COMPUTATIONA--NDDEETTEABLLIITI IN REPRESEN- TATION FOR FAIR DOWNSTREAM TASKS
Anonymous authors Paper under double-blind review
ABSTRACT
Deep representation learning methods are highlighted as they outperform classical algorithms in various downstream tasks, such as classification, clustering, genera- tive models, etc. Due to their success and impact on the real world, fairness concern is rising with noticeable attention. However, the focus of the fairness problem was limited to certain downstream task, mostly classification. We claim that the fairness problems to various downstream tasks originated from the input feature space, Le., the learned representation space. While several studies explored fair representation for the classification task, the fair representation learning method for unsupervised learning is not actively discussed yet. To fill this gap, define which suggests the fairness
new notion of fairness, of the representation as the distributional independence of the sensitive groups. We demonstrate motivating problems that achieving cnnpuutaiionally-mmdnnnniii representation is critical for fair downstream tasks. Moreover, we propose a novel fairness metric, Fair Fréchet distance (FFD), to quantify the computational- unidentifiability and address the limitation of a well-known fairness metric for unsupervised learning, i.e., balance. The proposed metric is efficient in computa- tion and preserves theoretical properties. We empirically validate the effectiveness of the
representations in various downstream tasks.
INTRODUCTION
Thanks to the outstanding performance and development of deep learning, it has been widely applied
to various domains, including natural language processing (NIP) (Devlin al., 2018), computer
vision (Krrra et al., 2019), and generative models (Goodfellow et al. (014). On the other hand, the
reliability and fairness concerns (Lee & Floridi 2020 Angwin et al. 2016 Dastin 2018 advanced
due to their impact on the real world applications. Such fairness concerns include credit limit estimation (Vigdor 2019), job application filtering (Dastin 2018), or crime prevention (Dressel & Farid 2018), etc. Accordingly, algorithmic faimess is getting growing attention prevent biased predictions.
Following the mainstream fairness literature, here focus on group faimess Dua & Grafi 2019 Zatar et al., 2015; Hardi et al.) 2016), which suggests the equality of certain statistical measures (e.g., true positive rate, positive prediction) between subgroups with different protected attribute (e.g.,
gender, race, religion, etc). It has been widely studied to mitigate fairness violations in downstream
tasks. Numerous studies (Hardt et al. 2016 Choi et al. 2020 Pleiss al. 2017 Madras al. 2018)
explore how to attain group fairness in classification tasks. The primary objective of this family of works is obtain the prediction independence of protected property. Hardt et al. (2016) suggest equal opportunity, which requires the same true positive rates for the subgroup. Calibration among the subgroups (Kleinberg et al. 2016) is to match the predicted probability and actual distribution
of favorable class. Moreover, some works (Kim al., 2020; Jang et al. 2021 study efficient multi-constraint optimization to satisfy multiple fairness notions.
However, most of the works mainly focus on the supervised setting, Even though deep learning has significant success in various unsupervised learning tasks, such as clustering (Xie et al. 2016 Guo et al. 2017), generative model (Karras et al., 2019: Radford et al.] 2019), and NLP (Hadifar
et al.] 2019), ) the fairness of unsupervised learning is relatively not actively studied (Buet-Golfoute Golfouse &
Utyagulov 2022), and how to quantify the fairness of unsupervised learning methods has not been

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well established yet. A widely used metric for fair clustering is called balance (Chierichetti et al. 2017). which is analogous to demographic parity (Barocas & Selbst 2016) in classification. However, the balance has some limitations since it quantifies fairness by computing the ratio of samples in different protected groups within a cluster. For instance, even in the ideal balance (the ratio of the samples from different groups matches the group truth), the sensitive groups can distribute samples separately within clusters. In this case, it is casy to determine which sensitive group the sample belongs to so that it might lead to a biased decision in downstream tasks. Especially in generative models, e.g., VAE (Kingma & Welling 2013). the generated samples can be imbalanced if the latent space is dependent on the sensitive attributes. This can cause critical problem as generative models
are widely applied to mitigate the imbalance of datasets (Guo et al. 2019 Fajardo et al. 2021 Mirza et al. 202T).
as a fairness notion in unsuper- vised learning- Analogous to the fact that biased data is responsible for the biased recision-making (Buolamwini & Gebru 2018 Mehrabi et al. 2021). we here claim that the learned representation itself plays a critical role in fair downstream tasks utilizing DNN, Even though deep representation has been appreciated for its superb performance (Eldan & Shamir 2016 Kozma et al. 2018), the fairnsss concerns in the space have been overlooked. Thus, we explore the fairness in representation space that could bridge DNN and the downstream tasks with fairness concerns. We validate claim on downstream tasks by comparing the performance and fairness of two distributions: fair and unfair representation.
