A F AIR G ENERATIVE M ODEL U SING T OTAL V ARIATION D ISTANCE
Anonymous authors
Paper under double-blind review
A BSTRACT
We explore a fairness-related challenge that arises in generative models.
The challenge is that biased training data with imbalanced representations of demographic groups may yield a high asymmetry in size of generated samples across distinct groups.
We focus on practically-relevant scenarios wherein demographic labels are not available and therefore the design of a fair generative model is particularly challenging.
In this paper, we propose an optimization framework that regulates the unfairness under such practical settings by employing one prominent statistical notion, total variation distance (TVD).
We quantify the degree of unfairness via the TVD between the generated samples and balanced-yet-small reference samples.
We take a variational optimization approach to faithfully implement the TVD-based measure.
Experiments on benchmark real datasets demonstrate that the proposed framework can signiﬁcantly improve the fairness performance while maintaining realistic sample quality for a wide range of the reference set size all the way down to 1% relative to training set.
I NTRODUCTION
High-quality realistic samples synthesized thanks to recent advances in generative models (Brock et al., 2019; Goodfellow et al., 2014;
Karras et al., 2019) have played a crucial role to enrich training data for a widening array of applications such as face recognition, natural language processing, and medical imaging (Wang et al., 2019; Chang et al., 2018;
Yi et al., 2019).
One challenge concerning fairness arises when generative models are built upon biased training data that preserve unbalanced representations of demographic groups.
Any existing bias in the dataset can readily be propagated to the learned model, thus producing generations that are biased towards certain demographics.
The unbalanced generated samples may often yield undesirable performances against underrepresented groups for downstream applications.
One natural way to ensure fair sample generation is to exploit demographic labels (if available) to build a fair generative model, e.g., via conditional GAN (Mirza & Osindero, 2014;
Odena et al., 2017; Miyato & Koyama, 2018) which employs such labels to easily generate an arbitrary number of samples for minority groups.
In many practically-relevant scenarios, however, such labels are not often available.
To address the challenge, one pioneering work (Choi et al., 2020) develops a novel debiasing technique that employs the reweighting idea (Ren et al., 2018; Kamiran & Calders, 2012; Byrd & Lipton, 2019) to put more weights to underrepresented samples, thereby promoting fair sample generation across demographic groups.
One key feature of the technique is to identify the bias (reﬂected in the weights) via a small and unlabelled reference dataset.
While it enjoys signiﬁcant fairness performance for moderate sizes of the reference dataset, it may provide a marginal gain for a more practically-relevant case of a small set size where the weight estimation is often inaccurate, as hinted by the meta-learning literature (Ren et al., 2018; Shu et al., 2020).
We also ﬁnd such phenomenon in our experiments; see
Table 2 for details.
On the other hand, one recent study (Roh et al., 2020) sheds lights on addressing the small set size issue.
Roh et al.
(2020) propose a robust training approach by employing the Jensen-Shannon divergence (or equivalently mutual information (Majtey et al., 2005)) between poisoned training and clean reference samples.
It then takes the divergence as a regularization term in order to promote robust training.
One key beneﬁt of Roh et al. (2020)in light of the reweighting-based approaches (Ren et al., 2018; Choi et al., 2020) is that the robustness performance is guaranteed even for a small size of the clean reference set down to 5% relative to the training set size.
This implies that the divergence-based regularization approach makes a more efﬁcient use of reference data for robustness, as compared to the reweighting technique.
Contribution: Inspired by the key beneﬁt featured in Roh et al.
(2020), we address the small set issue by another well-known statistical measure, in particular, total variation distance (TVD) .
See Section 3.2 for details on the rationale behind the use of TVD.
Similarly we introduce a reference dataset which is balanced and unlabelled.
We then employ it to formulate the TVD between the generated and reference sample distributions, which can serve as quantifying the degree of unfairness.
We then promote fair sample generation by adding the TVD into a conventional optimization (e.g., GAN-based optimization (Goodfellow et al., 2014;
Nowozin et al., 2016; Arjovsky et al., 2017)).
Motivated by the variational optimization technique w.r.t.
TVD (Villani, 2009;
Nowozin et al., 2016; Arjovsky et al., 2017), we translate the TVD-regularization term into a function optimization.
We also conduct extensive experiments on three benchmark real datasets: CelebA (Liu et al., 2015),
UTKFace (Zhang et al., 2017), and FairFace (Karkkainen & Joo, 2021).
We demonstrate via simulation that the proposed framework can signiﬁcantly boost up the fairness performance while offering high-quality realistic samples reﬂected in low FID.
We also ﬁnd that our approach outperforms the state of the art (Choi et al., 2020), particularly being robust to the balanced reference set size: the signiﬁcant improvements preserve for a wide range of the reference set size down to 1% relative to training data (more preferable in reality).
Related works: In addition to Choi et al.
(2020), Tan et al.
