1 Introduction
Federated learning (FL) yang2019federated refers to the paradigm of learning from fragmented data without sacrificing privacy. FL has aroused broad interests from diverse disciplines including highstakes scenarios such as loan approvals, criminal justice, healthcare, etc xu2020federated . 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 AfricanAmericans than Caucasians on average angwin2016machine . 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 different local clients
^{1}^{1}1We would like to emphasize that the algorithmic fairness we studied in this paper is not the federated fairness studied in li2019fair and mohri2019agnostic , as their fairness is referring to the performance at different clients, which is consistency.. Recently, Du et al. du2020fairness proposed a fairnessaware method, which considered the global fairness of the model learned through a kernel reweighting 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 COVID19 patients within a hospital system vaid2021federated , 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 tradeoff 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 li2019fair ; mohri2019agnostic trying to address the inconsistency without considering algorithmic fairness. In particular, Mohri et al. mohri2019agnostic proposed an agnostic federated learning (AFL) algorithm that maximizes the performance of the worst performing client. Li et al. li2019fair proposed a qFair Federated Learning (qFFL) approach to weigh different local clients differently by taking the 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 local clients, and represents the model utility for client , and is the model disparity quantified by some computational fairness metric (e.g., demographic parity dwork2012fairness or equal opportunity hardt2016equality ). Following the idea of AFL, we can maximize the utility of the worstperformed client to achieve performance consistency. We also propose to assign each client a "fairness budget" to ensure certain level of local fairness, i.e., with being a prespecified fairness budget for client . Therefore, we can formulate our problem as a constrained multiobjective optimization framework as shown in Figure 1, where each local model utility can be viewed as an optimization objective.
Since models with fairness and minmax 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 cannot be further optimized without degrading some others. A Pareto optimal solution of this problem cannot be achieved by existing linear scalarization methods in federated learning (e.g., federated average, or FedAve in mcmahan2017communication ), as the noni.i.d data distributions across different clients can cause a nonconvex 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 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 fairnessconstrained Pareto minmax model and experiments on both synthetic and realworld data sets show that FCFL can achieve a Pareto minmax utility distribution with fairness guarantees in each client.
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 dwork2012fairness requires the classification results to be independent of the group memberships and equalized opportunity hardt2016equality
seeks for equal false negative rates across different groups. Plenty of approaches have been proposed to reduce model disparity. One type of method is to train a classifier first and then postadjust the prediction by setting different thresholds for different groups
hardt2016equality . Other methods have been developed for optimization of fairness metrics during the model training process through adversarial learning beutel2017data ; louizos2015variational ; madras2018learning ; zemel2013learning ; zhang2018mitigating or regularization kamishima2011fairness ; zafar2015fairness ; beutel2019putting . Du et al. du2020fairness considered algorithm fairness in the federated learning setting and proposed a regularization method that assigns a reweighing value on each training sample for loss objective and fairness regularization to deal with the global disparity. This method cannot 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 multiobjective 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 worstperforming client. In particular, Li et al. li2019fair propose qFFL to obtain a minmax performance of all clients by empirically adjusting the power of the objectives, which cannot always guarantee a more consistent model utility distribution without sufficient searching for appropriate power values. Mohri et al. mohri2019agnostic propose AFL, a minmax 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 minmax performance distribution in this paper.
Multiobjective optimization aims to learn a model that gives consideration to all objectives involved. The optimization methods for multiobjective typically involve linear scalarization or its variants, such as those with adaptive weights chen2018gradnorm , but it is challenging for these approaches to handling the competing performance among different clients mahapatra2020multi . Martinez et al. martinez2020minimax
proposed a multiobjective optimization framework called MinMax Pareto Fairness (MMPF) to achieve consistency by inducing a minmax performance of all groups based on convex assumption, which is fairly strong as nonconvex objectives are ubiquitous. In this paper, we formulate the problem of achieving both fairness and consistency in federated networks through constrained multiobjective optimization. Previous research on solving this problem has been mostly focusing on gradientfree algorithms such as evolutionary algorithms
coello2006evolutionary , physicsbased and deterministic approaches evtushenko2014deterministic . Gradientbased methods are still underexplored zerbinati2011comparison. We propose a novel gradientbased 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 are local clients and each client is associated with a specific dataset , , where the input space and output space are shared across all clients. There are samples in the th client and each sample is denoted as . The goal of the federated learning is to collaboratively learn a global model with the parameters to predict the label as on each client. The classical federated learning aims to minimize the empirical risk over the samples from all clients i.e., where is the loss objective of the 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 th client as , where is the binary sensitive attribute, then we can define the multiclient fairness as follows:
Definition 1 (Multiclient fairness (MCF)).
