Title: Mobilizing Personalized Federated Learning in Infrastructure-Less and Heterogeneous Environments via Random Walk Stochastic ADMM
Venue: NeurIPS 2023
Accepted: yes
Keywords: Mobilized Federated Networks

Personalized Federated Learning

Random Walk

Stochastic ADMM
Abstract: This paper explores the challenges of implementing Federated Learning (FL) in practical scenarios featuring isolated nodes with data heterogeneity, which can only be connected to the server through wireless links in an infrastructure-less environment. To overcome these challenges, we propose a novel mobilizing personalized FL approach, which aims to facilitate mobility and resilience. Specifically, we develop a novel optimization algorithm called Random Walk Stochastic Alternating Direction Method of Multipliers (RWSADMM). RWSADMM capitalizes on the server's random movement toward clients and formulates local proximity among their adjacent clients based on hard inequality constraints rather than requiring consensus updates or introducing bias via regularization methods. To mitigate the computational burden on the clients, an efficient stochastic solver of the approximated optimization problem is designed in RWSADMM, which provably converges to the stationary point almost surely in expectation. Our theoretical and empirical results demonstrate the provable fast convergence and substantial accuracy improvements achieved by RWSADMM compared to baseline methods, along with its benefits of reduced communication costs and enhanced scalability.

Mobilizing Personalized Federated Learning in Infrastructure-Less and Heterogeneous Environments via Random Walk Stochastic ADMM

Abstract

This paper explores the challenges of implementing Federated Learning (FL) in practical scenarios featuring isolated nodes with data heterogeneity, which can only be connected to the server through wireless links in an infrastructure-less environment. To overcome these challenges, we propose a novel mobilizing personalized FL approach, which aims to facilitate mobility and resilience. Specifically, we develop a novel optimization algorithm called Random Walk Stochastic Alternating Direction Method of Multipliers (RWSADMM). RWSADMM capitalizes on the server's random movement toward clients and formulates local proximity among their adjacent clients based on hard inequality constraints rather than requiring consensus updates or introducing bias via regularization methods. To mitigate the computational burden on the clients, an efficient stochastic solver of the approximated optimization problem is designed in RWSADMM, which provably converges to the stationary point almost surely in expectation. Our theoretical and empirical results demonstrate the provable fast convergence and substantial accuracy improvements achieved by RWSADMM compared to baseline methods, along with its benefits of reduced communication costs and enhanced scalability.

1 Introduction

Federated Learning (FL) [1, 2, 3, 4] is a distributed machine learning paradigm that enables clients to learn a shared model without sharing their private data. Unlike traditional machine learning approaches that rely on central servers for model training, FL allows clients to collaborate and train the model in a distributed manner, overcoming privacy issues related to passing data to a central server. Despite its advancements, real-world applications in environments with insufficient network support continue to face challenges. a) Maintaining consistent and reliable connections between the central server and clients becomes exceedingly challenging in environments lacking network infrastructures, e.g., natural disasters or military warzones. While intermittent connectivity may be available through satellite networks, the instability and limited capacity of such networks prevent the transmission of large data volumes, making it difficult to collect model updates from soldiers or first responders. b) The non-IID (non-independent and identically distributed) nature of clients' data, characterized by heterogeneity across the network, can hinder the generalization of the global model

for each client. Addressing these challenges is crucial for practical FL in such environments. In this paper, we propose RWSADMM, a novel FL scheme that uses Random Walk (RW) algorithm to enable server mobility among client clients. These dynamic approach benefits scenarios with limited internet connectivity, where clients form clusters using local short-range transmission devices.

For example, in various contexts such as robotics, emergency response, or military operations, consider a scenario where individuals or entities are equipped with integrated visual augmentation systems (IVAS) [5]. To facilitate the collection of model updates from these entities, a mobile unit equipped with a powerful computer navigates the environment [6], communicating with locations through a network, which might be satellite-based or another suitable alternative. Upon reaching an entity, the mobile unit employs short-range communication technologies such as WiFi direct, Zigbee [7], or Bluetooth to establish connections with nearby IV AS devices. Through these connections, the unit collects model updates and distributes new models as necessary. A graph-based representation is utilized to determine the order of interactions, where entities are depicted as nodes, and the edge between an entity and its neighbor indicates that the neighbor is within the communication range of the unit that reaches the entity. This graph assists the mobile unit in making informed decisions about the order in which it engages with the entities.

