Addressing Algorithmic Disparity and Performance Inconsistency in Federated Learning

1 Introduction . Federated learning ( FL ) [ 1 ] refers to the paradigm of learning from fragmented data without sacrificing privacy . FL has aroused broad interests from diverse disciplines including high-stakes scenarios such as loan approvals , criminal justice , healthcare , etc [ 2 ] . An increasing concern is whether these FL systems induce disparity in local clients in these cases . For example , ProPublica reported that an algorithm used across the US for predicting a defendant ’ s risk of future crime produced higher scores to African-Americans than Caucasians on average [ 3 ] . This has caused severe concerns from the public on the real deployment of data mining models and made algorithmic fairness an important research theme in recent years . Existing works on algorithmic fairness in machine learning have mostly focused on individual learning scenarios . There has not been much research on how FL will impact the model fairness at 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . different local clients1 . Recently , Du et al . [ 6 ] proposed a fairness-aware method , which considered the global fairness of the model learned through a kernel re-weighting mechanism . However , such a mechanism can not guarantee to achieve fairness at local clients in FL scenario , since different clients will have different distributions across protected groups . For example , if we are building a mortality prediction model for COVID-19 patients within a hospital system [ 7 ] , where each individual hospital can be viewed as a local client . Different hospitals will have different patient populations with distinct demographic compositions including race or gender . In this case , the model fairness at each hospital is important because that ’ s where the model will be deployed , and it is unlikely that global model fairness can lead to local model fairness . Due to the potential trade-off between algorithmic fairness and model utility , one aiming to mitigate the algorithmic disparity on local clients can exacerbate the inconsistency of the model performance ( i.e. , the model performance is different at different local clients ) . There have been researches [ 4 , 5 ] trying to address the inconsistency without considering algorithmic fairness . In particular , Mohri et al . [ 5 ] proposed an agnostic federated learning ( AFL ) algorithm that maximizes the performance of the worst performing client . Li et al . [ 4 ] proposed a q-Fair Federated Learning ( q-FFL ) approach to weigh different local clients differently by taking the q-th power of the local empirical loss when constructing the optimization objective of the global model . In this paper , we consider the problem of enforcing both algorithmic fairness and performance consistency across all local clients in FL . Specifically , suppose we have N local clients , and ui represents the model utility for client i , and gi is the model disparity quantified by some computational fairness metric ( e.g. , demographic parity [ 8 ] or equal opportunity [ 9 ] ) . Following the idea of AFL , we can maximize the utility of the worst-performed client to achieve performance consistency . We also propose to assign each client a `` fairness budget '' to ensure certain level of local fairness , i.e. , gi ≤ i ( ∀i = 1 , 2 , · · · , N ) with i being a pre-specified fairness budget for client i . Therefore , we can formulate our problem as a constrained multi-objective optimization framework as shown in Figure 1 ( a ) , where each local model utility can be viewed as an optimization objective . Since models with fairness and min-max performance may be not unique , we also require the model to be Pareto optimal . A model is Pareto optimal if and only if the utility of any client can not be further optimized without degrading some others . A Pareto optimal solution of this problem can not be achieved by existing linear scalarization methods in federated learning ( e.g. , federated average , or FedAve in [ 10 ] ) , as the non-i.i.d data distributions across different clients can cause a non-convex Pareto Front of utilities ( all Pareto solutions form Pareto Front ) . Therefore , we propose FCFL , a new federated learning framework to obtain a fair and consistent model for all local clients . Specifically , we first utilize a surrogate maximum function ( SMF ) that considers the N utilities involved simultaneously instead of the single worst , and then optimize the model to achieve Pareto optimality by controlling the gradient direction without hurting client utilities . Theoretical analysis proves that our method can converge to a fairness-constrained Pareto min-max model and experiments on both synthetic and real-world data sets show that FCFL can achieve a Pareto min-max utility distribution with fairness guarantees in each client . The source codes of FCFL are made publicly available at https : //github.com/cuis15/FCFL . 2 Related Work . Algorithm fairness is defined as the disparities in algorithm decisions made across groups formed by protected variables ( such as gender and race ) . Some approaches have been proposed to mathematically define if an algorithm is fair . For example , demographic parity [ 8 ] requires the classification results to be independent of the group memberships and equalized opportunity [ 9 ] seeks for equal false negative rates across different groups . Besides , there are also recent works focusing on the interpretation of the source of model disparity [ 11 ] . Plenty of approaches have been proposed to reduce model disparity . One type of method is to train a classifier first and then post-adjust the prediction by setting different thresholds for different groups [ 9 ] or learning a transformation function [ 12 , 13 ] . Other methods have been developed for optimization of fairness metrics during the model training process through adversarial learning [ 14 , 15 , 16 , 17 , 18 ] or regularization [ 19 , 20 , 21 ] . Du et al . [ 6 ] considered algorithm fairness in the federated learning setting and proposed a regularization method that assigns 1We would like to emphasize that the algorithmic fairness we studied in this paper is not the federated fairness studied in [ 4 ] and [ 5 ] , as their fairness is referring to the performance at different clients , which is consistency . a reweighing value on each training sample for loss objective and fairness regularization to deal with the global disparity . This method can not account for the discrepancies among the model disparities at different local clients . In this paper , we propose to treat fairness as a constraint and optimize a multi-objective optimization with multiple fairness constraints for all clients while maximally maintain the model utility . As we introduced in the introduction , the global consensus model learned from federated learning may have different performances on different clients . There are existing research trying to address such inconsistency issue by maximizing the utility of the worst-performing client . In particular , Li et al . [ 4 ] propose q-FFL to obtain a min-max performance of all clients by empirically adjusting the power of the objectives , which can not always guarantee a more consistent model utility distribution without sufficient searching for appropriate power values . Mohri et al . [ 5 ] propose AFL , a min-max optimization scheme which focuses on the single worst client . However , focusing on the single worst objective can cause another client to perform worse , thus we propose to take all objectives into account and optimize a surrogate maximum function to achieve a min-max performance distribution in this paper . Multi-objective optimization aims to learn a model that gives consideration to all objectives involved . The optimization methods for multi-objective typically involve linear scalarization or its variants , such as those with adaptive weights [ 22 ] , but it is challenging for these approaches to handling the competing performance among different clients [ 23 ] . Martinez et al . [ 24 ] proposed a multiobjective optimization framework called Min-Max Pareto Fairness ( MMPF ) to achieve consistency by inducing a min-max performance of all groups based on convex assumption , which is fairly strong as non-convex objectives are ubiquitous . In this paper , we formulate the problem of achieving both fairness and consistency in federated networks through constrained multi-objective optimization . Previous research on solving this problem has been mostly focusing on gradient-free algorithms such as evolutionary algorithms [ 25 ] , physics-based and deterministic approaches [ 26 ] . Gradient-based methods are still under-explored [ 27 ] . We propose a novel gradient-based method FCFL , which searches for the desired gradient direction iteratively by solving constrained Linear Programming ( LP ) problems to achieve fairness and consistency simultaneously in federated networks . 3 Problem Setup . The problem to be solved in this paper is formally defined in this section . Specifically , we will introduce the algorithmic fairness problem , how to extend existing fairness criteria to federated setting , and the consistency issues of model utility in federated learning . 3.1 Preliminaries . Federated Learning . Suppose there areN local clients c1 , c2 , ... cN and each client is associated with a specific datasetDk = { Xk , Y k } , k ∈ { 1 , 2 , ... , N } , where the input spaceXk and output space Y k are shared across all N clients . There are nk samples in the k-th client and each sample is denoted as { xki , y k i } . The goal of the federated learning is to collaboratively learn a global model h with the parameters θ to predict the label Ŷ k as on each client . The classical federated learning aims to minimize the empirical risk over the samples from all clients i.e. , minθ 1∑N k=1 n k ∑N k=1 ∑nk i=1 lk ( hθ ( x k i ) , y k i ) where lk is the loss objective of the k-th client . Fairness . Fairness refers to the disparities in the algorithm decisions made across different groups formed by the sensitive attribute , such as gender and race . If we denote the dataset on the k-th client as Dk = { Xk , Ak , Y k } , where Ak ∈ A is the binary sensitive attribute , then we can define the multi-client fairness as follows : Definition 1 ( Multi-client fairness ( MCF ) ) . A learned model h achieves multi-client fairness if h meets the following condition : ∆Disk ( h ) − k ≤ 0 ∀k ∈ { 1 , . , N } ( 1 ) where ∆Disk ( h ) denotes the disparity induced by the model h and k is the given fairness budget of the k-th client . The disparity on the k-th client ∆Disk can be measured by demographic parity ( DP ) [ 8 ] and Equal Opportunity ( EO ) [ 9 ] as follows : ∆DPk = |P ( Ŷ k = 1|Ak = 0 ) − P ( Ŷ k = 1|Ak = 1 ) | ∆EOk = |P ( Ŷ k = 1|Ak = 0 , Y k = 1 ) − P ( Ŷ k = 1|Ak = 1 , Y k = 1 ) | ( 2 ) As data heterogeneity may cause different disparities across all clients , the fairness budgets k in Definition 3.1 specifies the tolerance of model disparity at the k-th client . Consistency . Due to the discrepancies among data distributions across different clients , the model performance on different clients could be different . Morever , the inconsistency will be magnified when we adjust the model to be fair on local clients . There are existing research trying to improve the model consistency by maximizing the utility of the worst performing client [ 4 , 5 ] : min θ max k∈ { 1 , . , N } 1 nk ∑nk i=1 lk ( hθ ( x k i ) , y k i ) where the max is over the losses across all clients .

Summary: This paper proposes a federated learning algorithm which is fair and gives consistent performance across different clients in the following sense.  To achieve fairness, some measure of disparity is considered and enforced as a constraint for each client (i.e. each client has an upper bound on how high the disparity measure can go).  To achieve consistency, the objective is to find model parameters that minimize the maximum loss incurred across the clients.  This contrasts with prior work which does not directly address the fairness locally at each client.  Compared to [21], this paper adds constraints on a disparity measure for each client, in addition to the min-max objective.  Compared to [6], this paper considers a fairness constraint at each client as opposed to a single, global fairness constraint.  The proposed gradient descent-based algorithm first aims to satisfy the fairness constraints, and then it aims to find a Pareto optimal solution (w.r.t the clients’ utilities) while continuing to satisfy the fairness constraints.  The proposed algorithm replaces the (non-smooth) max’s in the problem formulation with a smooth surrogate, which is basically the log-sum-exp trick.  The proposed method is evaluated on three datasets with respect to prediction accuracy as the loss/utility metric and two different fairness metrics: demographic parity and equal opportunity.  The results show that the proposed method typically performs significantly better with respect to the fairness measure, with a small loss in accuracy compared to other methods which do not achieve the same level of fairness.