A learned model achieves multiclient fairness if meets the following condition:
(1) 
where denotes the disparity induced by the model and is the given fairness budget of the th client. The disparity on the th client can be measured by demographic parity (DP) dwork2012fairness and Equal Opportunity (EO) hardt2016equality as follows:
(2)  
As data heterogeneity may cause different disparities across all clients, the fairness budgets in Definition 3.1 specifies the tolerance of model disparity at the th client.
Consistency. Due to the discrepancies among data distributions across different clients, the model performance on different clients could be different. Moreover, 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 li2019fair ; mohri2019agnostic :
where the is over the losses across all clients.
3.2 Fair and Consistent Federated Learning (FCFL)
Our goal is to learn a model which 1) satisfies MCF as we defined in Definition 3.1; 2) maintains consistent performances across all clients. We will use defined in Eq.(2) as measurement of disparity in our main text while the same derivations can be developed when adapting other metrics, so we have and is the function of calculating model disparity on the th client. Similarly, the model utility loss can be evaluated by different metrics (such as crossentropy, hinge loss and squared loss, etc). In the rest of this paper we will use () for () without causing further confusions. We formulate FCFLas the problem of optimizing the utilityrelated objectives to achieve Pareto MinMax performance with fairness constraints:
(3) 
The definitions of Pareto Solution and Pareto Front, which are fundamental concepts in multiobjective optimization, are as follows:
Definition 2 (Pareto Solution and Pareto Front).
Suppose
represents the utility loss vector on
learning tasks with hypothesis , we say is a Pareto Solution if there is no hypothesis that dominates : , i.e.,All Pareto solutions form Pareto Front .
From Definition 2, for a given hypothesis set and the objective vector , the Pareto solution avoids unnecessary harm to client utilities and may not be unique. We prefer a Pareto solution that achieves a higher consistency. Following the work in li2019fair ; mohri2019agnostic , we want to obtain a Pareto solution with minmax performance. Figure 1 shows the relationships among different model hypothesis sets, and we explain the meanings of different notations therein as follows:
(1) is the set of model hypotheses satisfying MCF, i.e.,
(2) is the set of model hypotheses achieving minmax performance (consistency) with MCF:
(4) 
(3) is the set of model hypotheses achieving Pareto optimality with MCF:
(5a)  
(5b) 
where Eq.(5a) satisfies meets MCF, and Eq.(5b) ensures that is a Pareto model with MCF.
(4) is our desired set of model hypotheses achieving Pareto optimality and minmax performance with MCF: .
In summary, our goal is to obtain a fair and consistent model to achieve Pareto optimality and minmax performance with MCF.
4 FairnessConstrained MinMax Pareto Optimization
4.1 Preliminary: Gradientbased MultiObjective Optimization
Pareto solutions of the multiobjective optimization problem can be reached by gradient descent procedures. Specifically, given the initial hypothesis with parameter , we optimize by moving to a specific gradient direction with a step size : . is a descent direction if it decreases the objectives (). Suppose is the gradient of the th objective with respect to the parameter , if we select which satisfies for all , is a descent direction and decreases after the iteration.
If we directly search for the descent direction to achieve , the computation cost can be tremendous when is a highdimensional vector. Désidéri et al. desideri2012multiple proposed to find a descent direction in the convex hull of the gradients of all objectives denoted as by searching for a dimension vector (typically in deep model), which is formulated as follows:
(6) 
4.2 Overview of the Optimization Framework
To obtain a fair Pareto minmax model, there are two optimization paths shown in Figure 1. The gray dotted line denotes the optimization path where we first achieve Pareto optimality with MCF, then we try to achieve minmax performance while keeping Pareto optimality. However, it’s hard to adjust a Pareto model to another mahapatra2020multi . Therefore, we propose to first achieve minmax performance with MCF then achieve Pareto optimality as the black solid line in Figure 1. In particular, we propose a twostage optimization method for this constrained multiobjective problem: 1) constrained minmax optimization to achieve a fair minmax model; 2) constrained Pareto optimization to continue to optimize the model to achieve Pareto optimality while keeping minmax performance with MCF.
Constrained MinMax Optimization We define a constrained minmax optimization problem on the hypothesis set :
(7)  
where is the given fairness budget on the th client. By optimizing the constrained objective in Eq.(7), we obtain a model that 1) satisfies MCF; 2) achieves the optimal utility on the worst performing client.
