Deep Fair Discriminative Clustering

1 INTRODUCTION . Clustering is essential as it is the basis of many AI tools and has been widely used in real-world applications involving humans ( Jain et al. , 1999 ) such as market research , social network analysis , and crime analysis . However , as AI tools augment and even replace humans in decision-making , the need to ensure clustering is fair becomes paramount . Here fairness is measured using protected status variables ( PSVs ) such as gender , race , or education level . Fairness takes two primary forms ( Binns , 2020 ) : i ) group-level fairness and ii ) individual-level fairness . In this paper , we study the former which ensures that no one cluster contains a disproportionately small number of individuals with protected status . Motivated by this goal , our work aims to add fairness rules to deep clustering . Recent works ( Rösner & Schmidt , 2018 ; Schmidt et al. , 2019 ; Kleindessner et al. , 2019b ; Backurs et al. , 2019 ; Bera et al. , 2019 ) have been proposed for non-deep fair clustering algorithms . To ensure group-level fairness , many of these works use the notion of the disparate impact doctrine encoded as a constraint , that instances from different protected groups must have approximately ( within a tolerance ) equal representation in a cluster compared to the population . Different geographic regions place this tolerance at different levels ( Chierichetti et al. , 2017 ) . These existing algorithms optimize the clustering quality by minimizing some well-known clustering objectives while satisfying the group-level fairness constraints . Previous examples of adding fairness to clustering algorithms include k-median based approaches ( Chierichetti et al. , 2017 ; Backurs et al. , 2019 ; Bera et al. , 2019 ) and spectral clustering based algorithm ( Kleindessner et al. , 2019b ) . However , all these works evaluate their performance on low-dimensional tabular data and ( Chierichetti et al. , 2017 ; Kleindessner et al. , 2019b ; Backurs et al. , 2019 ) study the problems only with binary PSV . Deep clustering ( Xie et al. , 2016 ; Hu et al. , 2017 ; Guo et al. , 2017 ; Wang et al. , 2019 ) has the ability to simultaneously cluster and learn a representation for problems with large amounts of complex data ( i.e. , images , texts , graphs ) . However , the representation learning ability sometimes makes the learner suffer from bias hidden in the data which can lead to unfair clustering results . For example , clustering of portraits may create clusters based on features which are surrogates for racial and other protected status information . One way to overcome this is by adding group-level fairness to deep clustering which is a challenging and understudied problem . A significant challenge is it is hard to translate the current fair clustering algorithms into an end-to-end deep clustering setting . For example , geometric pre-processing steps such as computing fairlets ( Chierichetti et al. , 2017 ) to ensure fairness will not work as the end-to-end learning of deep learners means the underlying features that clustering is performed on are unknown apriori . Similarly , another line of work that adds constraints into deep learning models such as ( Xu et al. , 2018 ; Zhang et al. , 2019 ) are not appropriate either as these constraints are at the instance level , whereas we require to apply fairness rules at a cluster level . The work on fair deep clustering is relatively new . The first work on fair deep clustering ( Wang & Davidson , 2019 ) studies deep fair clustering problem from a geometric perspective which aims to learn a fair representation with multi-state PSV . The most recent work ( Li et al. , 2020 ) proposes a deep fair visual clustering model with adversarial learning to encourage the clustering partition to be statistically independent of each sensitive attribute ( PSV ) . Although these deep clustering approaches demonstrate better clustering performance compared to the traditional fair clustering algorithms ( Table 1 ) , their fairness results are relatively poor compared to those fair clusterings with fairness guarantees ( Chierichetti et al. , 2017 ; Backurs et al. , 2019 ) . Our work can be seen as combining the benefits of deep learning and discrete optimization to produce guaranteed fair predictions on clustered data with PSVs while making out-of-sample fair predictions for data without PSVs . In this paper , we propose a novel deep fair clustering framework to address the above issues . We adopt a probabilistic discriminative clustering network and learn a representation that naturally yields compact clusters . To incorporate the group-level fairness rules in the deep learner , we first formulate our fairness objective as an integer linear programming ( ILP ) problem that guarantees group-level fairness . This ILP is efficient to solve as its constraint matrix is totally unimodular . Further , we propose a refinement learning algorithm to combine the solved fair assignments and the clustering objective . Experimental results on real-world datasets demonstrate that our model achieves guaranteed fair results with competitive clustering performance . We also examine the novel uses of our framework in predictive clustering , flexible fair clustering , and challenging tasks with multi-state PSVs . The major contributions of this paper are summarized as follows : • We optimize a general notion of fairness for multi-state PSVs which we prove is equivalent to optimize the general balance measure ( Chierichetti et al. , 2017 ) for disparate impact . • We formulate our fairness assignment sub-problem as an ILP which is NP-hard . We show that by relaxing our constraint matrix to be totally unimodular our sub-problem can be efficiently solved by an LP solver ( but still generate integer solutions ) . ( See Theorem 4.2 ) . • We propose an end-to-end refinement learning algorithm that combines deep learning and discrete optimization to learn fair representation for clustering . ( See Algorithm 1 ) . • Extensive experimental results show that our work can achieve guaranteed fairness with competitive clustering performance . We demonstrate our novel extensions for fair clustering tasks in predictive clustering , multi-state PSVs and flexible fairness rules . ( See Section 5.2 ) . In the next section 2 we discuss the related work . Then we outline our measure of fairness and how it relates to classic measures of disparate impact in section 3 . In our approach section 4 , we introduce our clustering framework and encode our fairness objective as an ILP which can be efficiently solved via our relaxation . A refinement learning algorithm is proposed for end-to-end fair clustering . Finally we empirically evaluate the effectiveness of our approach in section 5 and conclude in section 6 . 