Various applicable examples of such constrained network situations span across different domains, including ad hoc wireless learning [8, 9, 10], wildlife tracking [11, 12, 13], Internet of Underwater Things (IoUT) [14, 15], natural disaster management [16, 17], military operations [18], or fostering a digital democracy [19, 20] which assists in overcoming restrictions imposed by regimes that prohibit internet access to civilians.

Specifically, to address challenge a), we propose an algorithmic framework called RWSADMM (Fig. 1), short for Random Walk Alternating Directional method of Multipliers, which considers a dynamic reachability graph among distributed clients using a movable vehicle as the central server. Clients are represented as nodes in the graph, with edges denoting neighborhood connections. Personal devices, referred to as local clients, establish dynamic connectivity with the server when the server is nearby. The server connects with a selected client and its neighbors while moving between locations using a non-homogeneous RW algorithm for probabilistic navigation. In each computation round, the vehicle broadcasts and gathers local model updates from residing clients, who rely on short-range communication to interact with the moving server, when it's within the communication range. The received updates are aggregated and used to update the global model iteratively.

To tackle the second challenge (b) arising from the heterogeneity of data distribution among clients, RWSADMM incorporates model personalization through local proximity among adjacent clients using hard inequality constraints, as opposed to unconstrained optimization with regularization techniques that may induce model bias. By formulating the problem with these constraints, RWSADMM reduces the computational complexity for clients, effectively mitigating the limitations of local computational power. This is achieved by designing the solver to the stochastic approximation of the minimization subproblem within the typical ADMM algorithm.

(a) step

k

(b) step

k

+1

(c) Mobilized FL between clients

and the server

Mobilizedserverlocation

Mobilizedserverlocation

Clients participating in the update

Clients participating in the update

Clientsnotparticipatingin theupdate

Clientsnotparticipatingin the update

1815

12

64

16

1815

5

6

10

QQ@Q

13

0

11

1419

1419

Our research makes three main contributions. Firstly, our proposed RWSADMM algorithm is the first attempt to enable mobilizing FL with efficient communication and computation in an infrastructure-less setting . The RWSADMM framework involves a server dynamically moving between different regions of clients and receiving updates from one or a few clients residing in the selected zone, which reduces communication costs and enhances system flexibility. Secondly, to address the issue of data heterogeneity among clients, RWSADMM formulates local proximity among adjacent clients based on hard inequality constraints, avoiding the introduction of model bias via consensus updates . This approach provides an alternative realization of personalization, which is crucial when dealing with highly heterogeneous data distributions. Thirdly, to mitigate the computational burden on the clients, an efficient stochastic solver of the subproblem is designed in RWSADMM, which provably converges to the stationary point almost surely in expectation under mild conditions independent of the data distributions. Our theoretical and empirical results demonstrate the superiority of our proposed algorithm over state-of-the-art personalized Federated Learning (FL) algorithms, providing empirical evidence for the effectiveness of our approach.

2 Related Work

This paper is relevant to two distinct research areas, which are reviewed in two separate sections: FL frameworks that tackle data heterogeneity and ADMM-based FL frameworks.

FL with data heterogeneity FL was initially introduced as FedAvg, a client-server-based framework that didn't allow clients to personalize the global model to their local data [1]. This led to poor convergence due to local data heterogeneity, negatively impacting the global FL model's performance on individual clients. Recent works proposed a two-stage approach to personalize the global model. In the first stage, the FL global model is trained similarly to FedAvg. However, the second stage is included to personalize the global model for each local client through additional training on their local data. [21] demonstrated that FedAvg is equivalent to Reptile, a new meta-learning algorithm, when each client collects the same amount of local data. To learn a global model that performs well for most participating clients, [22] proposed an improved version, Per-FedAvg. This new variant aims to learn a good initial global model that can adapt quickly to local heterogeneous data. An extension of Per-FedAvg, called pFedMe [23], introduced an ℓ 2 -norm regularization term to balance the agreement between local and global models and the empirical loss. [24] proposed Ditto, a multi-task learning-based FL framework that provides personalization while promoting fairness and robustness to byzantine attacks. Ditto uses a regularization term to encourage personalized local models to be close to the optimal global model. [25] proposed to interpolate local and global models to train local models while also contributing to the global model. However, scaling these approaches can be challenging due to high communication costs, reliance on strong assumptions about network connectivity, or the requirement to compute second-order gradients. Additionally, there is a potential for enhancing the algorithms' overall performance.