(2020) propose a different way that promotes fair sample generation by smartly perturbing the input distribution of a pre-trained generative model with the help of a classiﬁer for sensitive attributes.
The key distinction w.r.t.
ours is that it relies upon the additional classiﬁer which requires the use of demographic labels to obtain.
Yu et al.
(2020) employ demographic labels for minority groups to generate a wide variety of samples with improved data converge by harmonizing GAN and MLE ideas.
A distinction w.r.t.
ours is that it requires the knowledge on demographic labels.
Jalal et al.
(2021) consider a fair generative model yet in a different context, image reconstruction.
The goal of the task is to ensure fair sample generation of restored images from degraded versions.
Since it relies upon the degraded images, it is not directly comparable to ours.
Another line of fair generative modeling focuses on label bias, instead of representation bias (Xu et al., 2018; 2019a;b; Sattigeri et al., 2019; Jang et al., 2021; Kyono et al., 2021).
The goal therein is to develop a generative model such that the generated decision labels are statistically independent of the given demographic labels.
Again, these are not directly comparable to ours, as they require the use of demographic labels.
The variational optimization technique w.r.t.
TVD that gives an inspiration to our work has originated from Villani (2003; 2009), wherein the author shows that the TVD can be expressed as a function optimization in the context of transport theory.
The technique was recently applied to the GAN context (Nowozin et al., 2016).
The TVD has also served as a useful tool for quantifying various fairness measures in fair classiﬁers that pursue individual fairness (Dwork et al., 2012; Dwork &
Ilvento, 2018) and group fairness (Gordaliza et al., 2019; Wang et al., 2020a; Farokhi, 2021).
P ROBLEM F ORMULATION
Setup: Fig.
1 illustrates the problem setting for a fair generative model that we focus on herein.
We consider a challenging yet practically-relevant scenario wherein demographic information (or that we call sensitive attribute), say z ∈ Z , is not available.
Under this blind setting, the goal of the problem is to construct a fair generative model that ensures the produced samples to have the same size across distinct demographics.
We assume that there are two types of data given in the problem: (i) training data D bias :=
{ x ( i ) bias } m bias i =1 ; (ii) reference data D ref :=
{ x ( i ) ref } m ref i =1 .
Since we consider the setting where training data is potentially biased, we use the word “ bias ” in the associated notations.
Here m bias denotes the number of training examples.
Let P bias be data distribution which each training data x ( i ) bias ∈ X is generated upon.
In a biased scenario having female-vs-male sensitive attribute, e.g., z = 0 (female) and z = 1 (male), we may have P bias ( Z = 0) >
P bias ( Z = 1) .
As in Choi et al.
(2020), we also employ a balanced yet small reference dataset for the purpose of promoting fair sample generation, which can be obtained via a set of carefully-designed data collection protocols employed in organizations like World Bank and biotech companies (Choi et al., 2020; 23&me, 2016; Hong, 2016).
Let P ref be the corresponding data distribution deﬁned on X :
= ≈ P the number of the samples than that examples m ref m bias
Denote by ˆ x := G ( w ) ∈
X the generated sample fed by a random noise input w ∈ W .
We assume that the generated samples have the same support X as training and reference samples. Let P G and P W be distributions w.r.t. generated samples and the random noise input respectively.
As a fairness measure that will be employed for the purpose of evaluating our framework to be presented in Section 3, we consider fairness discrepancy proposed by Choi et al.
It quantiﬁes how P G differs from P ref w.r.t.
a certain sensitive attribute, formally deﬁned below.
Deﬁnition 1 (Fairness Discrepancy (Choi et al., 2020)) .
Fairness discrepancy between P ref and P G w.r.t.
a sensitive attribute z ∈ { z 1 , . .
, z |Z| } is deﬁned as:
[Equation]
(1) where
[Equation]
 .
Here ˆ Z denotes the prediction of the sensitive attribute w.r.t.
a generated sample, yielded by a pre-trained classiﬁcation model which we call attribute classiﬁer as in Choi et al.
The attribute classiﬁer is employed only for the purpose of evaluation, and is trained based on another real dataset, e.g., like the one mentioned in Choi et al.
(2020): the standard train and validation splits of CelebA (Liu et al., 2015).
For faithful evaluation, we consider a vast number of generated samples (i.e., more than 10,000) as well as employ highly-accurate attribute classiﬁers, around 98% accuracy of gender classiﬁer for instance.
As a measure for the quality of generated samples that may compete with the fairness measure, we employ a well-known measure: Fréchet Inception Distance (FID) (Heusel et al., 2017).
It is deﬁned as the Fréchet distance (Fréchet, 1957) (also known as the second-order Wasserstein distance (Wasserstein, 1969)) between real and generated samples approximated via the Gaussian distribution.
The lower FID, the more realistic and diverse the generated samples are.
For a more precise measure that reveals sample quality of each sensitive group, we consider FID computed within each demographic, called intra FID (Miyato & Koyama, 2018;
Zhang et al., 2019; Wang et al., 2020b).