2 RELATED WORK . Fair clustering has received much attention recently ( Schmidt et al. , 2019 ; Kleindessner et al. , 2019a ; Ahmadian et al. , 2019 ; Chen et al. , 2019 ; Davidson & Ravi , 2020 ; Mahabadi & Vakilian , 2020 ; Brubach et al. , 2020 ) . Chierichetti et al . ( 2017 ) first addressed the disparate impact for clustering problems in the presence of binary PSVs . Their work apriori groups instances into many fairlets which are used as input into standard k-medians style algorithms . Their work is guaranteed to produce a specified level of fairness and achieve a constant factor approximation with respect to cluster quality . Backurs et al . ( 2019 ) improves the fair decomposition algorithm to linear run-time . Later on , Bera et al . ( 2019 ) propose a general fair clustering algorithm that allows human-specified upper and lower bounds on any protected group in any cluster . Their work can be applied to any clustering problems under ` p norms such as k-median , k-means , and k-center . Besides the centroid-based method , Kleindessner et al . ( 2019b ) extends the fairness notion to graph spectral clustering problems . Ziko et al . ( 2021 ) propose a general , variational and bound-optimization framework of fair clustering . Previous fair clustering approaches mainly focus on adding fairness constraints into traditional clustering algorithms . In our work , we aim to study the fairness problem for recently proposed deep clustering algorithms ( Xie et al. , 2016 ; Yang et al. , 2017 ; Hu et al. , 2017 ; Caron et al. , 2018 ; Shaham et al. , 2018 ; Tzoreff et al. , 2018 ; Shah & Koltun , 2018 ) . Deep clustering algorithms connect representation learning and clustering together and have demonstrated their advantages over the two-phase clustering algorithms which use feature transformation first and then clustering . The goal of deep fair clustering is to learn a fair and clustering-favored representation . We illustrate the basic intuitions behind traditional fair clustering methods and deep fair clustering approaches in Figure 1 . One of the earliest works ( Wang & Davidson , 2019 ) to address the deep fair clustering problem learns a latent representation such that the cluster centroids are equidistant from every “ fairoid ” ( the centroid of all the data belonging to the same protected group ) . Recently , Li et al . ( 2020 ) encodes the fairness constraints as an adversarial loss and concatenates the fairness loss to a centroid-based deep clustering objective as a unified model . Unlike previous deep fair clustering works , we translate the fairness requirements into an ILP problem that generates guaranteed fair solutions given the PSVs . Meanwhile , our formulation allows for a general notion of fairness that supports flexible fairness constraints and multi-state PSVs . Moreover , we propose a novel learning framework to train fair clustering models via simultaneous clustering and fitting the self-generated fairness signals . 3 DEFINITIONS OF GROUP-LEVEL FAIRNESS . We begin this section by overviewing the seminal definition of group-level fairness in clustering ( see equation 1 ) and then its extension to multi-state PSVs ( see equation 2 ) . We then go onto show a new measure that our deep clustering framework will optimize ( see equation 3 ) and equation 2 have the same optimal condition as shown in Theorem 3.2 . 3.1 NOTION OF FAIRNESS . Let X ∈ RN×D denote N data points with D dimension features . The prediction function φ assigns each instance to one unique cluster , φ : x→ { 1 , ... K } , which forms K disjoint clusters { C1 , ... CK } . Given the protected status variable ( denoted as PSV ) A with T states , X can be partitioned into T demographic groups as { G1 , G2 , ... GT } . Definition 1 . The seminal proposed measure of fairness for clustering with binary PSV ( Chierichetti et al. , 2017 ) encoded disparate impact as follows : balance ( Ck ) = min ( N1k N2k , N2k N1k ) ∈ [ 0 , 1 ] ( 1 ) Here N1k and N 2 k represent the populations of the first and second demographic groups in cluster Ck . Such a measure of fairness only works for binary PSV . To allow for multi-state PSVs , let Nmink = min ( N 1 k . . . N T k ) denotes the smallest ( in size ) protected group in cluster k and Nmaxk = max ( N 1 k . . . N T k ) denotes the largest group . We extend the balance measure for multistate PSV as : balance ( Ck ) = Nmink Nmaxk ∈ [ 0 , 1 ] ( 2 ) Recent works ( Rösner & Schmidt , 2018 ; Bera et al. , 2019 ) also propose a new fairness measure to allow for fair clustering problems with multi-state PSVs . Definition 2 . Let ρi be the representation of group Gi in the dataset as ρi = |Gi|/N , and ρi ( k ) be the representation of group Gi in the cluster Ck : ρi ( k ) = |Ck ∩Gi|/|Ck| . Using these two values , the fairness value for cluster Ck is : fairness ( Ck ) = min ( ρi ρi ( k ) , ρi ( k ) ρi ) ∈ [ 0 , 1 ] ∀i ∈ { 1 , . . . T } ( 3 ) The overall fairness of a clustering is defined as the minimum fairness value over all the clusters . Similarly , the overall balance is the minimum balance value of all the clusters .

Summary: This paper proposes a fair clustering algorithm that uses DL models to map the data into deep representations. The authors also show the equivalence between the practical fairness measure and the balance measure. The algorithms can be concluded as two steps:  1) find fair assignments $\hat y$ based on $y$; 2) tune the latent representations and $y$ according to the pseudo-label $\hat y$.