FIARSE: Model-Heterogeneous Federated Learning via Importance-Aware Submodel Extraction

Authors: feijie  wu
xingchen  wang
yaqing  wang
tianci  liu
lu  su
jing  gao
Venue: NeurIPS_cc_2024_Conference
Published: 2024-05-14 17:24:20

Abstract: In federated learning (FL), accommodating clients' varied computational capacities poses a challenge, often limiting the participation of those with constrained resources in global model training.  To address this issue, the concept of model heterogeneity through submodel extraction has emerged, offering a tailored solution that aligns the model's complexity with each client's computational capacity. In this work, we propose Federated Importance-Aware Submodel Extraction (FIARSE), a novel approach that dynamically adjusts submodels based on the importance of model parameters, thereby overcoming the limitations of previous static and dynamic submodel extraction methods.  Compared to existing works, the proposed method offers a theoretical foundation for the submodel extraction and eliminates the need for additional information beyond the model parameters themselves to determine parameter importance, significantly reducing the overhead on clients. Extensive experiments are conducted on various datasets to showcase the superior performance of the proposed FIARSE.

Main paper: 
 introduction: 
 Federated learning (FL) [36, 52] stands out as a promising distributed training paradigm, in which the clients enjoy mutual information without jeopardizing data privacy. Specifically, the FL server requests the clients to train a model with their local data and aggregates the models into a global one. Such a paradigm, however, may fail in a real-world FL system, where the clients usually have varying computation capacities [13, 42, 46] , likely preventing the clients with insufficient computation resources from being involved in training a large global model [73] .To tackle the challenge, a practical solution is to enable model heterogeneity, ensuring that the model deployed on each individual client aligns with its local computation capacity. This can be done by extracting a submodel for each client from the global model, which encompasses a subset of the parameters of the global model. During the model training period, the parameters of each submodel are thereby retrieved from the counterpart of the global model. Importantly, each parameter of the global model is exclusively averaged among the submodels containing it [13] .Depending on whether the submodels undergo reconstruction during the model training process, existing works fall into two categories: static submodel extraction [13, 25, 33, 35, 58, 67, 76] and dynamic submodel extraction [2, 7, 8, 21, 45] .Static submodel extraction creates a submodel for each client prior to the model training process. As illustrated in Figure 1a , the submodel remains unchanged throughout the training process. However, static submodel extraction suffers from certain limitations that affect both local clients and global performance. Locally, the metrics or knowledge used to extract a submodel for each client are likely to evolve during the training process. Since static submodels do not account for this evolution, they may fail to achieve optimal performance. Globally, the extractions of submodels from certain parts may incur client drift during the training, as pointed out by Alam et al. [2] and Liao et al. [45] . This phenomenon emerges when the clients update their submodels biased to their local optima, thereby deviating from the global optimum. As a result, it either degrades the training efficiency or leads to a surrogate convergence on a global scale [34, 64] .Dynamic submodel extraction updates each submodel dynamically in each training round, and thus is able to capture the evolution of the global model. For example, FedRolex [2] employs a rollingbased submodel extraction approach, addressing client drift by ensuring equal chances for training each parameter as shown in Figure 1b . While the method demonstrates significant performance improvement over static submodel extraction in terms of the global model, it sacrifices the submodels' performance. This is because FedRolex treats every parameter equally, leading to a lack of clear guidance on submodel extraction.To overcome the limitations of the above methods, in this paper, we propose Federated Importance-AwaRe Submodel Extraction, named FIARSE, for model-heterogeneous FL. Specifically, the proposed FIARSE method extracts the submodels based on the importance levels of model parameters.Here important parameters are the edges of the neural network that can induce dramatic changes in the final outputs when removed.Figure 1c visually illustrates the submodels extracted by FIARSE. As demonstrated, FIARSE constructs a submodel by sequentially incorporating parameters in descending order of their importance levels (represented by the thickness of the edges in the model), from highest to lowest, until the client's maximum computation capacity is reached. In contrast to static submodel extraction, our approach enables dynamic updates of the submodels, thereby effectively capturing the evolving nature of model parameters. When compared to rolling-based submodel extraction, FIARSE