Is Shapley Value fair?
Improving Client Selection for Mavericks in Federated Learning

Federated Learning (FL) [40, 24, 12, 27] enables privacy-preserving learning by deriving local models and aggregating them into one global model, meaning that sensitive private data, e.g., health records, never leaves the user’s personal devices. Thus, Federated Learning allows the construction of models that cannot be computed on a central server and is hence often the only alternative for medical research and other domains with high privacy requirements.

Concretely, a central server, the federator, selects clients that train local models using their own data. The federator then aggregates these local models into a global model. This process is repeated over several global rounds, typically with different sets of clients selected in each round. Consequently, clients have to invest resources — storage, computation, and communication — into a FL system. It is essential for the federator to measure the contributions of clients’ local models. Based on the contribution, the federator can select the most suitable clients to complete the learning fast and accurately, in addition to awarding rewards based on contribution.

There exist a number of proposals for contribution measurement, i.e., algorithms that determine the quality of the service provided by the clients, for Federated Learning [16, 1, 28, 33, 23, 31, 36, 29, 37]. In particular, previous work established that Shapley Value, which measure the marginal loss caused by a client’s sequential absence from the training, offer accurate contribution measurements. Yet, the prior art of contribution measurement focused on identical and independently distributed (i.i.d.) data [33]. Here, i.i.d. refers to i.i.d. data distributions whereas the data quantity can be heterogeneous. The assumption of i.i.d. is quite limiting in practice. For example, in the widely used image classification benchmark, Cifar-10 [18], most people can contribute images of cats and dogs. However, deer images are bound to be comparable rare and owned by few clients. Another relevant example arises from learning predictive medicine from clinics who specialize in different kinds of patients, e.g., AIDS and Amyotrophic Lateral Sclerosis, and own data of exclusive disease type.

We call such clients, who hold data sets from a majority of a subset of classes, Mavericks. Fig. 1 illustrates the different distributions: In a balanced distribution, all clients own data from all classes. If a system has Mavericks, they own one or more classes (almost) exclusively whereas the non-Maverick clients have a balanced distribution for the remaining classes. Multiple Mavericks could own separate classes or jointly own one class. The latter is referred to as Shared Mavericks. Mavericks are essential to learn high-quality global models in FL. Without Mavericks, it is impossible to achieve high accuracy on the classes for which they own the majority of all training data. Such classes could be, e.g., rare deceases, which can hence only be successfully classified if Mavericks are included.

Hypothesis on the Impact of Mavericks. While Shapley Value is shown to be effective in measuring contribution for the i.i.d. case, it is largely unknown if Shapley Value can assess the contribution of Mavericks fairly and effectively involve them via the selection strategy. In order to form a hypothesis, we first conduct experiments of learning a neural network classifier (See 3 in Supplement). It shows that random and Shapley Value-based selection both perform well in the scenarios that they are designed for. Yet, they both exhibit slow convergence for Mavericks, with Shapley Value not showing a clear advantage over the random selection. This result leads us to conjecture that Shapley Value is unable to fairly evaluate Mavericks’ contribution. Moreover, this example calls for a novel client selection strategy that can handle Mavericks appropriately.

Contributions. The initial experimental study motivates the two contributions of our paper: i) a thorough theoretical analysis that shows that indeed Shapley Value-based contribution measurements underestimate the contribution of Mavericks, and ii) FedEMD, a novel client selection based on the Wasserstein distances of a user’s data distribution to both the other users’ distributions and the global distribution.

Concretely, the theoretical analysis proves that Mavericks received low contribution measurement scores during the initial stage of the learning process. Intuitively, the reason for these scores lies in the difference of their data and hence model updates to the global model. In latter stages of the learning process, Mavericks’ scores converge towards the scores of other users. Intuitively, the decaying learning rate and the increased focus on small improvements for individual classes explain this increase in score for Mavericks. However, adapting a fairness measure that relates data quantity to contribution score, Mavericks, who tend to own more data than others, are still treated unfairly. The theoretical analysis is hence in line with our initial experimental results. More generally, the theoretical analysis reveals that clients with a skewed data distribution or a high data amount are not treated fairly by SVB.

We propose FedEMD, a light-weight Maverick-aware selection method for Federated Learning. As Mavericks are strategically involved when they can contribute the most, the convergence speed increases, meaning that the learning process can terminate faster and more efficiently. Our emulation results show that the proposed FedEMD has faster convergence in comparison to state-of-the-art algorithms [24, 3, 25], by at least 26.9% and 11.3% respectively, under FedAvg and FedSGD aggregations for a range of Maverick scenarios.

