Optimality and Stability in Federated Learning: A Game-theoretic Approach

1 Introduction . Recent advances of machine learning techniques has made it possible to apply powerful prediction algorithms to a variety of domains . However , in real-world situations , data is often distributed across multiple locations and can not be combined to a central repository for training . For example , consider patient medical data located at hospitals or student educational data at different schools . In each case , the individual agents ( hospitals or schools ) who hold the data wish to find a model that minimizes their error . However , the data at each location may be insufficient to train a robust model . Instead , the agents may prefer to build a model using data from multiple agents : multiple hospitals or schools . Collectively , the combined data may be able to produce a model with much higher accuracy , providing more powerful predictions to each agent and increasing overall welfare . However , it may be infeasible to transfer the data to some coordinating entity to build a global model : privacy , data size , and data format are all possible reasons that would make transferring data not a reasonable solution . Federated learning is a novel distributed learning paradigm that aims to solve this problem ( McMahan et al . [ 2016 ] ) . Data remains at separate local sites , which individual agents use to learn local model parameters or parameter updates . Then , only the parameters are transferred to the coordinating entity 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . ( for example , a technology company ) , which averages together all of the parameters in order to form a single global model , which all of the agents use . Federated learning is a rapidly growing area of research ( Li et al . [ 2020 ] , Kairouz et al . [ 2019 ] , Lim et al . [ 2020 ] ) . However , research has also noted that federated learning , in its traditional form , may not be the best option for each agent ( Yu et al . [ 2020 ] , Bagdasaryan and Shmatikov [ 2019 ] , Li et al . [ 2019 ] , Mohri et al . [ 2019 ] ) . In the real world , agents may differ in their true distribution : the true model of patient outcomes at hospital A may differ from the true model at hospital B , for example . If these differences are large enough , federating agents may see their error increase under certain situations , potentially even beyond what they would have obtained with only local learning . For example , a player with relatively few samples may end up seeing its model “ torqued ” by the presence of a player with many samples . For this reason , agents may not wish to federate with every other potential agent . Instead , each agent faces a choice : given the costs and benefits of federating with different players , it must determine which of the exponentially many combinations of players it would prefer to federate with . Simultaneously , every other agent is also attempting to identify and join a federating group that it prefers - and agents may have conflicting preferences . Prior work ( Donahue and Kleinberg [ 2021 ] , Hasan [ 2021 ] ) has formulated this problem as a hedonic game , which each player derives some cost ( error ) from the coalition they join . The aim of such research has been to identify partitions of players that are stable against deviations , for varying definitions of stability . A hedonic game in general may not have any stable arrangements , so the area ’ s contributions in the analysis of stability adds valuable insight into the incentives of federating agents . However , this framework also leaves open multiple game theoretic questions . While the federating agents have individual incentives to reduce their error , society as a whole also has an interest in minimizing the overall error . In the school example , individual schools wish to find coalitions that work well on their own sub-populations , while the overall district or state may have an interest in finding an overall set of coalitions that minimizes the overall error . This analysis of a coalition partition ’ s overall cost falls under the game theoretic notion of optimality . One natural question relates to the tension between these two goals : the self-interested goal of the individual actors ( stability ) and the overall goal of reducing total cost ( optimality ) . Given a that set of self-interested agents has found a stable solution , how far from optimal could it be ? This is reflected by the Price of Anarchy of a game , the canonical approach to study optimality and stability jointly ( Papadimitriou [ 2001 ] , Koutsoupias and Papadimitriou [ 1999 ] ) . The Price of Anarchy ( PoA ) is a ratio where the numerator is equal to the highest-cost stable arrangement and the denominator is equal to the lowest-cost arrangement ( the optimal arrangement ) . It is lower bounded by 1 , a bound that it achieves only if all stable arrangements are optimal . A higher Price of Anarchy value implies a greater trade off between stability and optimality , and bounding the Price of Anarchy for a particular game puts a limit on this trade-off . Federated learning is a situation where questions of stability have been analyzed , but to our knowledge there has been no systematic analysis of the Price of Anarchy in a model of federated learning . The present work : A framework for optimality and stability in federated learning In this work , we make two main contributions to address this gap . First , we provide an efficient , constructive algorithm for calculating an optimal federating arrangement . Secondly , we prove the first-ever constant bound on the Price of Anarchy for this game , showing that the worst stable arrangement is no more than 9 times the cost of the best arrangement . We begin Section 4 by defining optimality , drawing on a notion of weighted error derived from the standard objective in federated learning literature . The main contribution of this section is an efficient , constructive algorithm for calculating an optimal arrangement , along with a proof of its optimality . However , as demonstrated in Section 5 , optimality and stability are not always simultaneously achieved . This section analyzes the Price of Anarchy , which measures how far from optimal the worst stable arrangement can be . First , we demonstrate that the optimal arrangement is not always stable . Next , we show that there exist sub-regions where the Price of Anarchy is equal to 1 . Finally , this section proves an overall Price of Anarchy bound of 9 , the first constant bound for this game . It is worth emphasizing that , beyond the Price of Anarchy bound itself , part of the contribution of this work is the optimization and analysis to produce this bound . The proofs for this contribution are modular and illuminate multiple properties about the broader federated learning game under study . As such , these contributions could be useful for further investigating this model . For example , the modular structure of our proof is what enables us to establish stronger bounds for certain sub-cases . The results in this paper are theoretical and do not depend on any experiments or code . However , while writing the paper , we found it useful to write and work with code to check conjectures . Some of that code is publicly available at github.com/kpdonahue/model_sharing_games . 