Instead, we propose a novel approach, computtaiionll-unddenniffffiiaii
To measure fairness in representation space, we propose novel metric called FFD (Fair Fréchet distance) inspired by Fréchet distance (Dowson & Landau 1982) to efficiently quantify fairness in representation space by measuring distributional independence of the sensitive groups with computational identifiability ELébert-Johnson et al.] Lahoti et al. 2020). Unlike the balance, we not only consider statistical independence but also distributional independence between the sensitive groups. This can a good reference for future work to evaluate the fairness or distributional independence in the representation space of certain attributes of interest. Moreover, we propose a deep fair clustering framework to learn a fair representation that achieves comparable performance with other clustering methods while ensuring fairness. The contributions in the paper can be summarized as follows:
1. We study the motivating problem of why fair representation is important to achieve fair downstream tasks.
2. We propose novel metric that quantifies faimess in representation space. We provide rigorous analysis of the theoretical property and complexity of our fairness metric.
3. We propose framework for fair representation learning for downstream tasks.
4. We validate our method on various benchmark datasets comparing with state-of-the-art fair methods in the literature.
2 RELATED WORKS
GROUP FAIRNESS
As a class of definitions, group fairness measures the disparity of predicted outcomes among the subgroups with certain sensitive attributes. A number of works introduce fair notions to mitigate the bias and ensure the independence of the performance measures between the subgroups to achieve group fairness. Demographic parity (Barocas & Selbst 2016) suggests that positive prediction should be equalized and independent of the sensitive attribute. Equal opportunity (Hardt et al.) 2016) states that true positive rates should match. Likewise, Predictive equality (Chouldechoval 2011 7) states the equality of false positive rates. Group-wise calibration (Kleinberg et al. 2016:] Pleiss et al.) 2017) proposed to match the probability estimate with the actual ratio of positive distribution within the group. In an unsupervised setting, balance (Chierichetti et al. 2018) is introduced to have an equal number of samples from different protected groups within a cluster as fair clustering. However, the balance only considers statistical parity, which limits the utility metric since perfect balance (i.e., 1) does not guarantee fairness (as the base rate differs). Moreover, none of the works explore the fairness of the representation itself.
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FAIR SUPERVISED LEARNING
To assure group fairness, recent works in supervised learning reside in one of the three approaches: 1) pre-preoee pre-processing; 2) in-processing: 3) post- post-preessing.. Pre-processing method (Chen et al. 2018 suggests improving the skewed sample size problem. Adversarial learning methods (adrrs et al. 2018 Zhao al. 2019) are proposed to learn fair representation that is independent of sensitive attributes. Post-processing the output of the biased model (Hardt et al. 2016 Jang et al. 2021) with multiple fairness objectives are introduced as they are more efficient than training model from scratch. However, the aforementioned approaches minimize group fairess constraints specified for the classification task.
FAIR UNSUPERVISED LEARNING
Fairness in the unsupervised setting has recently got attention (But--Gllooue & Utyagulov 2022
Ghadiri et al. (021). A pioneering work (Chierichetti et al. 2018) in fair clustering method proposed fairlet decomposition pre-process data followed by classic clustering methods address disparate impact. Scalable fair clustering algorithm (Backurs et al. 2019) is the following work of fairlet decomposition by improving the efficiency with approximation. Variational framework (Ziko et al., 2019) is introduced to satisfy KL fairness objective. Wang & Davidson (2019) propose new concept called fairoid that enforces the centroids of each sensitive group feature space to have an equal distance to each cluster centroid. Adversarial objective (Li et al., 2020) is employed to learn a representation that is statistically independent w.r.f. sensitive attribute while cesstrrig--avorahhee utilizing individual clustering modules. However, to our best knowledge, previous works mostly focused on the predicted outcome to be independent of the sensitive attribute, i.e., statistical parity, We here study the independence of sensitive attributes in the learned representation in unsupervised learning.