Computing intra FID requires the knowledge on group identities of generated samples.
Since demographic labels are not available in our context, we rely upon the attribute classiﬁer (that we introduced above) for predicting demographic information of the generated samples.
GAN-based generative model: Our framework (to be presented soon) builds upon one powerful generative model:
Generative Adversarial Network (GAN) (Goodfellow et al., 2014).
The GAN comprises two competing players: (i) discriminator D ( · ) that wishes to discriminate real samples against generated samples; and (ii) generator G ( · ) that intends to fool the discriminator by producing realistic generated samples.
In particular, we consider a general optimization framework (Tseng et al., 2021) which subsumes many GAN variants as special cases:
[Equation]
where f D , f G , and g G indicate certain functions that vary depending on an employed GAN approach.
For instance, the choice of ( f D ( t ) , f G ( t ) , g G ( t )) =
( t, − t, t ) together with Lipschitz-1 condition on D leads to the prominent WGAN optimization (Arjovsky et al., 2017).
Another choice of ( f D ( t ) , f G ( t ) , g G ( t )) =
(min { 0 , − 1 + t } , min { 0 , − 1 − t } , t ) yields a hinge-loss-based GAN (Lim & Ye, 2017; Tran et al., 2017).
We adopt this as a base framework that we will add a fairness aspect into in the next section.
P ROPOSED F RAMEWORK 3. TVD- BASED APPROACH
One natural way to enforce a fairness constraint is to incorporate a fairness-regularization term into the base framework.
However, since the base framework consists of two distinct optimizations, how to add the regularization term is not clear.
To gain some insights into this, we focus on one instance of GAN, which allows us to express the two as only one optimization and then relate the optimization to a well-known statistical notion: total variation distance (TVD).
To this end, we ﬁrst consider the following mappings: ( f D ( t ) , f G ( t ) , g G ( t )) =
( t, − t, t ) .
Applying these into equation 2 gives:
[Equation]
We now employ the variational optimization technique presented in Villani (2003; 2009); Nowozin et al.
; Arjovsky et al. (2017) to translate equation 3 into the TVD between P bias and P G .
Details are described in Theorem 1 below.
Theorem 1 (Nowozin et al.
The optimization in equation 3 with the bounded constraint
D ( · ) ∈ [ − 1 , 1] is equivalent to:
[Equation]
where
[Equation]
Proof.
See appendix A.
3. F AIR GENERATIVE MODEL
Now we have the TVD-based framework with the single optimization.
So one natural way to impose a fairness constraint is to add a proper regularization term on top of equation 4.
Remember our framework employs the reference dataset containing balanced samples, reﬂected in P ref .
Hence, as the regularization term, we may consider a distance measure between P ref (target distribution) and P
G , since a large distance penalizes the objective.
To this end, we propose to use the same TVD as was used in equation 4:
[Equation]
where λ ∈ [0 , 1] denotes a normalized regularization factor that balances the sample quality against the fairness constraint.
The rationale behind the use of TVD regularization is that it is robust to the size of dataset compared to a set of f -divergences (Tseng et al., 2021). Moreover, it offers the best trade-off performances in fairness and sample quality compared to other divergence measures:
Jensen-Shannon divergence (Wong & You, 1985), Kullback-Leibler divergence (Kullback & Leibler, 1951), Pearson χ 2 -divergence (Pearson, 1900), and Wasserstein distance (Wasserstein, 1969).
We provide empirical evidence that demonstrates this performance beneﬁt in Section 4.2; see Table 3 for details.
One challenge that arises in equation 5 is that expressing the TVDs in terms of an optimization variable
G is not that straightforward.
For ease of implementation, we rely upon Theorem 1 again, which shows the equivalence between the TVD-based optimization and the two-player game (equation 3).
Since equation 5 includes one more TVD, we introduce the third player, who serves as another discriminator trying to distinguish generated samples from the balanced reference samples.
Based on this, we propose a three-player game which is equivalent to equation 5.
See Theorem 2 for the formal statement of the equivalence and its proof.
Theorem 2.
The optimization in equation 5 is equivalent to:
[Equation]
Proof.
Manipulating equation 6, we obtain:
[Equation]
where the equality is because we assume the same support X for all data samples.
Under bounded discriminator functions D ( · ) , D fair ( · ) ∈ [ − 1 , 1] , one can readily obtain the optimal discriminators as:
[Equation]
x ) = sign { P ref ( x ) − P G ( x ) } .
Substituting these into equation 7, we get equation 5.
This completes the proof.
The optimization structure of the new discriminator D fair is the same as that of D , except that we read P ref instead of P bias .
The generator optimization has an additional term w.r.t.
D fair ( X ) accordingly.
The translated three-player game can then be implemented, e.g., via parameterization of three neural networks.
We then employ a three-way alternating gradient descent (Goodfellow et al., 2014) for the parameterized neural networks.