Constrained Pareto Optimization Caring only about the utility of the worstperforming client can lead to unnecessary harm to other clients since the rest objectives can be further optimized. Therefore, we then continue to optimize to achieve Pareto optimality:
(8a)  
(8b)  
(8c)  
(8d) 
where we optimize without hurting the model utility on any client so that the converged Pareto model of Eq.(8) satisfies . Moreover, satisfies MCF as the constraint in Eq.(8c), so is a Pareto minmax model with fairness constraints on all clients.
4.3 Achieving MinMax Performance with MCF
Minimizing the current maximum value of all objectives directly in Eq.(7) can cause another worse objective and can be computationally hard when faced with a tremendous amount of clients. We will use () to denote () without causing further confusions for expression simplicity. We propose to use a smooth surrogate maximum function (SMF) polak2003algorithms to approximate an upper bound of and as follows:
(9) 
It is obvious that and . For , we can get a similar conclusion. The objective in Eq.(7) is approximated as follows:
(10a)  
(10b) 
Property 1.
There always exists an initial trivial model which satisfies the MCF criterion by treating all samples equally (e.g., ).
From Property 1, we can always initialize to satisfy in Eq.(10b). Then we optimize the upper bound of when ensuring MCF. As the hypothesis owns the parameter , we use and to represent the gradient and , respectively. We propose to search for a descent direction for Eq.(10) in the convex hull of in two cases where is defined in Eq.(6). For the th iteration:
(1) if satisfies the fairness constraint defined in Eq.(10b), we focus on the descent of :
(11) 
(2) if violates the fairness constraint, we aim to the descent of without causing the ascent of :
(12a) 
If the obtained gradient satisfies , we decrease the parameter as:
(13)  
where is the decay ratio that . From Eq.(13), we narrow the gap between and by decreasing the parameter as every time the algorithm approaches convergence. From Eq.(11) and Eq.(12), we optimize either or and keep without ascent in each iteration.
4.4 Achieving Pareto Optimality and MinMax Performance with MCF
As the model obtained from constrained minmax optimization cannot guarantee Pareto optimality, we continue to optimize to be a Pareto model without causing the utility descent on any client. We propose a constrained linear scalarization objective to reach a Pareto model and the th iteration is formulated as follows:
(14a)  
(14b)  
(14c) 
where and is the convex hull of . The nonpositive angle of with each gradient ensures that all objective values decrease. Similarly, if we aim to reach the Pareto solution without causing utility descent only on the worst performing client, the constraint in Eq.(14b) is replaced by .
Different from constrained minmax optimization in Section 4.3 where the objective to be optimized in each iteration depends on whether , in constrained Pareto optimization procedure, as we have achieved MCF, we optimize the objective in Eq.(14) for a dominate model in each iteration. Specifically, we restrict to keep fairness on each client given the reached hypothesis . Meanwhile, we constrain to descend or remain unchanged until any objective cannot be further minimized. Algorithm 1 in the Appendix shows all steps of our method. Moreover, the convergence analysis and the discussion on the time complexity of our framework are in Appendix.
5 Experiments
We intuitively demonstrate the behavior of our method by conducting experiments on synthetic data. For the experiments on two realworld federated datasets with fairness issues, we select two different settings to verify the effectiveness of our method: (1) assign equal fairness budgets for all local clients; (2) assign clientspecific fairness budgets. More experimental results and detailed implementation are in Appendix.
5.1 Experimental Setup
Federated Datasets (1) Synthetic dataset: following the setting in lin2019pareto ; mahapatra2020multi , the synthetic data is from two given nonconvex objectives; (2) UCI Adult dataset mohri2019agnostic : Adult contains more than 40000 adult records and the task is to predict whether an individual earns more than 50K/year given other features. Following the federated setting in li2019fair ; mohri2019agnostic , we split the dataset into two clients. One is PhD client in which all individuals are PhDs and the other is nonPhD client. In our experiments, we select race and gender as sensitive attributes, respectively. (3)eICU dataset: We select eICU pollard2018eicu , a clinical dataset collecting patients about their admissions to ICUs with hospital information. Each instance is a specific ICU stay. We follow the data preprocessing procedure in johnson2018generalizability and naturally treat 11 hospitals as 11 local clients in federated networks. We conduct the task of predicting the prolonged length of stay (whether the ICU stay is longer than 1 week, ) and select as the sensitive attribute.
Evaluation Metrics (1) Utility metric: we use to measure the model utility in our experiments; (2) Disparity metrics: our method is compatible with various of fairness metrics. In our experiments, we select two metrics (marginalbased metric Demographic Parity dwork2012fairness and conditionalbased metric Equal Opportunity hardt2016equality to measure the disparities defined in Eq.(2)(The results of Equal Opportunity are in the Appendix); (3) Consistency: following the work li2019fair ; mohri2019agnostic , we use the utility on the worstperforming client to measure consistency.