MOTIVATING PROBLEMS
In this work, we define a novel fairness notion called computationally unidentifiability that is more extensive than the existing task task-specific notions. Inspired by fair classification works (Hébert-Johnson as the maximum possible ability for external classifier distinguish which sensitive group the data belongs to. Two distributions are computationally unidentifiable if and only if they are identical, i.e., no external classifier can distinguish which sensitive group the sample is drawn from. We demonstrate motivating problems showing how such distributional independence affects fairness in downstream tasks.
le Lahoti et al.] 22020). we define caapuuationallideniffaab
3.1 CLASSIFICATION AND CLUSTERING
Consider data distribution with binary sensitive attribute, A = {0, 1). and binary label, Y {0, 1}- In Fig. I we illustrate synthetic data distributions similar to previous works Zatar et al. 2015 Kim et al. 2020) with two scenarios that both satisfy the perfect balance, i.e., base rate for each protected group is identical. The perfect balance
also be referred to statistical independence. We denote Xya as a set of instances with y E Y and E A. The detail of the synthetic data sampling process is in the appendix. When comparing two distributions in Fig. Th and Fig. Tal the distribution in Fig. lb eblliiiiy exposes which sensitive group a sample belongs to, i.e., computationally identifiable (CI) This has potential risk of discrimination in downstream tasks. Specifically, is unstable for maintaining good clustering performance since such data representation can be clustered by the sensitive group structure whether than the expected intrinsic features (Lee et al. 202T). In contrast, in Fig. la the representation satisfies not only statistical independence but also distributional independence w.l.l. sensitive attribute, i.e., computationally unidentifiable (CU). Therefore, models cannot casily identify which group sample belongs to and thus cannot discriminate against groups in the downstream tasks.
To validate our claim, evaluate two representations with classification and clustering, which are the most popular tasks in supervised and unsupervised learning. For the classification task, we test the logistic regression. To measure faimess in classification, adopt demographic parity (Barocas)
& Selbst 2016), DP-
P(Ý 1 0) P(ý 1|A 1)}, and equalized odds (Hardt et al.
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-10
(a) Computationally unidentifiable (fair) representation
(b) Computationally identifiable (unfair) representation
Figure 1: Illustration of synthetic data of computationally unidentifiable (fair) and identifiable (unfair) distribution with a binary sensitive group and class. Different color (resp. shape) indicates different class y (resp. sensitive group a). We denote Xy as a of instances with y, a {0. 1}.
Table 1: Evaluation of downstream tasks on different distributions. Computational Unidentifiable (CU) distribution achieves significantly fairer results with similar performance in both supervised and unsupervised learning tasks, while Computationally-Idenli Identfiiileei (CT) distribution has huge detriment of fairness. FFD is the proposed metric to measure the fairness of representation.
Compttational-Unndeetiiiaaaaa
Data
k-means++ (Unsup Learning)
Logistic Regression (Sup Learning)
FFD
Balance
ACC
AACC
Balance
ACC
AACC
DP
EOD
CU
6.876
0.931
0.846
0.002
0.851
0.865
0.005
0.029
0.031
CI
8.385
0.908
0.861
0.091
0.116
0.855
0,076
0.636
0.717
Y, = v)1, where y is predicted 2016), EOD E, IP(Ý y/A - v) P(Ý y y/A=
label. Both DP and EOD the lower, the better.
For fair clustering. we measure the balance by following the previous works Xie et al. 2016 ctt al. 2020 Bera et al. 2019), which is to satisfy Ex-D
(X) k) Ex~D
where C(X) indicates that the data X clustered to the k-th cluster by model C. Achieving the balanced clustering satisfies the statistical independence w.r.t. A, and balance 1 is perfect balance. However, we claim that statistical independence cannot fully examine fair clustering. To address the limitation of the previous fair unsupervised learning metric, we propose novel fairness metric for representation called Fair Fréchet Distance (FFD), which will be discussed the following section.
Table summarizes the evaluation of downstream tasks on the two distributions. Even though both CU and CI data are sampled from perfectly balanced distributions, fairness violations from CI are significantly worse than that of CU on both tasks. It is interesting to note that fairness is sensitive distributional independence; however, performance is not affected. This validates that fair representation itself has a substantial impact on fairness in downstream tasks while preserving utility. Moreover, FFD is good proxy to measure computational identifiability since smaller FFD indicates harder to identify sensitive information from the representation.
4 FAIR FRÉCHET DISTANCE
To quantify the proposed fairness notion in terms of computational identifiability, in this subsection we introduce a novel metric named Fair Fréchet Distance (FFD) to measure the distance between distributions from different sensitive groups.
Consider two sets of samples U gdx and E [Rd Suppose the samples U and V are drawn from multivariate Gaussian distributions, respectively. Define centering matrix Hn, Rax H I 11, 1,,1, I, where 1, E R is vector with all elements being 1; and I is the identity matrix. We first introduce two metrics in Definition and 4.2 that measure the distance between two distributions U and V.