This procedure is formally presented in Algorithm 1.
Remark 1 (Implementing bounded functions via hinge loss) .
Among a multitude of practices for implementing bounded functions D ( · ) , D fair ( · ) ∈ [ − 1 , 1] , we take an approximated approach that offers empirical beneﬁts: the use of hinge loss (Lim & Ye, 2017; Tran et al., 2017).
As expressed in
Steps 7 and 12 in Algorithm 1, we apply the hinge loss in both D and D fair so that the discriminators attempt to constrain themselves within the interval [ − 1 , 1] .
We found via extensive experiments that this choice enables more stabilized training, thereby offering greater performances relative to a direct and simple approach: employing tanh activations in D and D fair .
Actually this hinge-loss-based approximation is inspired by Nguyen et al.
Arjovsky et al.
Tan et al.
(2019) which explore in depth the equivalence between TVD and hinge-loss-based optimization.
(cid:4)
1: Input: Training data D bias = { x i ) bias } m bias i =1 , reference data D ref = { x ( i ) ref } m ref i =1 , noise input distribution P W , and several hyperparameters: λ ; the numbers of discriminator steps k and l ; and the number of iterations T 2: Initialize:
D ( φ discriminator
D fairness
) 3: for T steps do 4: for k do { Loop for training D (
Sample minibatch of m B noise inputs { w (1) , . .
, w ( m B ) } from P W 6:
Sample minibatch of m B training data points { x (1) bias , . . .
, x ( m B ) bias } from D bias at random 7:
Update D ( x ; φ ) by ascending its stochastic gradient: ∇ φ 1 − λ m B m B (cid:88) i
= (cid:2) min (cid:8) 0 , −
+ D ( x ( i ) bias ; φ ) (cid:9) + min (cid:8) 0 , − 1 − D ( G ( w ( i ) ; θ ); φ ) (cid:9)(cid:3) 8: end for 9: for l steps do { Loop for training D fair ( x ; ψ ) } 10:
Sample minibatch of m B fair noise inputs { w (1) , . .
, w ( m B fair ) } from P W 11:
Sample minibatch of m B fair reference data points { x (1) ref , . . .
, x ( m B fair ) ref } from D ref at random 12:
Update D fair ( x ; ψ ) by ascending its stochastic gradient: ∇ ψ λ m B fair m B fair (cid:88) i = (cid:2) min (cid:8) 0 , − 1 + D fair ( x ( i ) ref ; ψ ) (cid:9) + min (cid:8) 0 , − 1 − D fair ( G ( w ( i ) ; θ ); ψ ) (cid:9)(cid:3) 13: end for 14:
Sample minibatch of m B G noise inputs { w (1) , . .
, w ( m B G ) } from P W 15:
Update G ( w ; θ ) by descending its stochastic gradient:
[Equation]
16: end for Remark 2 (Three-way battles) . Fig.
2 illustrates the entire architecture of the translated three-level optimization.
Here we see interesting three-way battles . The ﬁrst is a well-known battle between the generator G and the 1st discriminator D . Remember D ∗ ( x ) = sign { P bias ( x ) − P G ( x ) } .
So one can interpret D ∗ as the strength of distinguishing real (potentially biased) samples against generated samples.
On the other hand, the generator intends to fool D , thus promoting realistic samples.
The second battle is in between the generator and the 2nd discriminator D fair .
The same interpretation can be made from D ∗ fair ( x )
= sign { P ref ( x ) − P G ( x ) } (the ability to distinguish balanced reference samples against the generated samples).
This way, the generator G is encouraged to produce balanced yet less realistic (due to the small-sized reference set) samples, thus pitting the 1st discriminator against the 2nd discriminator indirectly .
The last battle is in between the 1st and 2nd discriminators.
This tension is directly controlled by the fairness tuning knob λ ; see corresponding tradeoff curves presented in Fig.
4 in appendix E.4.
It turns out the three-way tradeoff relationships established via our TVD-based framework are greatly balanced, thus achieving signiﬁcant performances both in fairness and sample quality.
This is empirically demonstrated in Section 4.2; see Tables 1 and 3 for details.
(cid:4)
E XPERIMENTS
We conduct experiments on three benchmark real datasets: CelebA (Liu et al., 2015),
UTKFace (Zhang et al., 2017), and FairFace (Karkkainen & Joo, 2021).
We implement our algorithm in
PyTorch (Paszke et al., 2019), and all experiments are performed on servers with TITAN RTX and
Quadro RTX 8000 GPUs.
For our algorithm, all the simulation results (to be reported) are the ones averaged over ﬁve trials with distinct seeds.
4. S ETUP
Datasets: Our construction of D bias and D ref respects the method described in Choi et al.
Only for the purpose of data construction, we have an access to sensitive attributes z , so as to control the ratio of demographic group sizes.