ADMM-based FL The Alternating Direction Method of Multipliers (ADMM) is a widely recognized algorithm that effectively tackles optimization problems across multiple domains. In recent studies, ADMMhas been successfully employed in distributed learning, as demonstrated by several works [26, 27, 28, 29, 30, 31, 32, 33]. In Federated Learning (FL) context, researchers have proposed various methods to address specific challenges. For handling falsified data in Byzantine settings, [34] introduced a robust ADMM-based approach. To mitigate local computational burdens in FL, [35] developed an inexact ADMM-based algorithm suitable for edge learning configurations. FL itself enables local training without the need to share personal data between clients and the server. Despite the advantages of FL, there is still a concern regarding clients' privacy. Analyzing the parameter differences in the trained models uploaded by each client can compromise their privacy. To tackle this issue, [36] proposed an inexact ADMM-based federated learning algorithm that incorporates differential privacy (DP) techniques [37]. By leveraging DP, the algorithm enhances privacy protection during the FL process. These ADMM-based frameworks also have high communication costs, ranging from O ( n ) to O ( n 2 ) per iteration, depending on the network's density with n clients. [38] introduced a Proximal Primal-Dual Algorithm (Prox-PDA) to enable network nodes to compute the set of first-order stationary solutions collectively. Moreover, these algorithms do not account for data heterogeneity in their framework designs, leading to performance deterioration in such scenarios. The most similar algorithm to RWSADMM is called Walkman [39]. Walkman is an ADMM-based framework utilizing the random walk technique for distributed optimization. In

Walkman, the communication and computation costs are reduced by activating only one agent at each step. Compared to other ADMM-based approaches, including Walkman, RWSADMM has several distinctive features. RWSADMM leverages stochastic approximation to reduce computation costs per iteration and enforces hard inequality constraints instead of consensus to manage heterogeneous data, resulting in increased robustness. Additionally, RWSADMM considers the dynamic graph, allowing it to adapt to changing network conditions and potentially improve communication efficiency. RWSADMM also incorporates a hard constraint parameter ϵ to promote local proximity among clients instead of using a regularization term as Walkman does to promote client consensus. This approach better balances personalization and global optimization. Finally, while Walkman is fully distributed without server involvement, RWSADMM is a server-based approach in which the server aggregates information from a small group of clients in each computation round.

3 Random Walk Stochastic ADMM (RWSADMM)

Before delving into the specifics of the proposed algorithm, we present the key notation used throughout this research. x ∈ R d represents a vector with length d and e is defined as a vector with entries equal to 1 and X ∈ R l × d depicts a matrix with l rows and d columns. [ x ] i represents the i th element of vector x and [ X ] ij is the ( i, j ) th element of matrix X . [ X ] i , [ X ] j represent the i th row and j th column of matrix X , respectively. ( ∇ f ( x )) j is used to denote the j th entry of the gradient of f ( x ) . The inner product of A and B is shown as ⟨ A,B ⟩ . E t [ . ] indicates the expectation given the past ξ 1 , . . . , ξ t -1 . ⊙ represents the Hadamard product/element-wise product and ⊗ represents the Kronecker product between two matrices. Finally, Norm p of vector x is denoted as || x || p p =

d

i

∑

=1

|

x

i

|

,

x

∈

R

X

is written as

∥

X

∥

F

=

√ ∑

n

i

=1

∑

m

j

=1

|

x

ij

|

.

Let us first define our Mobilizing FL problem. Mobilizing FL, which involves a mobile server, can be formulated as an optimization problem on a connected graph G = ( V , E ) . The graph comprises a set of n clients, represented as V = v 1 , v 2 , . . . , v n , and a set of m edges, denoted as E . The objective is to minimize the average loss function across all clients while adhering to inequality constraints that ensure local proximity among the clients' respective local models. The optimization problem can be formulated mathematically as follows:

where f i ( x i ) represents the local loss function with the model parameter as x i for the client i , the vertex set N ( i ) contains client i and its neighboring clients, and ϵ i is the non-consensus relaxation between local neighboring clients to replace model consensus requirement in typical FL. In our proposed FL method, we model the server's movement as a dynamic Markov Chain, introducing a dynamic element to the traditional ADMM-based approach. This work is the first to consider a dynamic mobile server within the ADMM-based FL framework. In RWSADMM, client-server communication occurs only when the server is close to a client. The sequence of client indices that are updated, denoted as i k , evolves based on a non-homogeneous Markov Chain with a state space of 1 , . . . , n [40]. To describe the transition dynamics of the Markov Chain, we employ the nonhomogeneous Markovian transition matrix P ( k ) , which represents the probabilities of transitioning between clients at time k . Specifically, the conditional probability of selecting client j as the next client, given that client i is the current client, is defined as:

