Diverse Client Selection for Federated Learning via Submodular Maximization

1 INTRODUCTION . Federated learning ( FL ) involves collaboratively training of machine learning model across a large number of clients while keeping client data local . Recent approaches to this problem repeatedly alternate between device-local ( stochastic ) gradient descent steps and server-aggregation of the clients ’ model updates ( McMahan et al. , 2017 ) . In cross-device settings , a server and its model usually serves several thousands of devices . Therefore , the communication between clients and the server can be costly and slow , forming a huge impediment to FL ’ s viability . One property of the collection of clients that can mitigate these problems , however , is often not exploited , and that is redundancy . Specifically , many clients might provide similar , and thus redundant , gradient information for updating the server model . Therefore , transmitting all such updates to the server is a waste of communication and computational resources . How best to select a representative and more informative client set while adhering to practical constraints in federated learning is still an open challenge . Although several selection criterion have been investigated in recent literature , e.g. , sampling clients with probabilities proportional to their local dataset size ( McMahan et al. , 2017 ) , sampling clients of larger update norm with higher probability ( Chen et al. , 2020 ) , and selecting clients with higher losses ( Balakrishnan et al. , 2020 ; Cho et al. , 2020 ) , the redundancy and similarity of the clients ’ updates sent to the server is not represented and exploited in these approaches . In particular , communicating multiple clients ’ updates to the server may cause statistical and system inefficiency if too many of them are too similar to each other . The commonly studied modular score/probability for each individual client is incapable of capturing information as a property over a group of clients . Ideally , a diverse set of clients would be selected , thereby increasing the impact of under-represented clients that contribute different information , and thereby improving fairness . This , in fact , is a topic of increasing interest ( Mohri et al. , 2019 ; Cho et al. , 2020 ; Dennis et al. , 2021 ) . In this paper , we introduce diversity to client selection in FL , namely a strategy to measure how a selected subset of clients can represent the whole when being aggregated on the server . Specifically , in each communication round , we aim to find a subset whose aggregated model update approximates the aggregate update over all clients . By doing this , we aim to limit the impact of subset selection which introduces variance in the model updates across round , that could otherwise slow the learning process . Inspired by the CRAIG method of coreset selection for efficient machine learning training ( Mirzasoleiman et al. , 2020 ) , we derive an upper bound of the approximation error as a supermodular set function ( in particular , the min-form of the facility location function ( Cornuéjols et al. , 1977 ) ) evaluated on the selected subset . We can then apply submodular maximization ( Fujishige , 2005 ; Iyer et al. , 2013 ; Wei et al. , 2014 ) on a complement submodular function to ( approximately ) minimize the error upper bound . We employ the greedy selection ( Nemhauser et al. , 1978 ) of a subset of clients according to the marginal gain of the submodular function to achieve a solution with provable approximation guarantee ( Conforti & Cornuejols , 1984 ) . By integrating the diverse client selection into the most commonly studied FL scheme , i.e. , Federated Averaging ( FedAvg ) ( McMahan et al. , 2017 ) , we propose DivFL that applies global model aggregation over a selected subset of clients after multiple local steps on every client . We present theoretical convergence analysis of DivFL and show its tolerance to the heterogeneity of data distributions across clients and large number of local steps . However , our method differs from the CRAIG method where selection is performed based on model updates ( involving multiple epochs at the clients ) . In addition , our approach allows for partial device participation where the server does not have access to all data at any communication round , as is standard in FL ( McMahan et al. , 2017 ) . In experiments , we compare DivFL with other client selection approaches on both synthetic dataset and FEMNIST , wherein our method excels on convergence , fairness and learning efficiency . 