For CelebA, we consider two scenarios depending on the number of focused attributes: (i) CelebA-single (gender); (ii) CelebA-multi (two attributes: gender and hair color).
Training data D bias is constructed to have 9 : 1 ratio (female vs. male) samples where m bias = 67507 .
We take balanced samples for D ref ( 1 : 1 ratio).
For CelebA-multi, we have four groups: (i) (female, non-black); (ii) (male, non-black); (iii) (female, black); (iv) (male, black).
For D bias , we take 85 : 15 ratio samples (non-black hair vs. black hair) where m bias = 60000 .
For
UTKFace dataset, we consider a race attribute: white vs. non-white.
We take 9 : 1 ratio biased samples with m bias ≈ 10000 .
For FairFace dataset, we consider another type of race categorized as white vs. black.
We also take the 9 : 1 ratio biased samples yet with m bias ≈ 20000 .
A wide range of the reference set size is taken into consideration.
We focus mainly on two sizes: (i) 10% ( m ref ≈ 0 .
1 m bias ) ; (ii) 25% ( m ref ≈ 0 .
25 m bias ) .
To demonstrate the robustness of our proposed approach to the reference set size, we also consider small sizes of the reference set down to 1%.
See appendix B.1 for more details.
Baselines: We consider three baselines.
The ﬁrst baseline, say Baseline I, is a non-fair algorithm building upon the base framework described in equation 2.
It is trained on the aggregated dataset
D bias ∪ D ref .
The second baseline, say Baseline II, is the same non-fair algorithm yet trained only with a small balanced reference set D ref .
The last is the state of the art, Choi et al.
For all three baselines, we employ the hinge loss optimization (Lim & Ye, 2017; Tran et al., 2017): choosing the mapping functions in equation 2 as ( f D ( t ) , f G ( t ) , g G ( t ))
= (min { 0 , − 1 + t } , min { 0 , − 1 − t } , t ) .
Attribute classiﬁers: As mentioned in Section 2 (near Deﬁnition 1), we employ attribute classiﬁers, only for the purpose of evaluating our twin measures: (i) fairness discrepancy (deﬁned in equation 1); (ii) intra FID.
We introduce four different attribute classiﬁers for predicting senstive attributes in the following scenarios: (i) gender for CelebA-single; (ii) gender and hair-color for CelebA-multi; (iii) white-vs-non-white race for UTKFace; (iv) white-vs-black race for FairFace.
For all the classiﬁers, we use a variant of ResNet18 (He et al., 2016).
CelebA and FairFace classiﬁers are trained over the standard train and validation splits of CelebA (Liu et al., 2015) and FairFace (Karkkainen & Joo, 2021), respectively.
For training UTKFace classiﬁer, we use 8 : 1 : 1 splits of UTKFace (Zhang et al., 2017) dataset.
We found that our evaluation is sensitive to the performances of the attribute classiﬁers; see appendix C.1 for a detailed discussion.
Hyperparameter search: For implementation of all three baselines (Baselines I, Baseline II, and Choi et al.
and the proposed framework, we all employ the BigGAN architecture (Brock et al., 2019).
In other words, we parameterize G , D , and D fair with the neural-net architecture introduced in Brock et al.
We leave details in appendix B.2.
We also provide a complexity analysis of our algorithm with a comparison to the state of the art (Choi et al., 2020); see appendix D for details.
4. R ESULTS
Table 1 provides performance comparison with the three baselines on CelebA dataset.
Notice for all the considered settings that our approach exhibits better (or comparable) performances than the state of the art (Choi et al., 2020) both in fairness (“Fairness discrepancy”) and sample quality (“Intra
FID”).
The lower, the better for all the measures.
The performance gaps are more apparent in the small reference set size.
Notice in the last row w.r.t.
CelebA-single in Table 1 that the fairness performance exhibits slight degradation with an increase in the reference set size.
See appendix C. for an in-depth discussion on this counter-intuitive adverse effect of the increased reference set size.
Due to space limitation, we leave in appendix E.1 experimental results conducted on the other two datasets: UTKFace and FairFace.
The performance trends are similar to those in CelebA.
Fig.
3 visualizes generated samples on CelebA-single with the 10% reference set size.
The top ﬁgure corresponds to Choi et al.
(2020), while the bottom is due to the proposed algorithm.
For each ﬁgure, faces above the yellow lines are female samples, while the rest are male samples.
Here the gender is [Table] predicted via the attribute classiﬁer.
While yield well-balanced samples ( for Choi et al.
(2020), and ours; see Table 1 for a more comparison), our algorithm produces more realistic sample images.
This is reﬂected in lower intra FID values, around 9 .
(female group) and 14 . [-
 (male-]29 ( group).
On the hand, et
(2020) offers intra FID values of around 21 .
07 and 25 .
74 for female and male groups, respectively.
See Table(in appendix E) for the values of intra FID.
In appendix F, we also provide generated samples for UTKFace and FairFace datasets.