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Definition 4.1. Fréchet distance (FD) (Dowson & Landau 1982 between U and V is defined as: FD V) a pv 12 + Tr (Ev + Viv 2v
where ju, PV and Ev, Σy are the means and covariance matrices of U and V, respectively. Definition 4.2. We define the Fair Fréchet Distance within Cluster (FFDC) between U and as follows:
U1mg V1n
+ | |UHno
I 1a Vni-1
) TT(VTT
FFDC FFDCC(U.V)
no
VTO
Vni
Vno IVn--1
when no, n1 1, else FFDC* 0C.
Next, we define Fair Fréchet Distance (FFD) in Definition For simplicity, we define FFD for the case with binary sensitive feature E {0, 1}. We will introduce how extend such measure the case with valui- sensitive features at the end of this subsection.
For a clustering assignment of m samples into с clusters X2. XR]. where Xş
pdxxa, k 2, C, contains the ny samples in the k-th cluster that sums to Σ n = m. Within
kim
each cluster Xg, define UR ddx and VA Rdx=s as follows: if =1
X
if a else
(1)
else
where x is the i-th sample in XRi aj E 40, the sensitive feature of the i-th sample in X5; and is the zero vector. Thus we have UA + Vv2 = X2. k
1,2,...,e
Definition 4.3. With the definition of Ux and Vv Vilū=1 in equation define FFD for the m samples with the clustering assignment 12, X2,....... Xa as:
F X2,....,AA!
X²)) = max FFDC VR.)
Theorem 4.4. With the definition of Uk/ f= and Valū== in equation ! the following inequality holds:
FFD?( X Xa)) max
Tr(X,X;)
ni
I max (U,, VA) (FX' Xe..,,,..
Proof of Theorem 4.4 is in the appendix. In the case of the multi-valued sensitive feature, we can extend the definition FFDC Definition44.2) with the max FFDC value among all pairs of sensitive groups in a cluster, and thus extend the definition of FFD in Definition 4.3 We can easily verify that Theorem still holds in the case with the multi-valued sensitive feature.
4.1 INSIGHT FROM THEOREM
Consider clustering problem that partitions the samples into clusters, where each data sample formulated as a d-dimensional vector. The FFD metric we proposed in Definition 4.3) is efficient in computation, which requires linear time w.E.f the number of features and number of samples. The calculation of FFD in Definition 4.3 has a time complexity of ((ddc) (since we only need to
calculate the trace of matrix UUT and V VT in Definition 4.2 requires linear instead of quadratic time w.r.. d). In contrast, traditional FD metric in Definition
has a cubic time complexity w.r.t. number of features. The time complexity for calculating FD is On dd)) (since it requires the computation of exact covariance matrices Eu and Σy in Definition and the corresponding square root).
Theorem indicates that the FD metric is upper bounded by our proposed FFD metric, thus minimizing FFD indicates the minimization of the upper bound of FD. Further, the gap between the FD and our FFD metric is bounded by max; n Tr( ( xT)) Note that FFD is minimized if and only if the two following conditions are met in each cluster X k = 1,2.
U&1nz = Villg || 11? = H 11?
in which case we have FD² = 0. Thus FD value minimized if and only our proposed FFD metric minimized.
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5 FAIR CLUSTERING FRAMEWORK
In this section, we present our deep fair clustering framework and its objective functions. For simplicity, we consider sensitive attribute as binary feature. However, this can be easily extended to multiple sensitive attribute problems. Consider the c-clustering problem, given the i.dd sampled m data samples X' E Rdx= where each sample is represented by a d-dimensional vector. Encoder E learns representation 2 E Rixm and a clustering module C' takes as an input and outputs probability P E Rexm of the predicted cluster computationally unidentifiable fairness and high clustering performance. Given a matrix X' E RRdm with samples, denote the i-th data point from X' its bold lower case letter with index in the superscript, e.g., x", and the k-th entry of a vector as a lower case letter e.g., Z
soft label. The goal of E and C' is to achieve
5.1 CLUSTERING LOss
Inspired by the previous works (ii et al. 2016 Li et al. 2020), we employ clustering loss learn the representation that is concentrated in the cluster centroids. Clustering module C' assigns probability that sample z' Exx') belongs each cluster k'' by comparing with trainable cluster centroids eg on Student t-distribution as:
(1 (1+ || c₁ /1))))
Ph E1 |z cR' 122 -
where p indicates the probability that x' belongs to k-th cluster and is the degree of freedom of Student t-distribution. Then, assign the target cluster of by sharpening the soft assignment pi. within a sensitive group a as
( PK
of EN ((rk)) ExEx, which reinforce the confidence of the predicted cluster and prevent large clusters regularizer. We set empirical clustering loss Let as KL divergence between pk and q
(3)
Les = KL(PIIQ) = ΣΣ
Pk log
Pk:
(4)
9k
NEX
5.2 FAIRNESS LOSS
Our goal is to further improve fairness in the clustering task that sensitive group is not identifiable by the samples in cluster. Recent work proposed to use fairoid (fair-centroid) (Wang & Davidson 2019) that the centroid of each sensitive group should have equal distance all cluster centroids. We claim that fairoid cannot guarantee fair representation since equidistance centroids can be perfectly separated by the cluster centroids.
