Central Server Free Federated Learning over Single-sided Trust Social Networks

1 INTRODUCTION . Federated learning has been well recognized as a framework able to protect data privacy Konečnỳ et al . ( 2016 ) ; Smith et al . ( 2017a ) ; Yang et al . ( 2019 ) . State-of-the-art federated learning adopts the centralized network architecture where a centralized node collects the gradients sent from child agents to update the global model . Despite its simplicity , the centralized method suffers from communication and computational bottlenecks in the central node , especially for federated learning , where a large number of clients are usually involved . Moreover , to prevent reverse engineering of the user ’ s identity , a certain amount of noise must be added to the gradient to protect user privacy , which partially sacrifices the efficiency and the accuracy Shokri and Shmatikov ( 2015 ) . To further protect the data privacy and avoid the communication bottleneck , the decentralized architecture has been recently proposed Vanhaesebrouck et al . ( 2017 ) ; Bellet et al . ( 2018 ) , where the centralized node has been removed , and each node only communicates with its neighbors ( with mutual trust ) by exchanging their local models . Exchanging local models is usually favored to the data privacy protection over sending private gradients because the local model is the aggregation or mixture of quite a large amount of data while the local gradient directly reflects only one or a batch of private data samples . Although advantages of decentralized architecture have been well recognized over the state-of-the-art method ( its centralized counterpart ) , it usually can only be run on the network with mutual trusts . That is , two nodes ( or users ) can exchange their local models only if they trust each other reciprocally ( e.g. , node A may trust node B , but if node B does not trust node A , they can not communicate ) . Given a social network , one can only use the edges with mutual trust to run decentralized federated learning algorithms . Two immediate drawbacks will be : ( 1 ) If all mutual trust edges do not form a connected network , the federated learning does not apply ; ( 2 ) Removing all single-sided edges from the communication network could significantly reduce the efficiency of communication . These drawbacks lead to the question : How do we effectively utilize the single-sided trust edges under decentralized federated learning framework ? In this paper , we consider the social network scenario , where the centralized network is unavailable ( e.g. , there does not exist a central node that can build up the connection with all users , or the centralized communication cost is not affordable ) . We make a minimal assumption on the social network : The data may come in a streaming fashion on each user node as the federated learning algorithm runs ; the trust between users may be single-sided , where user A trusts user B , but user B may not trust user A ( “ trust ” means “ would like to send information to ” ) . For the setting mentioned above , we develop a decentralized learning algorithm called online pushsum ( OPS ) which possesses the following features : • Only models rather than local gradients are exchanged among clients in our algorithm . This scheme can reduce the risk of exposing clients ’ data privacy Aono et al . ( 2017 ) . • Our algorithm removes some constraints imposed by typical decentralized methods , which makes it more flexible in allowing arbitrary network topology . Each node only needs to know its out neighbors instead of the global topology . • We provide the rigorous regret analysis for the proposed algorithm and specifically distinguish two components in the online loss function : the adversary component and the stochastic component , which can model clients ’ private data and internal connections between clients , respectively . Notation We adopt the following notation in this paper : • For random variable ξ ( i ) t subject to distribution D ( i ) t , we use Ξn , T and Dn , T to denote the set of random variables and distributions , respectively : Ξn , T = { ξ ( i ) t } 1≤i≤n,1≤t≤T , Dn , T = { D ( i ) t } 1≤i≤n,1≤t≤T . Notation Ξn , T ∼ Dn , T implies ξ ( i ) t ∼ D ( i ) t for any i ∈ [ n ] and t ∈ [ T ] . • For a decentralized network with n nodes , we use W ∈ Rn×n to present the confusion matrix , where Wij ≥ 0 is the weight that node i sends to node j ( i , j ∈ [ n ] ) . N outi = { j ∈ [ n ] : Wij > 0 } and N ini = { k ∈ [ n ] : Wki > 0 } are also used for denoting the sets of in neighbors of and out neighbors of node i respectively . • Norm ‖ · ‖ denotes the ` 2 norm ‖ · ‖2 by default . 2 RELATED WORK . The concept of federated learning was first proposed in McMahan et al . ( 2016 ) , which advocates a novel learning setting that learns a shared model by aggregating locally-computed gradient updates without centralizing distributed data on devices . Early examples of research into federated learning also include Konečný et al . ( 2015 ; 2016 ) , and a widespread blog article posted by Google AI McMahan and Ramage ( 2017 ) . To address both statistical and system challenges , Smith et al . ( 2017b ) and Caldas et al . ( 2018 ) propose a multi-task learning framework for federated learning and its related optimization algorithm , which extends early works SDCA Shalev-Shwartz and Zhang ( 2013 ) ; Yang ( 2013 ) ; Yang et al . ( 2013 ) and COCOA Jaggi et al . ( 2014 ) ; Ma et al . ( 2015 ) ; Smith et al . ( 2016 ) to the federated learning setting . Among these optimization methods , Federated Averaging ( FedAvg ) , proposed by McMahan et al . ( 2016 ) , beats conventional synchronized mini-batch SGD regarding communication rounds as well as converges on non-IID and unbalanced data . Recent rigorous theoretical analysis Stich ( 2018 ) ; Wang and Joshi ( 2018 ) ; Yu et al . ( 2018 ) ; Lin et al . ( 2018 ) shows that FedAvg is a special case of averaging periodic SGD ( also called “ local SGD ” ) which allows