[Table] Table 2 demonstrates the robustness of our algorithm to the reference set size.
Notice even for the 1% reference set size, our algorithm offers still respectable performances both in fairness and sample quality.
On the other hand, Choi et al.
(2020) suffers from performance degradation starting from 2 .
Refer to appendix E.2 for intra FIDs w.r.t.
the female group.
Table 3 demonstrates the rationale behind the use of TVD-based regularization.
Observe that among the considered regularization methods, our TVD-based approach offers the best (or the second best) performances both in fairness and sample quality.
It also yields the smallest discrepancy between intra
FIDs of different groups.
Another noticeable observation is that our divergence-based regularization approach outperforms Choi et al.
(2020) for a variety of other divergence measures not limited to
TVD; also see Table 1 for detailed comparison.
For implementing other measures, e.g., JSD, we replace the TVD regularization term in equation 5 with the considered measure (e
[Table]
g., JSD) while maintaining the ﬁrst term in the objective function.
We also conduct more extensive experiments which take different divergence measures even for the ﬁrst term in the objective function in equation 5.
Even in such cases, we obtain similar results which exhibit good performances when employing TVD for fairness regularization.
See appendix E.3 for the experimental results.
[Table]
C ONCLUSION
We introduced a TVD-based optimization framework for a fair generative model that well tradeoffs the fairness performance (quantiﬁed as fairness discrepancy) against sample quality (reﬂected in intra
FID).
Inspired by the equivalence between the TVD and function optimization, we also developed an equivalent three-player optimization which can readily be implemented via neural-net parameterization.
Our algorithm offers better performances than the state of the art both in fairness and intra
FID, exhibiting more signiﬁcant performances particularly for practically-relevant scenarios where the access to balanced dataset is limited.
One future work of interest is to push forward for more challenging scenarios where the reference dataset is not available.
E THICS S TATEMENT
One deﬁning feature of our proposed framework is to offer fair sample generation of demographic groups while maintaining realistic high quality of generated samples.
Hence, it is expected to play a crucial role to enrich well-balanced training data for a widening array of downstream applications, thereby ensuring great performances even for underrepresented groups which often suffer from degraded performances due to biased data.
One disadvantage of our approach requires the use of the balanced reference set.
So it may not be directly applicable to more challenging scenarios where gathering such non-biased samples are very limited.
Another ﬂip side is that our framework builds upon the adversarial training approach which often incurs training instability.
Hence, we believe one promising future direction is to push the boundary towards such challenging settings while addressing the training stability issue.
R EPRODUCIBILITY S TATEMENT
To ensure the reproducibility of experiments, we provide a thorough description regarding our three benchmark real datasets: CelebA (Liu et al., 2015),
UTKFace (Zhang et al., 2017), and FairFace (Karkkainen & Joo, 2021), along with data construction methods in Section 4.1 and appendix B.1.
All the hyperparameter settings w.r.t.
the three real datasets are presented in Appendix B.2.
We also provide the average running time of our algorithm together with speciﬁc computer conﬁguration details in Appendix D. Lastly, we provide the codes in the supplementary material.
A A PPENDIX : P ROOF OF T HEOREM
Theorem 1 (Nowozin et al.
The optimization in equation 3 with the bounded constraint
D ( · ) ∈ [ − 1 , 1] is equivalent to:
[Equation]
where
[Equation]
Proof.
We ﬁrst manipulate equation 3 as:
[Equation]
where the equality is because we assume the same support X for training and generated samples.
Under the bounded constraint D ( · ) ∈ [ − 1 , 1] , the optimal D ∗ maximizing the above summation becomes (Nowozin et al., 2016):
[Equation]
Substituting this into equation 8, we obtain:
[Equation]
where the ﬁrst equality is due to the optimal D ∗ .
This completes the proof.
B A PPENDIX : I MPLEMENTATION D ETAILS
We ﬁrst present m bias (the number of samples for training set) and m ref (w.r.t.
reference set) employed in a variety of considered scenarios.
Next we provide details on the model architectures (together with hyperparameters) employed for ( G ( · ) , D ( · ) , D fair ( · )) (in our framework) as well as for attribute classiﬁers.
B. T HE NUMBER OF SAMPLES FOR TRAINING AND REFERENCE SETS
Tables 4 and 5 show m bias and m ref employed for our experiments.
Notice that m bias (or m ref ) in
UTKFace and FairFace is much smaller than that of CelebA. As expected, this leads to the overall performance degradation of UTKFace and FairFace, as reported in appendix E; see Tables, 15, 16, and 17 therein.
[Table]
[Table] [Table]
A RCHITECTURES AND HYPERPARAMETER CONFIGURATIONS
We employ attribute classiﬁers, only for the purpose of evaluating our twin measures: (i) fairness discrepancy (deﬁned in equation 1); (ii) intra FID.