Additionally, it is assumed that the server determines the probability of all possible locations for its next destination based on the transition matrix P ( k ) at time k . This provides a probabilistic approach to server navigation, allowing it to move around the network more effectively. To guarantee convergence, RWSADMM depends on the frequency of revisiting each agent. This quality is described by the mixing time of the algorithm. An assumption for the mixing time is as follows:

Assumption 3.1. The random walk ( i k ) k ≥ 0 , v i k ∈ V forms an irreducible and aperiodic (ergodic) Markov Chain with transition probability matrix of P ( k ) ∈ R n × n defined in Eq. (2) and stationary distribution π satisfying lim k →∞ π T P ( k ) = π T . The mixing time (for a given δ > 0 ) is defined as the smallest integer τ ( δ ) such that ∀ i ∈ V ,

p

d

2

where π ∗ := min i ∈V π i . This inequality states the fact that regardless of the current state i and time k , the probability of visiting each state j after τ ( δ ) steps is ( δ π ∗ ) -close to π j , that is, ∀ i, j ∈ V ,

Eq. (4) is used to prove the sufficient descent of a Lyapunov function L β in Section 3.1. Let's also define

from which one can further obtain ∥ P ( k ) ∥ ≤ ∥ P max ∥ for all k . Namely, the matrix P max is computed as the element-wise maximum matrix among all the matrices P ( k ) , for k = 0 , . . . , ∞ . Therefore, the mixing time requirement in Eq. (3) is guaranteed to hold for

where σ ( P ) := sup { ∥ ∥ f T P ∥ ∥ / ∥ f ∥ : f T 1 = 0 , f ∈ R n } . Using Eq. (5), we have ∀ P , σ ( P ) ≤ σ ( P ) max and the inequality ( a ) can be inferred.

3.1 Algorithm

In this section, we derive RWSADMM by integrating random walk and stochastic inexact approximation techniques into ADMM. Considering X := row ( x 1 , x 2 , . . . , x n ) ∈ R p × n , F ( X ) := ∑ n i =1 f i ( x i ) , where the operation row ( . ) refers to row-wise stacking of vectors x i 's. The mobilizing FL problem (1) can be expressed as:

where 1 = [1 1 . . . 1] ∈ R n i , n i denotes the volume of the vertex set N ( i ) . The constraint implies that | x i -x j | ≤ ϵ i , ∀ i = 1 . . . n and ∀ j ∈ N ( i ) through the triangle inequality. y i stored on the server is necessarily introduced as a local proximity of N ( i ) . We can obtain the augmented Lagrangian for problem (7)

where ε i = ϵ i / 2 and Z i ∈ R n i p are the dual variable and β > 0 is the barrier parameter. The RWSADMM algorithm minimizes the augmented Lagrangian L β ( y 1: n , X , Z 1: n ) in an iterative manner. At each iteration k , only a subset of clients covered by the mobilized server, the clients in N ( i k ) , participate in the federated update. The following updates are performed:

where y ′ i k , x i k , and z ′ i k denote the groups of variables of the local parameters stored by client i k at the ( k -1) th update. After solving these subproblems, we update the multiplier z i k as follows:

Next, we derive the solver of each subproblem. The three steps are noted as Updating x i k , Updating y i , and Updating z i .

The Problem (9) can be solved iteratively, consuming significant computational resources for the local clients. Furthermore, the computational complexity increases as the local dataset grows, as is often true in real-world applications. By utilizing the stochasticity and first-order subgradient expansion, we arrive at a more computationally efficient approximation of the original problem in Eq. (10).

In Eq. (10), ξ i k denotes one or a few samples randomly selected by client i k from its feature set and their ground truth labels in pairs at the k -th iteration. The function g i k ( x ′ i k , ξ i k ) is defined as the stochastic gradient of f i k ( x ′ i k ) at x ′ i k . The stochastic approximation can tremendously reduce memory consumption and save computational costs in each iteration. By setting the subgradient of the objective function in Eq. (10) to zero, we can derive the closed-form solution in Eq. (11).

where the signum function sgn ( · ) extracts the signs of a vector and t ′ i k = y ′ i k -x ′ i k .