nodes to perform local updates and infrequent synchronization between them to communicate less while converging quickly . However , they can not be applied to the single-sided trust network ( asymmetric topology matrix ) . Decentralized learning is a typical parallel strategy where each worker is only required to communicate with its neighbors , which means the communication bottleneck ( in the parameter server ) is removed . It has already been proved that decentralized learning can outperform the traditional centralized learning when the worker number is comparably large under a poor network condition Lian et al . ( 2017 ) . There are two main types of decentralized learning algorithms : fixed network topology He et al . ( 2018 ) , and time-varying Nedić and Olshevsky ( 2015 ) ; Lian et al . ( 2018 ) during training . Wu et al . ( 2017 ) ; Shen et al . ( 2018 ) shows that the decentralized SGD would converge with a comparable convergence rate to the centralized algorithm with less communication to make large-scale model training feasible . Li et al . ( 2018 ) provides a systematic analysis of the decentralized learning pipeline . Online learning has been studied for decades . It is well known that the lower bounds of online optimization methods are O ( √ T ) and O ( log T ) for convex and strongly convex loss functions respectively Hazan et al . ( 2016 ) ; Shalev-Shwartz et al . ( 2012 ) . In recent years , due to the increasing volume of data , distributed online learning , especially decentralized methods , has attracted much attention . Examples of these works include Kamp et al . ( 2014 ) ; Shahrampour and Jadbabaie ( 2017 ) ; Lee et al . ( 2016 ) . Notably , Zhao et al . ( 2019 ) shares a similar problem definition and theoretical result as our paper . However , single-sided communication is not allowed in their setting , restricting their results . 3 PROBLEM SETTING . In this paper , we consider federated learning with n clients ( a.k.a. , nodes ) . Each client can be either an edge server or some other kind of computing device such as smart phone , which has local private data and the local machine learning model xi stored on it . We assume the topological structure of the network of these n nodes can be represented by a directed graph G = ( nodes : [ n ] , edges : E ) with vertex set [ n ] = { 1 , 2 , . . . , n } and edge set E ⊂ [ n ] × [ n ] . If there exist an edge ( u , v ) ∈ E , it means node u and node v have network connection and u can directly send messages to v. Let x ( i ) t denote the local model on the i-th node at iteration t. In each iteration , node i receives a new sample and computes a prediction for this new sample according to the current model x ( i ) t ( e.g. , it may recommend some items to the user in the online recommendation system ) . After that , a loss function , fi , t ( · ) associated with that new sample is received by node i . The typical goal of online learning is to minimize the regret , which is defined as the difference between the summation of the losses incurred by the nodes ’ prediction and the corresponding loss of the global optimal model x∗ : R̃T : = T∑ t=1 n∑ i=1 ( fi , t ( x ( i ) t ) − fi , t ( x∗ ) ) , where x∗ = arg minx ∑T t=1 ∑n i=1 fi , t ( x ) is the optimal solution . However , here we consider a more general online setting : the loss function of the i-th node at iteration t is fi , t ( · ; ξi , t ) , which is additionally parametrized by a random variable ξi , t . This ξi , t is drawn from the distribution Di , t , and is mutually independent in terms of i and t , and we call this part as the stochastic component of loss function fi , t ( · ; ξi , t ) . The stochastic component can be utilized to characterize the internal randomness of nodes ’ data , and the potential connection among different nodes . For example , music preference may be impacted by popular trends on the Internet , which can be formulated by our model by letting Di , t ≡ Dt for all i ∈ [ n ] with some time-varying distribution Dt . On the other hand , function fi , t ( · ; · ) is the adversarial component of the loss , which may include , for example , user ’ s profile , location , etc . Therefore , the objective regret naturally becomes the expectation of all the past losses : RT : = E Ξn , T∼Dn , T { T∑ t=1 n∑ i=1 ( fi , t ( x ( i ) t ; ξ ( i ) t ) − fi , t ( x∗ ; ξ ( i ) t ) ) } ( 1 ) with x∗ = arg minx EΞn , T∼Dn , T ∑T t=1 ∑n i=1 fi , t ( x ; ξ ( i ) t ) . One benefit of the above formulation is that it partially resolves the non-I.I.D . issue in federated learning . A fundamental assumption in many traditional distributed machine learning methods is that the data samples stored on all nodes are I.I.D. , which fails to hold for federated learning since the data on each user ’ s device is highly correlated to that user ’ s preferences and habits . However , our formulation does not require the I.I.D . assumption to hold for the adversarial component at all . Even though the random samples for the stochastic component still need to be independent , they are allowed to be drawn from different distributions . Finally , one should note that online optimization also includes stochastic optimization ( i.e. , data samples are drawn from a fixed distribution ) and offline optimization ( i.e. , data are already collected before optimization begins ) as its typical cases Shalev-Shwartz et al . ( 2012 ) . Hence , our setting covers a wide range of applications .

Summary: The paper proposed Online Push-Sum (OPS) method, which aims at solving decentralized federated optimization problems under a social network scenario where the centralized authority does not exist in a federated learning (FL) system. A social network application scenario is assumed by OPS where the graph is of single-sided trust. The author further extends the proposed OPS method to the online setting and provide regret analysis. The experimental study indicates that OPS is effective and converges faster than other decentralized online methods.