We introduce four different attribute classiﬁers for predicting sensitive attributes in the following scenarios: (i) gender for CelebA-single; (ii) gender and hair-color for CelebA-multi; (iii) white-vs-non-white race for UTKFace; (iv) white-vs-black race for FairFace.
For all the attribute classiﬁers, we use a modiﬁed version of ResNet18 (He et al., 2016); see Table 6 for architectural details.
We train the classiﬁers for CelebA and FairFace scenarios with the standard train and validation splits of CelebA (Liu et al., 2015) and FairFace (Karkkainen & Joo, 2021), respectively.
UTKFace (Zhang et al., 2017) does not provide such standard dataset splits, so we manually divide the full UTKFace dataset into 8 : 1 : 1 ratio of train/valid/test splits and use them for training the attribute classiﬁer for UTKFace.
We use the same hyperparameters for training all of the classiﬁers.
We use a batch size of 64 and adam optimizer with the learning rate 0 .
001 and ( β 1 , β 2 ) =
We also employ early stopping.
We adopt the BigGAN (Brock et al., 2019) architecture for implementing all the baselines (Baselines
I, II, and Choi et al.
and the proposed framework.
See Table 7 for architectural details.
For D fair employed in our framework, we use the same architecture as D .
Tables 8, 9, andexhibit hyperparameters used for our approach.
Tables 11 and 12 exhibit hyperparameters used for experiments with other regularizations.
For implementing JSD-based regularization, we rely upon the original GAN with non-saturating loss (Goodfellow et al., 2014).
For Kullback-Leibler divergence, we employ the loss function proposed in f -GAN (Nowozin et al., 2016).
Least Squares GAN loss (Mao et al., 2017) is employed for implementation of Pearson- χ 2 divergence.
For implementing Wasserstein distance, WGAN-GP (Gradient Penalty) (Gulrajani et al., 2017) is employed. For all the experimental results, λ was chosen as the one that yields the lowest FID among all the candidates respecting a good fairness performance, e.g., fairness discrepancy < 0 .
1 on CelebA dataset.
For a batch size, we considered the following candidates { 8 , 16 , 32 , 64 , 128 , 256 } .
We controlled the learning rates for G and D among { 5 · 10 − 5 , 10 − 4 , 2 · 10 − 4 , 4 · 10 − 4 } and { 2 · 10 − 4 , 4 · 10 − 4 } respectively.
For the number of D -step per G -step, we considered a number among { 2 , 3 , 4 } . [Table]
[Table] For the second discriminator w.r.t fairness, we swept the learning rate through { 10 − 4 , 2 · 10 − 4 , 4 · 10 − 4 } and the number of D -step per G -step through { 1 , 2 , 3 } .
For Choi et al.
, we found for UTKFace and FairFace that a simple 3 or 5-layer CNN classiﬁer better estimates example weights than the ResNet-based architecture used in Choi et al.
All baseline performances are measured using the same approach as in Choi et al.
(2020): average over 10 independent evaluation sets of 10,000 samples each drawn from the corresponding baseline model.
[Table]
C A PPENDIX : A DDITIONAL DISCUSSIONS
We ﬁrst investigate the evaluation sensitivity w.r
[Table]
t. the attribute classiﬁers that we employ for performance evaluation.
Next [Table] we discuss the impact of the reference set size on the fairness performance in our framework. Lastly
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we explore a connection between our interested setting and transfer learning framework.
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[Table]
to a regularization method based on Jensen-Shannon divergence (Wong & You, 1985) implemented via Goodfellow et al.
“KLD-based” is the one built upon Kullback-Leibler divergence (Kullback & Leibler, 1951; Nowozin et al., 2016).
“ χ 2based” represents the one implementing Pearson- χ 2 divergence (Pearson, 1900;
Mao et al., 2017).
“WD-based” refers to a regularization with Wasserstein distance (Wasserstein, 1969; Gulrajani et al., 2017).
[Table]
[Table]
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size
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[Table] rate [Table] G
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5e5e- 5e1e-
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rate
[Table] 2e2e- 2e1e- ( [Table] [Table]
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step C. E VALUATION SENSITIVITY W . R . T . ATTRIBUTE CLASSIFIERS As mentioned in Sections 2 and 4.1, we rely upon the attribute classiﬁer for evaluating our two performance measures, fairness discrepancy (see Deﬁnition 1 in Section 2) and intra FID.
One natural question is then to ask how sensitive the evaluation is to the choice of the attribute classiﬁer.
We address this question by investigating a gap between performances measured with two differentlyaccurate classiﬁers.
We found that fairness discrepancy varies signiﬁcantly depending on the accuracy of the attribute classiﬁer.
This is in contrast with intra FID which is not much affected with the choice [Table] of the attribute classiﬁer.