To achieve compuiationallunideetfffii ture (Kingma & Welling 2013) for the encoder to leverage the reparameterization trick. Then we can formulate the latent feature of an instance x² as z' = E'(x)) μ + EOT, where NO( I). where μr and o are the mean and variance respectively. To enforce the learned representation inde- pendent of the sensitive attribute, minimize the distance between distributions from different protected group within a cluster, i.e., K KL(Pia.k) ( .e)), where P(a,k) is probability distribution of the samples in sensitive group with predicted cluster k. Assume the distribution follows the Gaussian distribution as P(a,k) = =N(Ma(a
we employ variational autoencoder (VAH) struc-
(a.,)) Then our fairness objective to minimize
KL divergence can be written as:
2 2 2og (a.kk) )
o (a,k) + (M(a)) - 14(0 1))
+1)
Lfair =
(5)
(a'
o car
For the empirical loss Lyarr we use (((,k) T... Eiex,, lh and 8 (a.k) the empirical mean and variance where Xa is denoted as set of instances predicted as cluster lo in group a. since we assume all samples are i.i.d.
To. Exex,, o' as
To sum up, our final objective is to minimize the loss as follows:
min Late + Laair.
(6)
E,C

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EXPERIMENTS
In this section, we compare fairness and the performance of the proposed method with the state-of- the-art methods.
6.1 EXPERIMENTAL SETUP
Benchmark Dataset. We use two image datasets and two tabular datasets to evaluate the methods. MNIST-USPS dataset consists of 60,000 MNIST I and 7,291 USP hand written gray scale digits. We consider the source of the image i.e., MNIST, USPS as sensitive attribute with c 10 clustering problem. MTFL Zhang et al. 2014) consists 12,995 facial images and its landmark information. also provides information such gender and wearing glasses. By following (Li et al. 2020), wearing glasses or not as sensitive attribute and c = clustering problem with desired clustering attribute is gender.
We pre-process the image dataset by normalizing the pixel value. The normalization parameters mean 0.1307, std 0.3081 for MNIST-USPS, and mean= (0.3527, 0.3902, 0.4697), and std= (1, 1, for MTFL respectively.
Comparing Methods. To evaluate method, compare with the following related methods in the experiments. ScFC (Backurs et al. 2019) is non-deep fair clustering method that approximates fairlet decomposition algorithm
linear run time. ALG (Bera al. 2019) is non-deep fair clustering method that is based on k-median approach. DFC (Li et al.) 2020) is a deep fair clustering method learn fair and clustering-favoree favorable representation by adversarial loss and cluster modules with an individual group. VFC (Ziko et al. 2019) is a variational framework for fair clustering with KL fairness as clustering objective.
As baseline and reference, we use k-means++ and perfect clustering. We use the same backbone structure for deep fair clustering methods for the fair evaluation. For USPS-MNIST, we pretrain the encoder to reconstruct the original image VAE following DFC (Li et al. 2020). For MTFL, we adopt ResNe150 (He et al. 2016) pretrained with ImageNet for the encoder. We used Adam optimizer (Kingma & Ba 2014 with learning rate as 10-5, We implement all experiments Nvidia Quadro RTX 6000 and Intel 19.9 99-9960X with 128GB RAM.
Evaluation Metric. For the evaluation, measure performance with accuracy and NMI (Strchl & Ghosh 2002), and fairness with accuracy difference between sensitive groups, balance, and FFD. The four metrics can be computed as:
Ev. mi log man
Accuracy Exwx alarg P/ y']
NMI
n
n V ae log nk )E) n log
Tak
Rukk
Balance = min
min
Dak Touk
FFD2
max
Up Unlnx
Va Vilnz
12+
|I
Vi || 1 TH(UUU TT(VA
vig
n2
7ik
We denote n, T&g Pukk nj., and n as total number of samples, number of samples predicted as cluster kt cluster with group u, has ground truth label j. and samples intersected with & and j. Also, y indicates the true label of x', which is matched the clusters by the linear sum assignment problem to find the best pair between the predicted cluster and the true label for calculating accuracy. The lower bound of Fréchet Distance (FD) can be calculated by simply omitting the term the above equation.
http://yann... lecun. wwtp/o/.//mccccccoct
https:// kaggle com/ www.uaagga.uusssaaaeaeaa
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MNIST-USPS
MTH.