Baselines As we do not find prior works proposed for achieving fairness in each client, we select FA du2020fairness , MMPF martinez2020minimax and build FedAve+FairReg as baselines in our experiments. For all baselines, we try to train the models to achieve the optimal utility with fairness constraints. If the model cannot satisfy the fairness constraints, we keep the minimum of disparities with reasonable utilities. (1) MMPF martinez2020minimax , Martinez et al. develop MMPF which optimizes all objectives on convex assumption to induce a minmax utility of all groups; (2) FA du2020fairness , Du et al. propose FA, a kernelbased modelagnostic method with regularizations for addressing fairness problem on a new unknown client instead of all involved clients; (3) FedAve+FairReg, we build FedAve+FairReg, which optimizes the linear scalarized objective with the fairness regularizations of all clients.
5.2 Experiments on Synthetic Dataset
Following the setting in lin2019pareto ; mahapatra2020multi , the synthetic data is from the two nonconvex objectives to be minimized in Eq.(15) and the Pareto Front of the two objectives is also nonconvex.
(15) 
Nonconvex Pareto Front means that linear scalarization methods (e.g., FedAve) miss any solution in the concave part of the Pareto Front. In this experiment, we optimize under the constraint . Considering the effect of the initialization in our experiment, we conduct experiments when the initialization satisfies the constraints and violates the constraints.
From the results in Figure 2, when the initialization violates the constraints in Figure 2(a), the objective decreases in each step until it satisfies the constraint and finally FCFL reaches the constrained optimal . As the initialization satisfies the constraints in Figure 2(b), our method focuses on optimizing until it achieves the optimal with the constraint .
5.3 Experiments on Realworld Datasets with Equal Fairness Budgets
5.3.1 Income Prediction on Adult Dataset
We show the results with the sensitive attribute being race in our main text and the results when gender is the sensitive attribute are in Appendix. We set the fairness budgets defined in Eq.(1) in two different cases, (1) looser constraint: ; (2) stricter constraint: .
5.3.2 Prolonged Length of Stay Prediction on eICU Dataset
The length of the patient’s hospitalization is critical for the arrangement of the medical institutions as it is related to the allocation of limited ICU resources. We conduct experiments on the prediction of prolonged length of stay(whether the ICU stay is longer than 1 week) on eICU dataset. We use race as sensitive attribute and set the fairness budgets defined in Eq.(1) in two cases: (1) looser constraint: ; (2) stricter constraint:
From Figure 4, our method achieves minmax performance with fairness budget compared to the baselines. When we constrain , all baselines fail to satisfy the constraints and the disparities are about 0.1 while our method significantly decreases the disparities and the maximum of the disparities is . Besides, we maintain comparable utilities on all clients compared to baselines.
5.4 Experiments with ClientSpecific Fairness Budgets
Data heterogeneity encourages the different levels of the disparities on different clients. Consistent fairness budgets can cause unexpected hurt on some specific clients with severe disparities. We explore the performance of our method given clientspecific fairness budgets. Specifically, we firstly conduct unconstrained minmax experiments and measure the original disparities on all clients, then we constrain the model disparities based on the original disparities of all clients, i.e., .
From Figure 5(a) and Figure 5(b), we show the effect of the clientspecific fairness budgets on model disparities and utilities. The decreasing means the stricter constraints on both clients and FCFL reduces the disparities significantly as shown in Figure 5(a). With the stricter clientspecific constraints on both clients, the utilities on both clients decrease slightly as shown in Figure 5(b), which implies that FCFL achieves a great balance between the fairness and the utility of all clients. FCFL is compatible with clientspecific fairness budgets which enhances its flexibility and avoids severe hurt to the specific clients.
The results of LoS prediction task with clientspecific fairness budgets are shown in Figure 5(c) and Figure 5(d). As decreases from 1.0 to 0.2, the maximum of all client disparities in Figure 5(c) decrease from 0.2 to 0.05 which means the model becomes fairer on all clients. Figure 5(d) shows the minimum of the utilities which slightly decreases from 0.62 to 0.6 and our method achieves an acceptable tradeoff between model fairness and utility as the amount of clients increases in this task.
6 Conclusion
In this paper, we investigate the consistency and fairness issues in federated networks as the learned model deployed on local clients can cause inconsistent performances and disparities without elaborate design. We propose a novel method called FCFL to overcome the disparity and inconsistency concerns in the favored direction of gradientbased constrained multiobjective optimization. Comprehensive empirical evaluation results measured by quantitative metrics demonstrate the effectiveness, superiority, and reliability of our proposed method.
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