For instance
[Table]
10% accuracy of the attribute classiﬁer yields around 25% deviations in fairness discrepancy [Table] while exhibiting marginal deviations in intra FID values less than 1% [Table]
C. I MPACT OF REFERENCE SET SIZE ON FAIRNESS PERFORMANCE
Our framework heavily depends on reference data in regulating unfairness, and the size of the reference data indeed affects our performance of fairness. If a given reference dataset is too small to well-represent the underlying reference distribution, the measured TVD (via such small-sized reference set) would then be inaccurate, degrading the fairness performance accordingly.
On the other hand, such degradation can occur with a large reference dataset. For implementing the fairness regularization, we rely upon GAN framework that often yields a model focused on learning only a speciﬁc part of representations in a given dataset, rather than covering all the representations inside (Metz et al., 2016; Arjovsky et al., 2017). Due to this property, with a larger reference set augmented with vast amount of representations, our models (generator and fairness discriminator) can be more encouraged toward learning such augmented (yet less relevant to representations w.r.t. demographic groups) representations and therefore less focused on respecting balanced representations of demographic groups, degrading the fairness performance accordingly.
This phenomenon is well exhibited in our experiments; see the last rows in Tables 1 and 15 for instance.
C. R ELATION TO TRANSFER LEARNING
Our problem setting can be interpreted as a transfer learning framework, where we wish to exploit the knowledge learned from a large training dataset D bias to augment learning on a small reference dataset D ref . We explore such connection herein with a basic transfer learning technique, ﬁne-tuning. Employing only one discriminator, we ﬁrst train our models on D bias until convergence, and then ﬁne-tune the models on D ref .
Table 13 compares the performance of the ﬁne-tuning method with all the other approaches, evaluated on CelebA-single with 10% reference set.
Notice that the ﬁne-tuning method well-balances between Baseline I and II, yet still inferior than the proposed approach both in fairness and sample quality.
[Table]
D A PPENDIX : C OMPLEXITY ANALYSIS
We also offer complexity analysis of our approach while making a comparison to Choi et al. [Table]
As performance measures, we consider the total number of parameters as well as the running time measured under Pytorch on Xeon Silver 4210R CPU and TITAN RTX GPU.
Table 14 presents the interested measures (together with fairness and intra FID performances) for one certain scenario:
CelebA-single with 10% reference set size.
While our approach provides better intra FID and fairness performances, it comes at a cost of an increased complexity, around 16 .
7% and 40% for the running time and model complexity, respectively.
[Table]
E A PPENDIX : A DDITIONAL RESULTS
We ﬁrst provide performance comparison with the three baselines (Baseline I, Baseline II, and Choi et al.
on UTKFace and FairFace datasets.
Next we present the complete results on CelebA dataset, where intra FID values for all demographics are included.
We then compare fairness regularization performances with distinct base optimizations.
Lastly, we provide tradeoff curves that illustrate the tension between fairness and sample quality in our framework.
E. P ERFORMANCES ON UTKF ACE AND F AIR F ACE
Tables 15 and 16 concern UTKFace and FairFace datasets.
One signiﬁcant distinction w.r.t.
CelebA is that training and reference set sizes are much smaller, around 7 times.
Hence, as expected, the overall performances are worse than those on CelebA dataset.
Even in this small data regime, we observe the same trends on the performance beneﬁts of ours relative to the baselines.
[Table]
[Table]
E. C OMPLETE RESULTS ON C ELEB A
Tables 17 to 19 exhibit the complete results w.r.t.
CelebA experiments, where intra FID values for all associated groups are provided.
Observe that in all considered settings, our approach offers better (or comparable) sample quality than the state of the art (Choi et al., 2020) for all demographics, reﬂected in low intra FID values.
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[Table]
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Table 20 compares regularization performances with different divergence measures for the ﬁrst term in equation 5.
Even in this comparison, we observe that our TVD-based approach is still prominent among the considered methods, yielding well-balanced performances of fairness and sample quality.
We do not include the performance regarding Kullback-Leibler divergence, where we failed to get meaningful results due to numerical problem.
This is also reported in other works (Song & Ermon, 2020) which relies upon the same implementation (Nowozin et al., 2016).
[Table]
E. F AIRNESS - QUALITY TRADEOFF CURVES
Fig.
4 visualizes a tradeoff relationship between sample quality and fairness in our framework.
Despite slight ﬂuctuations, one can readily see that as lambda increases, fairness performance improves (smaller fairness discrepancy) at the expense of the degraded sample quality, reﬂected in larger intra FID.
This validates the role of λ as a tuning knob for controlling fairness.
F A PPENDIX : A DDITIONAL G ENERATED S AMPLES
We provide generated samples obtained from many scenarios under the three real datasets, not included in the main paper due to space constraint.
Unlike the main paper, we also present samples generated from Baseline I and Baseline II.
Remember that Baseline I is a non-fair algorithm building upon the base framework (equation 2 in Section 2) with the hinge loss (Lim & Ye, 2017; Tran et al., 2017), while being trained with the aggregated dataset D bias ∪ D ref .
Baseline II is the same non-fair algorithm yet trained only with D ref .
We leave all the samples from below.