Acc
NMI
Balance
FFD
Acc
NMI
Balance
FFD
Perfect
1.0
1.0
0.12
14.90
1.0
1.0
0.145
7.57
k-mcans++
0.593
0.503
0.054
16.24
0.727
0.228
0.076
7.57
ScFC Backurs et al. 2019
0.353
0.258
0.072
13.63
0.731
0.182
0.080
8.72
ALG (Bera el al. 2019
0.581
0.487
0.091
15.80
0.747
0.227
0.124
8.80
VFC
Ziko ct al.
2019p
0,580
0.522
0.115
14.67
0.756
0.227
0.140
8.79
DFC Li ct al. 2020
0.824 0.828
0.053
14.13
0.731
0.163
0.137
48.02
Ours (only Lata)
0.768
0.762
0.058
2.04
0.762
0.211
0.081
46,30
Ours
0.831 0.837
0.091
1.82
0.728
0.154
0.113
43.58
Table Evaluation of clustering methods on two datasets: MNIST-USPS and MTFL. For accuracy and NMI, it is higher the better. Balance is better if it is closer to perfect clustering i.e. original data statistic. FFD" measures distributional independence of sensitive attribute, i.e., the lower, the better. FFD is measured in the learned representation (resp. input space) space for the deep (resp. non-deep) models. FFD measurement the input space is underlined.
6.2 QUANTITATIVE EVALUATION
In Table we report the quantitative evaluation of two image datasets. For accuracy and NMI, the higher, the better, and balance is better if it is close to that of perfect clustering. For FFD, it is lower the better. To calculate FFD, we set all comparing deep models (DFC and ours) to have the same dimension in the representation space. In addition, we preprocessed the latent features from each model by normalizing the maximum magnitude to I for a fair comparison. For non-deep models, we measure FFD in the original input space, and the values underlined. Note that we do not directly compare FFD from deep and non-deep models since they are calculated in different spaces.
In the table, observe some non-deep fairness methods achieve lower accuracy than classical k-means++, which is sacrificed to have better balance. With the proposed method, we achieve comparable or better results on both accuracy and balance compared with the baselines. Moreover, we could achieve a significantly lower FFD than the other deep fair method, DFC (Li al. 2020). As an ablation study, we evaluate our framework with the same structure without the fair loss term. We empirically found that integrating faiir in training sometimes favorably contributes to not only fairness but also performance.
It is interesting to note that ScFC (Backurs et al. 2019) got lower FFD than the perfect clustering in MNIST-USPS. Thus FFD can be also a good measure reveal how biased the dataset itself is against some demographic groups, e.g., imbalanced data or under-represeentaiion analysis.
6.3 QUALITATIVE ANALYSIS
In this subsection, we qualitatively evaluate fairness of the learned representation proposed in the paper comparing with other deep fair clustering methods. Fig. 2 illustrates t-SNE (Van der Maaten & Hinton] 2008) visualization of the original data, the learned representation of our model, and DFC on MNIST-USPS dataset. The colors in top and bottom rows indicate different ground truth labels and sensitive attributes, respectively. The first two columns show the progress of our model in the training process. The last column in the figure is the visualization of DFC after converges. phase, as in Fig. 2al we observe that representation is clustered based on the sensitive attribute. This shows that for the pretraining of the encoder or some downstream tasks, sensitive information takes an important role, which is not desirable. At last, as in Fig. 2b we could achieve similar distribution between different sensitive groups within a cluster. This can be explained by the proposed objective functions that our Lfair aims to learn the representation that follows the multivariate normal distribution for all sensitive groups meanwhile the centroids of a different sensitive group within the cluster. It is noticeable that the representation from DFC is highly identifiable compared with ours. This would result in potential bias in downstream tasks or possibly generating clusters with the same sensitive group.
the starting
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68
(a) Ours (iteration=0)
(b) Ours (iteratio=100000)
(c) DFC Li al. 2020
Figure 2: 1-SNE visualization of FFC and DFC from 1024 randomly selected samples MNIST- USPS dataset. Samples with different colors indicate different ground truth labels (10 digits) and different sensitive groups (MNIST, USPS) for the top and the bottom row.
