A Fair Generative Model Using Total Variation Distance

1 INTRODUCTION . 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 ) . 2 PROBLEM FORMULATION . 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 Dbias : = { x ( i ) bias } mbias i=1 ; ( ii ) reference data Dref : = { x ( i ) ref } mref i=1 . Since we consider the setting where training data is potentially biased , we use the word “ bias ” in the associated notations . Here mbias denotes the number of training examples . Let Pbias 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 Pbias ( Z = 0 ) > Pbias ( 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 Pref be the corresponding data distribution defined on X : i=1 ( potentially biased ) and balanced reference data { x ( i ) ref } mref i=1 . The entire structure of the proposed model will be illustrated in detail in Fig . 2 . Here mbias ( or mref ) denotes the number of training samples ( or reference samples ) . Pref ( Z = 0 ) ≈ Pref ( Z = 1 ) . Since the balanced reference set is often challenging to collect , typically the number of the reference samples is much smaller than that of training examples : mref mbias . Denote by x̂ : = 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 PG and PW 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 . ( 2020 ) . It quantifies how PG differs from Pref w.r.t . a certain sensitive attribute , formally defined below . Definition 1 ( Fairness Discrepancy ( Choi et al. , 2020 ) ) . Fairness discrepancy between Pref and PG w.r.t . a sensitive attribute z ∈ { z1 , . . . , z|Z| } is defined as : F ( Pref , PG ) : = ‖pref ( z ) − pG ( z ) ‖2 ( 1 ) where pref ( z ) : = Pref ( Z = z1 ) Pref ( Z = z2 ) ... Pref ( Z = z|Z| ) and pG ( z ) : = PG ( Ẑ = z1 ) PG ( Ẑ = z2 ) ... PG ( Ẑ = z|Z| ) . Here Ẑ 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 . ( 2020 ) . 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 : ( Discriminator ) max D EPbias [ fD ( D ( X ) ) ] + EPG [ fG ( D ( X ) ) ] ; ( Generator ) min G −EPG [ gG ( D ( X ) ) ] ( 2 ) where fD , fG , and gG indicate certain functions that vary depending on an employed GAN approach . For instance , the choice of ( fD ( t ) , fG ( t ) , gG ( t ) ) = ( t , −t , t ) together with Lipschitz-1 condition on D leads to the prominent WGAN optimization ( Arjovsky et al. , 2017 ) . Another choice of ( fD ( t ) , fG ( t ) , gG ( 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 .

Summary: In their paper, the authors propose a novel generative adversarial network (GAN) which, more than realistic outputs, promotes fairness in the generative distribution. Specifically, when the training dataset is trademarked by some _unfair_ property (the authors' example being unevenly distributed gender or ethnicity), their proposed method-- a "TVD based" approach, which I will call TVD-GAN going forward --will rectify this in their generative distribution. The generative distribution is evaluated using an $L_2$ norm between the likelihoods of some samples given a reference distribution and the generative distribution ($\mathbb{P}_{\text{ref}}$ and $\mathbb{P}_G$, respectively in eq.(1)).  TVD-GAN minimises the "total variational distance" (TVD), through regularisation in eq.(5). As claimed on p.4, TVD "offers the best trade-off performances in fairness and sample quality compared to other divergence measures", as the results indeed seem to suggest. The main goal-- I believe --is to provide a more realistic sample using the larger training dataset, while still maintaining a fair distribution using the smaller reference dataset.