2 BACKGROUND AND RELATED WORK . We consider a typical federated learning objective : min w f ( w ) = N∑ k=1 pkFk ( w ) , where for each client k ∈ [ N ] , pk is a pre-defined weight ( such that ∑N k=1 pk = 1 ) that can be set to 1 N or the fraction of training samples , and Fk is the client-specific empirical loss . While there are various possible modeling approaches , we consider this canonical objective of fitting a single global model to the non-identically distributed data across all clients ( McMahan et al. , 2017 ) . Client Selection in Federated Learning . Client1 sampling is a critical problem particularly for cross-device settings where it is prohibitive to communicate with all devices . Two common ( or default ) strategies are ( a ) sampling the clients based on the number of local data points and uniformly averaging the model updates , and ( b ) sampling the clients uniformly at random and aggregating the model updates with weights proportional to the local samples ( Li et al. , 2020 ) . There is also recent work proposing advanced sampling techniques to incorporate dynamic systems constraints , accelerate the convergence of federated optimization , or to obtain a better model with higher accuracy ( Nishio & Yonetani , 2019 ; Ribero & Vikalo , 2020 ; Cho et al. , 2020 ; Lai et al. , 2020 ) . We investigate client selection through the lens of encouraging client diversity at each communication round which largely remains unexplored in previous work . The closest client selection method to ours is based on clustering ( e.g. , selecting representative clients from separate clusters ( Dennis et al. , 2021 ) ) . We note that performing ( private ) clustering in federated settings is still an open problem , and our method can be viewed as a soft version of dynamic clustering at each round ( discussed in the next paragraph ) . The benefits of gradient ( or model ) diversity has been demonstrated in other related contexts , such as scaling up mini-batch stochastic gradient descent ( SGD ) ( Yin et al. , 2018 ) . Enforcing sample or gradient diversity during optimization also implicitly places more emphasis on the underrepresented sub-population of clients , and can promote fairness defined as representative disparity ( Hashimoto et al. , 2018 ) . Similar to previous work ( e.g. , Cho et al. , 2020 ; Balakrishnan et al. , 2020 ) , we observe our approach yields more fair solutions across the network in Section 5 . Diverse Subset Selection via Submodularity . Modular scores have been widely studied for subset selection in machine learning and federated learning , e.g. , a utility score for each sample or client often measured by the loss . However , the diversity of a subset can not be fully captured by such 1Following conventions , we use the term ‘ client ’ for the problem of client selection . Throughout the paper , we use ‘ devices ’ and ‘ clients ’ interchangeably . modular scores since there is no score interaction . Diversity is often well modeled by a diminishing return property , i.e. , the ( marginal ) gain an element brings to a subset diminishes as more elements added to the subset . There exists a rich and expressive family of functions , all of which are natural for measuring diversity , and all having the diminishing returns property : given a finite ground set V of size n , and any subset A ⊆ B ⊆ V and a v /∈ B , a set function F : 2V → R is submodular if F ( v ∪A ) − F ( A ) ≥ F ( v ∪B ) − F ( B ) . ( 1 ) This implies v is no less valuable to the smaller set A than to the larger set B . The marginal gain of v conditioned on A is denoted f ( v|A ) , f ( v ∪A ) − f ( A ) and reflects the importance of v to A. Submodular functions ( Fujishige , 2005 ) have been widely used for diversity models ( Lin & Bilmes , 2011 ; Batra et al. , 2012 ; Prasad et al. , 2014 ; Gillenwater et al. , 2012 ; Bilmes & Bai , 2017 ) . Maximizing a submodular function usually encourages the diversity and reduces the redundancy of a subset . This