Acc (Diff)
NMI
Balance
FFD
Perfect
1.0 (0.0)
1.0
0.12
DFC
0.824 (0.160)
0.828
0.053
14.13
DFC (k-means++)
0.812 (0.115)
0.754
0.044
7.61
Ours
0.831 (0.015)
0.837
0.091
1.82
Ours (k-means+++
0.831 (0.016) 0.834
0.090
1.82
Table 3: Evaluation the learned representations from deep networks on MNIST-USPS dataset. We compare the end-to-end deep model and adopt k-means++ clustering method. The representation with lower FFD achieves more stable and fair results.
6.4 JUSTIFICATION OF FAIR FRÉCHET DISTANCE AS FAIRNESS METRIC
Representation learning for clustering using deep networks can benefit from their structure of discovering intrinsic features that are difficult to observe in raw data. However, as we mentioned in (unfair), they are more vulnerable being
the motivation, if samples clustered with extrinsic features i.e., sensitive attribute.
compurationalll--deetfffaa
To validate this claim, conduct classical k-means++ algorithm cluster the learned representa- tion from our method and DFC. In Table we summarize the results. As expected, DFC lost more accuracy and NMI compared to ours when the learned features are clustered with k-means++ because the FFD was higher than ours. In contrast, we observe almost identical results by k-means++ when we train with our representation. Also, we achieve better balance and NMI compared with DFC variant. This confirms that FFD is good metric of fair clustering as the representation with lower FFD consistently outcomes fair clusters. This is also shown qualitatively by t-SNE representation. When the representation is computationally-ide -ineeniillll-deeniiiiabllll and easily separable by the sensitive attribute, this can result in subsequent unstable and unfair clustering.
CONCLUSION AND DISCUSSION
this paper, we define computationally unidentifiable fairness as novel notion fairness to measure distributional independence of sensitive attributes by leveraging Fréchet distance. Furthermore, elaborate on the theoretical analysis of the proposed metric and find some interesting properties. We integrate contrastive learning and distributional constraint achieve state-of-the ofte--of-the-att the-ttt-rt performance while maintaining ppppuaaiionlluunddenfffaaaaaaiii other fair clustering methods various benchmark datasets to validate claim.
We report experimental results comparing with

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Under review as a conference paper at ICLR 2023
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12


Reviews:
[{"review_id": "Cp3rrKDMzPA", "reviewer": "Reviewer_mwVu", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper introduces computational-unidentifiability, a concept for fair representation learning which essentially makes it challenging to identify the sensitive attribute of a sample within cluster, by ensuring that the two distributions remain the same. An associated metric called the fair fretchet distance is proposed. An algorithm for clustering that minimizes the KL-divergence based fairness loss (ensuring the within cluster samples) have low KL divergence under gaussian assumptions is demonstrated to have better fair fretched distance on benchmark datasets with improvement in downstream fairness tasks", "summary_of_the_review": "I am concerned that the contribution is lacking some conceptual nuances and choices made for the algorithm are not fully justified. Hence I will review the rebuttal to see if authors are able to address all concerns. ", "strength_and_weaknesses": "1. Definition of distributional independence that is distinct from statistical independence is not clear, and seems unnecessary given that downstream task is mostly only minimizing loss terms in terms of the KL divergence.\n\n2. The choice of KL-divergence to impose a fairness constraint is odd, and very specific. Why isn't a non-parametric distance like MMD preferred?\n\n3. Authors don't provide any intuition on why the fair fretchet distance is a good metric beyond bounding it with the fretched distance.\n\n4. Computational unidentifiability seems to imply that the distribution for the sensitive attributes within cluster are essentially the same. Hence I am a bit concerned about why only a limited clustering algorithm based on KL is proposed. Many distributions don't have closed form estimators, and non-parametric estimates are challenging. On the other hand, scaling MMD seems much easier.\n\n5. More baselines could be added such as the ones authors themselves cite - fair-centroid?\n\n6. Overall clustering being ill-defined, one could also compare all fair representation learning methods that impose direct independence between the learned representation and the outcome (since the goal seems to be to improve downstream classification fairness)\n\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "9wlO9uk3iP", "reviewer": "Reviewer_Tw9y", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "In this paper, the authors propose a method for deep clustering which attempts to match the distributions of several sensitive groups within each cluster. They propose a metric/loss based on Frechet distance, and incorporate this into a learning framework. Empirically, they show how this loss can result in clusters with the desired property.", "summary_of_the_review": "I think this paper could use some improvement in terms of the communicating the usefulness of the fairness notion they are considering, as well as general clarity improvements around the exposition, method and experiments. As such, I'm recommending a rejection.", "strength_and_weaknesses": "I think fair clustering is a useful direction and there are some novel ideas in here, but overall I'm not sure I totally grasp the direction this paper is heading in.