Updating y i k : We solve the following problem

one can readily derive a closed-form solution for the problem (12) as:

where t i k = y ′ i k -x i k is similar to that of Eq. (11) except the updated x . Specifically, via mathematical induction, we can attain the new updated form of y i k below, which can also reduce the communication cost from O ( n ) to O (1) :

The κ coefficient used in Eq. (15) is decayed in each process step to achieve better convergence. Please refer to Appendix A for the entire RWSADMM algorithm framework.

4 Theoretical Analysis

In this section, we present the theoretical convergence guarantee of RWSADMM. To ensure its convergence, certain common assumptions are made regarding the properties of the loss functions. The assumptions are as follows:

Assumption 4.1. The objective function f ( x ) is bounded from below and coercive over R p , that is, for any sequence { x k } k ≥ 0 ⊂ R p ,

Assumption 4.2. The objective function f i ( x ) 's are L-smooth, that is, f i are differentiable, and its gradients are L-Lipschitz, that is, ∀ u , v ∈ R p [39],

Remark: In consequence it also holds that ∀ u , v ∈ R p

Assumption 4.3. The objective function f is M-Lipschitz, that is, ∀ u , v ∈ R p [41],

Assumption 4.4. The first-order stochastic gradient is sampled, which returns a noisy but unbiased estimate of the gradient of f at any point x ∈ R p , that is, ∀ x ∈ R p ,

Remark: Substituting Eq. (20) into Eq. (17), one can obtain that for i = 1 , . . . , n , we have

Substituting Eq. (20) into Eq. (18), for i = 1 , . . . , n , we can obtain

Assumption 4.5. The noise variance of the stochastic gradient is bounded as:

This condition bounds the expectation of ∥∇ f ( x t ) -g ( x t , ξ t ) ∥ 2 . Using Jensen's inequality, this condition implies a bounded variance [41].

We revisit the related crucial properties of the Markov Chain. The first time that the Markov Chain ( i k ) k ≥ 0 hits agent i is denoted as T i := min { k : i k = i } , and maximum value of T over all clients is defined as T := max { T 1 , . . . , T n } . For k > T , let τ ( k, i ) denote the iteration of the last visit to agent i before k , mathematically we have

To prove the convergence of our proposed algorithm, two Lyapunov functions defined for RWSADMMare required to be investigated:

where L β ( y k , X k ; Z k ) is defined in Eq. (8). The M k β is utilized in the convergence analysis. To guarantee the convergence of our algorithm, first, we refer to the asymptotic analysis of the nonhomogeneous Markov chain presented in [42]. Define Φ ( k, l ) with k ≥ l as the product of the transition probability matrices for the Markov chain from time l to k , i.e., Φ ( k, l ) = P ( k ) . . . P ( l ) with k ≥ l . Then we have the following convergence result:

Lemma 4.6. Consider

∀ s , lim k →∞ Φ ( k, l ) = 1 n ee T .

The convergence of Φ is geometric and the rate of convergence considering ∀ k, l, with k ≥ l ≥ 0 , is given by

Using Lemma 4.6, the convergence analysis of the algorithm is as follows.

Lemma 4.7. Under Assumptions 4.1 and 4.2, if β > 2 L 2 + L +2 , ( M k β ) k ≥ 0 is lower bounded and convergent, the iterates ( y k , X k , Z k ) k ≥ 0 generated by RWSADMM is bounded.

The proof sketch and the detailed convergence proof are presented in Appendix B. Using Lemma 4.7 and B.6, we can present the convergence of RWSADMM in Theorem 4.8.

Theorem 4.8. Let Assumption 4.5 hold. For β > 2 L 2 + L +2 , it holds that any limit point ( y ∗ , X ∗ , Z ∗ ) of the sequence ( y k , X k , Z k ) generated by RWSADMM satisfies y ∗ = x i ∗ , i = 1 , . . . , n where y ∗ is a stationary point of Eq. (7) , with probability 1 , that is,

If the objective function of Eq. (7) is convex, then y ∗ is a minimizer.

Next, Theorem 4.9 further presents that the algorithm converges sublinearly. This is comparable to the convergence rate of other FL methods [43, 44, 24, 25], but the existing methods didn't consider the dynamic graph and infrastructure-less environment. The detailed proof is offered in Appendix C.