Contribution Measurement. Existing work on contribution measurement can be categorized in two classes: i) local approach: clients exchange the local updates, i.e., model weights or gradients, and measure the contribution of each other, e.g., by creating a reputation system [16], and ii) global approach: all clients send all their model updates to the federator who in turn aggregates and computes the contribution via the marginal loss [28, 33, 23, 31, 36, 29, 37]. The main drawbacks of local approaches are the excessive communication overhead and the lower privacy due to directly exchanged model updates [1]. In contrast, the global approach has lower communication overhead and avoids the privacy leakage to other clients by communicating only with the federator. Prevailing examples of globally measuring contribution are Influence [28, 29] and Shapley Value [33, 37, 36, 30]. The prior art demonstrates that Shapley Value can effectively measure the client’s contribution for the case when clients’ data is i.i.d. or of biased quantity [30]. Yet, a recent experimental study[41] demonstrates that the correlation between a user’s data quality and its Shapley Value is limited. The results raise doubts whether Shapley Value is really a suitable choice for contribution measurement. However, there is no rigorous analysis on whether Shapley Value can effectively evaluate the contribution from heterogeneous users with skewed data distributions.

Client Selection. Selecting clients within a heterogeneous group of potential clients is key to enabling fast and accurate learning based on high data quality. The state-of-the-art client selection strategies focus on the resource heterogeneity [26, 39, 15] or data heterogeneity [3, 5, 22, 4]. In the case of data heterogeneity, which is a focus of our work, selection strategies [5, 13, 3] gain insights on the distribution of clients’ data and then select them in specific manners. Goetz et. al [13] apply active sampling and Cho et. al [5] use Power-of-Choice to favor clients with higher local loss. TiFL [3] considers both resource and data heterogeneity to mitigate the impact of straggler and skewed distribution. TiFL applies a contribution-based client selection by evaluating the accuracy of selected participants each round and choose clients of lower accuracy. FedFast [25] chooses classes based on clustering and achieves fast convergence for recommendation systems. However, there is no selection strategy that addresses the Maverick scenario.

Data Heterogeneity. As an alternative to client selection strategies, multiple methodologies have been suggested to properly account for data heterogeneity in FL systems [21, 7, 8, 11, 14]. Early solutions require the federator to distribute a shared global training set [42], which is demanding and violates data privacy. Later studies either focus on the local learning stage [22, 17] or improved aggregation [35]. For instance, FedProx [22] improves the local objective by adding an additional $L_{2}$ regularization term, whereas FedNova [35] first normalizes the local model updates based on the number of their local steps and aggregates the local models. The downside of aforementioned solutions is the requirement of additional computation on clients.

The objective is to rigorously and analytically answer the question if Shapley Value is a fair contribution measurement for Mavericks. First, we formalize FL, our assumptions, Maverick, and fairness. Since the main building block of Shapley Value is the Influence Index [28], we then derive the fairness of Influence Index. Further, we derive the fairness of Shapley Value from the Influence Index and show that Shapley Value-based contribution measurements underestimate the contribution of Mavericks but more generally the contribution of other clients with skewed data distributions or large amounts of data. We empirically verify the lack of fairness for Mavericks in existing contribution measurements.

Notation: In this paper, we focus on classification tasks. We denote the set of possible inputs as $\mathcal{X}$ and the set of $C$ class labels that can be associated with the inputs as $\mathcal{Y}=\{Y_{1},Y_{2},...,Y_{C}\}$. In agreement with other works, we let $f\colon\mathcal{X}\xrightarrow{}\mathcal{P}$ assign a probability distribution of potential classes to inputs. With $\omega$ denoting the learned weights of the machine learning tasks, the empirical risk is $\mathcal{L}(\boldsymbol{\omega})$ for a SGD-based learning process under cross-entropy loss. Furthermore, the learning objective is then generally defined as: $\min\mathcal{L}(\boldsymbol{\omega})=\min\sum_{c=1}^{C}p(y=c)\mathbb{E}_{\boldsymbol{x}\mid y=c}\left[\log f_{c}(\boldsymbol{x},\boldsymbol{\omega})\right]$.