\n\nSome feedback:\n- Introduction: the introduction makes some claims about novelty that I don't think are supported. The authors introduce \"computational-unidentifiability\" which they later define as the property that \"no external classifier can distinguish which sensitive group the sample is drawn from\" (Sec 3, Motivating Problems). They claim that this is a novel approach, and that in general, \"fairness concerns in the [unsupervised learning] space have been overlooked\". I think these novelty claims are misguided given the literature on adversarial fair representation learning (see Edwards & Storkey [1] for the first instance of these ideas, and Madras et al [2] for a thorough examination of an adversarial fairness metric which encapsulates this idea. There is also a literature on fair representation learning outside of adversarial notions which these novelty claims seem to ignore. Now, as far as I know, the adversarial fair representation learning literature doesn't usually consider clustering, and I believe that application is novel - however, the introduction and novelty claims of the paper should be rescoped accordingly.\n- Introduction: the exposition is a little bit confusing. Re: clarity, the introduction really only mentions clustering in passing, instead discussing unsupervised learning/representation learning more generally. However, I think it would be good to focus the introduction much more clearly on clustering, given that this paper is only concerned with clustering.\n- Fair Frechet Distance: I'm a little bit confused by the metric Fair Frechet Distance, and which corresponding notion of fairness it is intended to enforce. As far as I can tell, it seems like FFD is intended to minimize distributional distance between groups within a cluster. However, for the notion of a fair representation, and if we are concerned about a downstream classifier identifying these groups, shouldn't we also be concerned about the distribution of groups *between* clusters? I think there should be a little more work done in the paper to justify why this is a good metric for fairness specifically.\n- Sec 5: I think reading between the lines I can figure out how the encoder and clustering objective work together, but I think it would be helpful to have some more explanation here - up to this point there is no mention of an encoder, and it would be helpful for those not so familiar with the literature to explain how it is incorporated (rather than just doing clustering on fixed representations).\n- Table 2: I'm not quite sure what the takeaway is from the MTFL column - it seems that the proposed methods are much worse than a number of the baselines on FFD. There should probably be a bit more explanation about what is happening here.\n\nNotes:\n- Sec 6.1, 2nd paragraph: \"desired clustering attribute is gender\" - can you explain what a desired clustering attribute is here? \n- some sloppiness in language throughout - for instance, bottom of page 8: \"It is noticeable that the representation from DFC is highly identifiable compared with ours\". I think what the authors are trying to say is that it is easier to identify, within a cluster, the sensitive attribute of interest from the DFC representations than from their proposed method's. Being a little bit more precise with expressing these notions I think will go a long way to improving the clarity of the paper\n- I find section 6.4 a bit confusing - I'm not quite sure what the experimental protocol is here, and what is being compared \n\n\n[1] Edwards, Harrison, and Amos Storkey. \"Censoring representations with an adversary.\" arXiv preprint arXiv:1511.05897 (2015).\n[2] Madras, David, et al. \"Learning adversarially fair and transferable representations.\" International Conference on Machine Learning. PMLR, 2018.\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works."}, "extra_text": {}}, {"review_id": "7QjE3ZhOWbQ", "reviewer": "Reviewer_7PFC", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper attempts to address the fair representation learning method for unsupervised learning. They define a notion of fairness, computational-unidentifiability, which suggests the fairness of the representation as the distributional independence of the sensitive groups. They also propose a new fairness metric, FFD, to quantify the computational-unidentifiability. Then, some comparison experiments are also conducted to validate the effectiveness of the proposed method in various downstream tasks. ", "summary_of_the_review": "This paper provides a set of solutions for fair representations, including a novel metric and a framework. Some experiments are also conducted. I think it is good enough for publication. ", "strength_and_weaknesses": "Strengths:\n1. They propose a novel metric that quantifies fairness in representation space. And they provide rigorous analysis of the theoretical property and complexity of the fairness metric. \n2. They propose a framework for fair representation learning for downstream tasks.\n3. They compare the proposed method with the state-of-the-art fair method in the literature (Balance). \n\nWeaknesses:\n\n\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}]

Author rebuttal:

Decision: Reject