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 first introduces a new task about federated learning in which fairness and consistency are jointly considered, then proposes a solution named FCFL to solve the problem. After introducing fairness, the problem is transferred to a min-max optimization problem, and the two-stage optimization algorithm works well on it. The main contributions are as follows:  1) Introduce a new task which jointly considers model fairness and consistency in federated learning, 2) Propose an optimization method called FCFL to address the problem, 3) Conduct experiments on a synthetic dataset and two public datasets and compare the proposed algorithm with three baseline methods, and show that the proposed method outperforms the rest.