Theorem 4.9. Under Assumptions (3.1) , (4.1) , and (4.2) , with given β in Lemma 4.7, and local variables initiated as ∇ f i ( x 0 i ) = β x 0 i = z 0 i , ∀ i ∈ { 1 , . . . , n } , there exists a sequence { g k } k ≥ 0 with { g k } ∈ ∂L k +1 β satisfying

where C is a constant depending on β , L , and γ , n , and τ ( δ ) .

Communication Complexity Using Theorem 4.9, the communication complexity of RWSADMM for nonconvex nonsmooth problems is as follows. To achieve ergodic gradient deviation E t := min k ≤ K E ∥ ∥ g k ∥ ∥ 2 ≤ ω for any K > τ ( δ ) + 2 , it is sufficient to have

(a) is achieved by taking L 0 β and f -as constants and independent of n and the network structure. Using the τ ( δ ) definition from (6), by setting δ = 1 / 2 and assuming the reversible Markov chain with P ( k ) T = P ( k ) , the communication complexity is

Communication Comparison Among the baseline frameworks, Per-FedAvg [22] and APFL [25] have addressed the communication complexity of their respective frameworks. By assuming that Assumption 3.1 holds and utilizing Eq. (30), we can determine the communication complexity of RWSADMM as O ( ω -1 ) for K iterations. In comparison, Per-FedAvg exhibits a higher communication complexity of O ( ω -3 / 2 ) . In the case of APFL, all clients are assumed to be used in each computation round to ensure convergence in nonconvex settings. The communication complexity of APFL is determined as O ( n 3 / 4 ω -3 / 4 ) , where n represents the total number of clients. Consequently, when n is large, APFL exhibits a significantly higher communication rate than RWSADMM. Overall, the communication complexity analysis suggests that RWSADMM offers superior scalability and communication efficiency compared to existing methods.

5 Experimental Results

Setup We evaluate the performance of RWSADMM using heterogeneous data distributions. All the experiments are conducted on a workstation with Threadripper Pro 5955WX, 64GB DDR4 RAM, and NVidia 4090 GPU. All frameworks are performed on standard FL benchmark datasets (MNIST [45], Synthetic [23], and CIFAR10 [46]) with 10-class labels and convex and non-convex models. Multinomial logistic regression (MLR), multilayer perceptron network (MLP), and convolutional neural network (CNN) models are utilized for strongly convex and two non-convex settings, respectively. We create a moderately dynamic connected graph of randomly placed nodes where each node has at least 5 neighboring nodes at k -th update. We set the probability transition matrix P ( k ) as [ P ( k )] ij = 1 /deg ( i k ) and set up the experiments for N = 20 clients with a regeneration frequency of 10 steps for the dynamic graph. The data is split among clients using a pathological non-IID setting. The data on each client contains a portion of labels (two out of ten labels), and the allocated data size for each client is variable. For the Synthetic data, we use the same data generative procedures of [23] with 60 features and 100 clients. All local datasets are split randomly with 75% and 25% for training and testing, respectively. The models' details, the rationale behind graph construction, and hyperparameter tuning for β , κ , and selected ε value are further described in Appendix D.

Performance Comparison The performance of RWSADMM is compared with FedAvg [1] as a benchmark and several state-of-the-art personalized FL algorithms such as Per-FedAvg [22], pFedMe [23], APFL [25], and Ditto [24]. The test accuracy and training loss for the MNIST dataset is depicted in Fig. 2. (Synthetic and CIFAR10 figures are presented in Appendix D). Test accuracy and time cost for all the datasets are reported in Table 1.

The test accuracy progress curves of RWSADMM for all the models (2a-2c) have a significantly faster convergence. For the non-convex models (2b), RWSADMM reaches convergence after 200 iterations, while the rest of the algorithms, except Ditto, work toward convergence until 600 iterations. The performance curves are shown for 100 iterations for consistency. When tested on MNIST with MLP, RWSADMM demonstrated comparable performance against pFedMe. In the test on Synthetic data with MLR models, RWSADMM exhibited a significant advantage over the other methods, with an improved margin of 14 . 95% . Regarding computational efficiency, RWSADMM is slower than FedAvg and Per-FedAvg, but faster than pFedMe. Furthermore, RWSADMM converges in fewer iterations (200 iterations) than pFedMe (600 iterations). RWASDMM is also run for more extensive networks with 50 and 100 nodes as a separate set of experiments. The performance comparison results and diagrams are also in Appendix D.