2 Related work . Federated learning As we mentioned previously , federated learning has recently seen numerous advances . In this section , we highlight a few papers in federated learning that are especially related to our work . The idea that agents might differ in their true models ( that data might be generated non-i.i.d . across multiple agents ) is commonly acknowledged in the federated learning literature . For example , Yu et al . [ 2020 ] , Bagdasaryan and Shmatikov [ 2019 ] empirically demonstrate that federated learning , and especially privacy-related additions , can cause a wide disparity in error rates . Some techniques have been developed specifically towards this problem . For example , hierarchical federated learning adds an additional layer of hierarchical structure to federated learning , which could be used to reduce latency or to cluster together similar players ( Lin et al . [ 2018 ] , Liu et al . [ 2020 ] ) . Many other works also relate to clustering , such as ( Lee et al . [ 2020 ] , Sattler et al . [ 2020 ] , Shlezinger et al . [ 2020 ] , Wang et al . [ 2020 ] , Duan et al. , Jamali-Rad et al . [ 2021 ] , Caldarola et al . [ 2021 ] ) . Nagalapatti and Narayanam [ 2021 ] uses tools from cooperative game theory to selectively combine gradient updates from multiple agents . These works , which tend to be more applied than our work , may also differ in that they analyze situations where additional information is known , such as the data distribution at each location . Other work aims to improve accuracy rates by selecting acquiring additional data ( Blum et al . [ 2017 ] , Duan et al . [ 2021 ] ) . Some papers specifically analyze federated learning for high-stakes situations such as medical settings ( Xia et al . [ 2021 ] , Guo et al . [ 2021 ] , Vaid et al . [ 2021 ] , Kumar et al . [ 2021 ] , Zhang et al . [ 2021 ] ) . In general , all of these works have the goal of reducing the average error over all federating agents , which we will use to motivate our definition of optimality in later sections . Game theory in federated learning The closest work to this current paper is our prior paper Donahue and Kleinberg [ 2021 ] , which we discuss in greater detail in Section 3 . Another paper using hedonic game theory to analyze federated learning games is Hasan [ 2021 ] , which gives conditions for Nash stability in federated learning . Other works analyze the incentives of players to contribute resources towards federated learning : Blum et al . [ 2021 ] analyzes fairness and efficiency in sampling additional points for federated learning and Le et al . [ 2021 ] analyzes incentives for agents to contribute computational resources in federated learning . Interestingly , multiple works take a game theoretic approach towards coalition formation in cloud computing , but with the aim of minimizing some cost besides error , such as electricity usage Guazzone et al . [ 2014 ] , Anglano et al . [ 2018 ] . 3 Model and assumptions . We assume that there are M total agents ( sometimes referred to as players ) . Each agent i 2 [ M ] has drawn ni data points from their true local distribution g ( ✓i ) , where ✓i are their true local parameters and g ( · ) is some true labeling function . A player ’ s goal is to learn a model with low expected error on its own distribution . If a player opts for local learning , then it uses its local estimate of these parameters g ( ✓̂i ) to predict future data points , obtaining error erri ( { i } ) . If a set of players C are federating together , we say that they are in a coalition or cluster together . They combine their local estimates of parameters into a single federated estimate , governed by the weighted average of their parameters : ✓̂C = 1P i2C ni · X i2C ni · ✓̂i ( 1 ) A federating player i obtains error erri ( C ) : note that this value may differ between players in the same coalition . For example , if player j has more samples than player k , then ✓̂C will be weighted more towards player j , meaning that player j will have lower expected error than k. The weighted average method in Equation 1 is commonly used in federated learning ( McMahan et al . [ 2016 ] ) . Because it is the most straightforward method , it is sometimes called “ vanilla ” federated learning . Alternative ways of federation might involve customizing the model for individuals , as in domain adaptation . For example , Donahue and Kleinberg [ 2021 ] models three methods of federation : vanilla ( called “ uniform ” ) , as well as two models of domain adaptation . There are multiple reasons why we opted to analyze the federation method in Equation 1 in this work . First of all , this federation method is the most straightforward method , and as such it is the natural candidate to begin analysis . Secondly , this federation method is the most interesting to analyze technically . Domain adaptation serves to increase the incentives of an individual agent to participate in federation : it reduces the tension between an individual ’ s incentives and the optimal overall arrangement . Because of this , for Price of Anarchy it is more valuable and challenging to explore the case in Equation 1 , where incentives are more opposed .

Summary: This paper studies a basic federated learning problem, in which some agents, each one with their own dataset, have to jointly agree on a global model. The paper focuses on the trade-off between stability (enforcing that agents are incentivized to collaborate in the global model) and optimality (maximizing social welfare). The paper studies this trade-off by means of a Price of Anarchy analysis.