We are solving the learning objective in a Federated Learning scenario. In an FL system, there is a set $\mathcal{C}$ of $N$ clients. Enumerate the $K$ clients selected in a round by $C_{1},\ldots,C_{K}$. Each client $C_{k}$ selected in round $t$ computes local updates $\omega^{k}_{t}$ and the federator aggregates the results. Concretely, with $\eta$ being the learning rate, $C_{k}$ updates their weights in the t-$th$ global round by

The most common aggregation method is an average of the client updates, weighted by the amount of data one owns. Let $n^{k}$ denote the data quantity, then FedAvg aggregation is $\boldsymbol{\omega}_{t}=\sum_{k=1}^{K}\frac{n^{k}}{\sum_{k=1}^{K}n^{k}}\boldsymbol{\omega}_{t}^{k}$.

Assumptions: i) All parties are honest and follow the protocol. ii) Clients are homogeneous with respect to other resources, such as computation and communication. iii) The network is reliable with all messages being delivered within a maximal delay. iv) The global distribution has high similarity with the real-world (test dataset) distribution.

The relation between being a Maverick and the amount of owned data can be essential. Learning success on a class relates closely to the amount of data available for that class. Thus, for the Maverick to have a positive contribution when they are the only one owning a class, they should own more data than others. Our evaluation hence focuses on Mavericks that own considerably more data than other clients, e.g., more than half of the data of all selected clients. However, the theoretical results regarding fairness hold as long as the Maverick owns at least the average amount of data.

Our aim is to show that Shapley Value-based contribution measures achieve a low fairness if Mavericks are involved. Concretely, we show that the value of Eq. 4 is lower if $C_{k}$ is a Maverick than if they are a non-Maverick.

Here, we propose the novel client selection algorithm FedEMD, which enables FL systems with Mavericks to achieve faster convergence. First, we have established a trivial solution to empirically show that involving Mavericks in every round does not result in better accuracy. Quite the contrary, always including a Maverick has a detrimental effect in the long run.(See A.4 in Supplement). Further, we design FedEMD based on the Wasserstein Distance (EMD) [2] of client data distributions as a light-weight dynamic client selection strategy.

Overview. FedEMD (summarized in Alg. 1) selects clients each round using weighted random sampling, where the weights of each client are dynamically updated every round. To obtain the weights, we compute the global and current Wasserstein distance for each client. We structure FedEMD in three steps : i) Distribution Reporting and Initialization (Line 2–7): Clients perform distribution profile reporting so that the federator is able to sum up the global distribution and initialize the current distribution. ii) Dynamic Weights Calculation (Line 10–18): In this key step, we utilize a light-weight measure based on EMD to calculate dynamic selection probabilities over time, which achieve faster convergence, yet avoid overfitting. iii) Weighted Client Selection (Line 19–23): According to the probabilities computed in the previous step, FedEMD applies weighted random sampling to select $K$ clients.

Motivation of choice of Wasserstein Distance: FedEMD considers both global and current distance on client data distribution to increase the chance of selecting Mavericks at the early stages and profit from their diverse data. Later on, the selection probability is reduced to avoid skewing the distribution towards the Maverick classes. To measure the distance of distributions, the weight divergence defined by [42] of $\boldsymbol{\omega_{t}}$ and $\boldsymbol{\omega_{t}^{k}}$ is:

Note that $\|\sum_{k=1}^{K}\frac{n^{i}}{\sum_{i=1}^{K}}P^{i}(y=c)-P^{k}(y=c)\|$ is the EMD of $C_{k}$ and the global distribution, which directly leads to the drift of $\boldsymbol{\omega_{t}}$ by $C_{k}$. Thus, we use EMD as the distance measure in our distribution-based FedEMD (Design principle and details see A.5 in Supplement).

Dynamic Weights Calculation: Let $EMD(\mathcal{P_{1}},\mathcal{P_{2}})$ be the EMD of distributions between $\mathcal{P_{1}}$ and $\mathcal{P_{2}}$ and $\mathcal{D}$ be the self-reported data distribution matrix of $N$ clients. It is only reported once at the beginning of the learning. Also, we assume a limit of $R$ on the global communication rounds. The dynamic selection probability of clients in round $t$, $Proba_{t}$, is based on both the global distance $emd_{g}$ and current distance $emd_{c}$ between the local distribution $\mathcal{D}_{i}$ and the global/current $\mathcal{D}_{g}$, $\mathcal{D}_{c}^{t}$ (Line 23). Assume that $c_{1}$ is one class randomly chosen by federator except for the Maverick class¹¹1the federator can tell from the reported data distributions which classes are Maverick classes from $\mathcal{D}$, here we apply normalization method $Norm(P,\mathcal{D})=\frac{\sum_{c=1}^{C}P(y=c)}{\sum_{i=1}^{N}P^{i}(y=c_{1})/N}$ for distributions $P$ (Line 7, 21). The current distribution is the accumulated of selected clients over rounds (Line 18).