6 Conclusion and Future Work

This study proposes a novel approach called RWSADMM, designed for systems with isolated nodes connected via wireless links to the mobile server without relying on pre-existing communication

(a) MLR acc

(b) MLP acc

(c) CNN acc

(d) MLR loss

(e) MLP loss

(f) CNN loss

1.00

Strongly convexMLR

1.00

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0.95

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Global rounds

Global rounds

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Non-convex MLP

Non-convex CNN

0.70

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1

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0.60

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acc(%)

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98.63

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167

99.29

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295

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0.04

3857

96.44

±

0.12

473

97.17

±

0.18

692

CIFAR10

MLR

MLP

CNN

acc(%)

t(s)

acc(%)

t(s)

acc(%)

t(s)

FedAvg

40

.

84

±

0

.

01

160

41

.

02

±

0

.

05

69

38

.

65

±

0

.

05

78

PerAvg

47

.

43

±

0

.

09

192

60

.

25

±

0

.

07

253

83

.

52

±

0

.

01

800

pFedMe

67

.

53

±

0

.

34

515

78

.

12

±

0

.

38

340

83

.

56

±

0

.

05

3480

Ditto

75

.

2

±

0

.

01

225

81

.

37

±

0

.

13

259

83

.

86

±

0

.

02

2189

APFL

75

.

17

±

0

.

32

50

78

.

00

±

0

.

18

55

66

.

23

±

0

.

03

702

RWSADMM

(our method)

80.72

±

0.11

131

84.99

±

0.20

253

87.08

±

0.03

3759

infrastructure. The algorithm enables the server to move randomly toward a local client, establishing local proximity among adjacent clients based on hard inequality constraints, addressing the challenge of data heterogeneity. Theoretical and experimental results demonstrate that RWSADMM is fast-converging and communication-efficient, surpassing current state-of-the-art FL frameworks. This study primarily focuses on the methodological framework for RWSADMM. Future research directions should explore essential techniques such as incorporating differential privacy techniques and examining scalability in more extensive network and dataset scenarios. Further investigation is needed to assess the implementation in physical networks and evaluate the effect of communication delays in the real world.

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nAuthors consider in this paper a Federated Learning framework with a mobile server. They propose an algorithm named Random Walk Stochastic Alternating Direction Method of Multipliers (RWSADMM) for networks with instability and limited capacity. To overcome the non-IID nature of clients' data, they ensure local proximity among adjacent clients using hard inequality constraints. Their theoretical and empirical results demonstrate the effectiveness of RWSADMM.\n\nSTRENGTHS:\nThe mobile Federated Learning in infrastructure-less environment is an important problem. The paper is well organized with rigorous theoretic analysis.\n\nWEAKNESSES:\nThe random walk mobility model is over simplified and is not a realistic model for wireless adhoc network. The random walk model was initially developed to emulate the unpredictable movement of particles in physics. It is a memoryless mobility model that does not consider temporal and spatial correlation. In addition, topology control technique is a  must to ensure reliable operation of a wireless adhoc network in reality and therefore such a memoryless assumption is hardly the case in many real life applications.   \nAlthough the concept of the Federated Learning (FL) framework with a mobile server is interesting, it seems to be a special case of asynchronous FL, where only a subset of clients is selected to participate in the update in each iteration. Authors are recommended to elaborate the difference between this work and existing asynchronous FL work in the literature.\n\nQUESTIONS:\nCan authors elaborate more regarding employing the dynamic Markov Chain to model the server's movement?\nThe introduction of the y-variable in Eq. (7) is unclear.\nIf one of the neighboring nodes of the selected node in an iteration fails to respond, does the algorithm still work?\n\nLIMITATIONS:\nThe mobility model considered in this paper is of limited applicability.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-06"}

comments: SUMMARY:
This paper proposes random walk stochastic method based on traditional ADMM, which enables personalized federated learning considering the mobility of clients and using a mobilized server. The authors solve the optimization of the stored client’s model parameters through Lagrangian and dual problem. The theoretical analysis proves the convergence performance of the proposed algorithm based on assumptions 4.1 to 4.5. The paper validates the performance of the algorithm by comparing its accuracy and loss with APFL/Ditto/PerAvg/FedAvg/pFedMe.

STRENGTHS:
1.This paper realizes the mobilizing federated learning in a in an infrastructure-less environment for the first time.
2.This paper has rigorously proposed the RWSADMM algorithm and rigorously proves its convergence performance.
3.The paper solves the model update faster by solving the dual problem and achieves better communication cost.