property has been utilized for data selection in active learning ( Guillory & Bilmes , 2011 ) , curriculum learning ( Zhou & Bilmes , 2018 ) , mini-batch partitioning ( Wang et al. , 2019 ) , gradient approximation ( Mirzasoleiman et al. , 2020 ) , etc . Although the number of possible subsets A is ( n k ) , enumerating them all to find the maximum is intractable . Thanks to submodularity , fast approximate algorithms ( Nemhauser et al. , 1978 ; Minoux , 1978 ; Mirzasoleiman et al. , 2015 ) exist to find an approximately optimal A with provable bounds ( Nemhauser et al. , 1978 ; Conforti & Cornuejols , 1984 ) . Despite its success in data selection , submodularity has not been explored for client selection in federated learning . Encouraging diversity amongst local gradients ( or model updates ) of selected clients can effectively reduce redundant communication and promote fairness . Moreover , it raises several new challenges in the FL setting , e.g. , ( 1 ) it is unclear which submodular function to optimize and in which space to measure the similarity/diversity between clients ; ( 2 ) What convergence guarantee can be obtained under practical assumptions such as heterogeneity among clients , and ( 3 ) What are the effects of outdated client selection due to communication constraints ? 3 DIVERSE CLIENT SELECTION . In this section , we introduce “ federated averaging with diverse client selection ” ( or DivFL ) , a method that incorporates diverse client selection into the most widely studied FL scheme , federated averaging ( FedAvg ) . We will first derive a combinatorial objective for client selection via an approximation of the full communication from all clients , which naturally morphs into a facility location function in the gradient space that can be optimized by submodular maximization . We then present the standard greedy algorithm that optimizes the objective by selecting a diverse subset of clients at every communication round . 3.1 APPROXIMATION OF FULL COMMUNICATION . We aim to find a subset S of clients whose aggregated gradient can approximate the full aggregation over all the N clients V = [ N ] . To formulate this problem , we start by following the logic in Mirzasoleiman et al . ( 2020 ) . Given a subset S , we define a mapping σ : V → S such that the gradient information ∇Fk ( vk ) from client k is approximated by the gradient information from a selected client σ ( k ) ∈ S. For i ∈ S , let Ci , { k ∈ V |σ ( k ) = i } be the set of clients approximated by client-i and γi , |Ci| . The full aggregated gradient can be written as∑ k∈ [ N ] ∇Fk ( vk ) = ∑ k∈ [ N ] [ ∇Fk ( vk ) −∇Fσ ( k ) ( vσ ( k ) ) ] + ∑ k∈S γk∇Fk ( vk ) . ( 2 ) Subtracting the second term from both sides , taking the norms , and applying triangular inequality , we can obtain an upper bound for the approximation to the aggregated gradient by S , i.e. , ∥∥∥∥∥∥ ∑ k∈ [ N ] ∇Fk ( vk ) − ∑ k∈S γk∇Fk ( vk ) ∥∥∥∥∥∥ ≤ ∑ k∈ [ N ] ∥∥∥∇Fk ( vk ) −∇Fσ ( k ) ( vσ ( k ) ) ∥∥∥ . ( 3 ) The above inequality holds for any feasible mapping σ since the left hand side does not depend on σ . So we can take the minimum of the right hand side w.r.t . σ ( k ) , ∀k ∈ [ N ] , i.e. , ∥∥∥∥∥∥ ∑ k∈ [ N ] ∇Fk ( vk ) − ∑ k∈S γk∇Fk ( vk ) ∥∥∥∥∥∥ ≤ ∑ k∈ [ N ] min i∈S ∥∥∇Fk ( vk ) −∇Fi ( vi ) ∥∥ , G ( S ) . ( 4 ) The right hand side provides a relaxed objective G ( S ) for minimizing the approximation error on the left hand . Minimizing G ( S ) ( or maximizing Ḡ , a constant minus its negation ) equals maximizing a well-known submodular function , i.e. , the facility location function ( Cornuéjols et al. , 1977 ) . To restrict the communication cost , we usually limit the number of selected clients to be no greater than K , i.e. , |S| ≤ K. This resorts to a submodular maximization problem under cardinality constraint , which is NP-hard but an approximation solution with 1− e−1 bound can be achieved via the greedy algorithm ( Nemhauser et al. , 1978 ) .

Summary: The paper studies a problem of improving the efficiency of federated learning by selecting a subset of clients whose gradients are representative. This is formulated as a submodular maximization problem, and the authors analyze a greedy algorithm for it. This is evaluated through experiments on synthetic as well as real datasets.