Distance-based Aggregation: The larger $\widetilde{emd}_{g}$ is, the higher the probability $Proba_{i}^{t}$ that a client $C_{i}$ is selected, since $C_{i}$ brings more distribution information to train $\boldsymbol{\omega_{t}}$. However, the current distance is also taken into consideration to improve the selection of Mavericks. Therefore, $Proba^{t}_{i}=softmax(\alpha\widetilde{emd}_{g}[i]-t\beta\widetilde{emd}_{c}[i])$ for $C_{i}$ in round $t$, where $\alpha,\beta$ is the distance coefficient to weigh the global and current distance. $\alpha,\beta$ shall be adapted for different initial distributions, i.e., different dataset and distribution rules (See A.10 in Supplement). Clients with larger global distance and smaller current distance have a high probability to be selected, and thus Mavericks are involved more often during the early period and less later on of training to gain faster convergence. The sampling procedure for selecting $K$ clients based on $Proba^{t}$ (Line 9–17) has complexity of $O(K\log(\frac{N}{K}))$ [9], so comparably low.

In this section, we evaluate the convergence speed of FedEMD in comparison to four state-of-the-art selection strategies and FedProx as the representative of SOTA algorithm for data heterogeneity. The evaluation considers four different classifiers and two Maverick types.

Datasets and Classifier Networks We use public image datasets: i) Fashion-MNIST [38] for bi-level image classification; ii) MNIST [20] for simpler tasks which need less data to do fast learning; iii) Cifar-10 [18] for more complex task such as colored image classification; iv) STL-10 [6] for applications with small amounts of local data for all clients. We note that light-weight neural networks are more applicable for federated learning scenarios, where clients typically have limited computation and communication resources [25]. Thus, here we apply light-weight CNN for each dataset correspondingly, neural network details are provided in A.9 in Supplement.

Federated Learning System The system considered has 50 participants with homogeneous computation and communication resources and 1 federator. At each round, the federator selects 5 (a common practise in related studies [32]) clients according to a client selection algorithm. The federator uses FedSGD or FedAvg to aggregate local models from selected clients. As FedSGD always considers one local epoch, we choose the same for FedAvg to enable a fair comparison of the two approaches. Two types of Mavericks are considered: exclusive and shared Mavericks. We consider the case of single Maverick owning an entire class of data in most of our experiments.

Evaluation Metrics i) Global test accuracy for all classes; ii) Source recall (related results see A.4 in Supplement) for classes owned by Mavericks exclusively; iii) $R@99$: the number of communication rounds required to reach 99% of test accuracy of random selection based results; iv) Normalized Shapley Value ranging between $[0,1]$ to measure the contribution of Mavericks; 5) Computation time (results see A.7 in Supplement): the computation time measured by the federator, including training, aggregation and client selection time.

Baseline We consider four selection strategies: Random [24], Shapley Value-based, FedFast [25], and recent TiFL [3]²²2When implementing, we focus on their client selection and leave out other features. For example, we do not implement the features related to the communication acceleration in TiFL and the aggregation in FedFast. under both FedSGD and FedAvg aggregation methods. Although Shapley Value itself is a contribution measure rather than client selection strategy, it has been widely adopted in incentive design to attract the participants of clients with high Shapley Value [30]. Thus, the Shapley Value-based algorithm here is a variant of weighted random selection whose weights are decided by the relative Shapley Value, i.e., the probability of a client to be selection corresponds to their relative Shapley Value. Further, in order to compare with state-of-the-art solution for heterogeneous FL which focus on the optimization design, we evaluate FedProx [22] as one of the baselines.

Client selection is key to successful Federated Learning as it enables maximizing the usefulness of different diverse data sets. In this paper, we highlighted that existing schemes fail when clients have heterogeneous data, in particular if one class is exclusively owned by one or multiple Mavericks. We proposed and evaluated an alternative algorithm that encourages the selection of diverse clients at the opportune moment of the training process, accelerating the convergence speed by up to at least 26.9% in average for multiple dataset under FedAvg aggregation, compared to the state of the art.