WEAKNESSES:
1.The paper assume the mobilized server wanders in the region following a random walk pattern, and uses the warzone as an example. In a drastic combat, the mobilized server should usually follow a more regularized route instead of randomly wandering in the field. Also, the client are also set to be stationary in the paper, which can not represent usual moving soldiers and devices. Therefore, I think the system model is not very realistic, and the applicability of the system in a real war may be limited.

QUESTIONS:
1.On page 8, the authors claim RWSADMM’s O(\omega^{-1}) communication complexity is better than Per-FedAvg’s O(\omega^{-3/2}) complexity. This hold when \omega ranges from 0 to 1, but the authors seem not to discuss or define the range of the parameter.

LIMITATIONS:
N/A
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2023-07-06

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a new FL setting that involves a dynamically moving server among clients residing in different regions and receiving updates from clients of the selected region through wireless links. The proposed algorithm RWSADMM addresses personalization and data heterogeneity by minimizing the local proximity among neighbor nodes of each client. Additionally, the authors apply a first-order subgradient expansion for computational complexity reduction, which is useful for real-world applications. The algorithm is demonstrated to converge provably faster than baseline methods through theoretical analysis and experimental comparisons.\n\nSTRENGTHS:\nThis paper introduces a novel asynchronous mobilizing federated learning setting that can be \nutilized in realistic war situations where communication infrastructure is unavailable. Additionally, \nit establishes a mathematical framework based on dynamic graph-based representation, allowing the server to move through the field and interact with various clients to update the neural network.\n\nThe proposed algorithm surpasses the baseline methods by a significant margin, demonstrating notably faster convergence rates, particularly in realistic non-IID scenarios.\n\nThe methodology employed in this study is a theoretically sound solution derived from the analysis of dynamic Markov Chain and Alternating Direction Method of Multipliers.\n\nThe proposed RWSADMM algorithm ensures convergence based on commonly assumed properties of the loss function. Furthermore, the communication complexity of RWSADMM is compared to existing methods like Per-FedAvg and APFL, providing theoretical evidence of its superior scalability.\n\nIn the proposed setting, both convex and non-convex models converge faster and exhibit higher accuracy performance compared to the baseline methods.\n\nWEAKNESSES:\nThe proposed setting (mobilized federated learning) seems somewhat unrealistic. In a war situation where long-range communication is not supported, is there a need for federated learning? Considering that neural networks are large and transmitting them via tactical vehicles would be challenging, wouldn't it be better to transmit data for distributed learning or perform centralized learning on the server, rather than transmitting model updates? \n\nThe proposed Markov Chain does not seem to effectively represent the problem situation (mobilized federated learning). In real scenarios, nodes within the communication range of the communication vehicle should be selected, but the proposed algorithm does not consider the physical distances between nodes. Additionally, considering realistic situations like Figure 3, where nodes (walking soldiers) move and the topology keeps changing, the Markov Chain used in this paper cannot reflect spatial rearrangements of participating nodes.\n\nI am concerned about privacy issues. In the proposed algorithm, each node needs to receive model updates from neighboring nodes to calculate the proximity term. However, this process can expose the data that the node possesses, as it can be extracted from the updates of other nodes. Furthermore, simultaneous reception of model updates from multiple nodes can result in communication overhead.\n\nThere is a lack of ablation study on the proposed components. There is no analysis on how well the proposed local proximity term resolves data heterogeneity, and no trade-off analysis on how much computational resources are saved through first-order expansion and the corresponding impact on performance.\n\nQUESTIONS:\nIn the current problem, the server moves through a random walk. However, in practical situations, shouldn't the server actively modify its routing to reach unseen nodes? \n\nLooking at the Experimental setup in Section 5, there are only 20 clients. Considering the trend in recently published Federated learning works, shouldn't the proposed algorithm be compatible with scenarios involving a larger number of clients?\n\nIn Figure 2, although PerAvg and APFL are FL works geared to data heterogeneity, their performances are significantly lower compared to FedAvg. Why does this discrepancy occur?\n\nWhat are the confidence intervals for the reported performance in Figure 2 and Table 1? It is important to report confidence intervals, as performance can vary significantly  depending on how the spatial relationship between nodes is initialized.\n\nThe proposed setting (mobilized federated learning) seems somewhat unrealistic. In a war situation where long-range communication is not supported, is there a need for federated learning? Considering that neural networks are large and transmitting them via tactical vehicles would be challenging, wouldn't it be better to transmit data for distributed learning or perform centralized learning on the server, rather than transmitting model updates?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-06"}