adaptively identifies important parameters, ensuring outstanding performance for both the global model and local submodels. Referring back to the model shown in Figure 1 , the edges connecting the leftmost neuron at the second layer are indicative of important parameters. However, the rolling-based submodel extraction method would roll these parameters out of the submodel in round t + 1 (as shown in Figure 1b ), resulting in inadequate training on these crucial parameters. In contrast, our proposed method can identify and retain these important parameters in the training process, as illustrated in Figure 1c .Contributions. The contributions of this paper can be highlighted from the following perspectives:• Algorithmically, we propose an importance-aware framework for model-heterogeneous federated learning. This framework can construct a client-specific submodel calibrated to each client's computation and storage capacity. It achieves this by representing the importance level of each model parameter with its magnitude, thereby avoiding additional storage or computational overhead needed to explicitly maintain the importance scores.• Theoretically, we prove that the proposed algorithm converges at a rate of O 1/ √ T , where T is the number of communication rounds. This convergence rate is consistent with that of the state-of-the-art FL algorithms, indicating that the proposed submodel construction mechanism does not undermine the convergence properties. To the best of our knowledge, this is the first study to provide a theoretical analysis for model-heterogeneity FL under partial-client participation.• Empirically, we conduct extensive experiments on image and text classification tasks, employing a training-from-scratched model ResNet, and a pretrained model RoBERTa. The results verify that FIARSE significantly outperforms existing approaches, particularly on the clients with limited capacity. The superior performance on resource-constrained devices demonstrates FIARSE's advantages in efficiently adapting submodels to meet diverse capabilities. 
related work: 
 This section discusses the state-of-the-art works that are most relevant to our research. Appendix A provides a more comprehensive review.Computation Heterogeneity in FL. Computation heterogeneity in FL refers to the varying computational capacities among clients, including differences in hardware capabilities and resource availability. One typical solution is to allow faster clients to perform more local updates, while slower ones update their local models fewer times [43, 50, 53, 57, 64, 72] . However, these approaches require clients to train the full model, which becomes infeasible when some clients cannot load the full model due to limited computation resources. In our work, we extract a submodel for each client that fits within their computational capacity.Model Customization in FL. Model customization allows the clients to build their local models that align with their local computation resources [11, 28, 46, 66, 77, 80] . To aggregate these heterogeneous models, the method often employs distillation for knowledge transfer, which requires a shared public dataset. However, this approach becomes infeasible when a shared public dataset is unavailable [3, 59] . In our work, we eliminate the need for a public dataset, broadening its applicability to a wider range of scenarios.Model Sparsification in FL. Model sparsification, also known as model pruning, removes the unimportant model parameters from a deep learning model, reducing computation overhead and tailoring model sizes to suit clients with varying computational resources [8, 45, 81] . For example, Flado [45] achieves model sparsification by requiring each client to maintain the importance levels of model parameters and extracting a submodel that encompasses the most important model parameters. While effective, this method incurs considerable overhead on each client in terms of both storage and computation, posing significant challenges for resource-constrained devices. In contrast, our work implicitly represents parameter importance through their values, eliminating the need to maintain separate importance scores for each parameter. M (i) ∈ {0, 1} d such that ∥M (i) ∥ 1 ≤ γ i d.Let M be the collections of the clients' masks M (i) , i.e., M = ∪ i∈[N ] M (i) ∈ {0, 1} N ×d . To simultaneously optimize the model parameter and the clients' masks, the model-heterogeneity FL system is formulated asEQUATIONLet D i be the local dataset of client i ∈ [N ], ℓ be the loss function which calculates the loss for a model on a given data sample (including an input and a target). Therefore, the local objective F i (•) in Equation ( 1) indicates the expected loss for client i on the local dataset. For simplicity, we consider all N clients to carry equal weights (i.e., 1/N ) in Equation ( 1), and the proposed approach can be extended to the scenario where the clients are with different weights.A Generic Solution: Partial Averaging. There are numerous submodel extraction approaches, which are categorized into static and dynamic submodel extractions. However, these methods adopt partial averaging to aggregate clients' models into a global one, and the details are outlined as follows: At round t ∈ {0, 1, . . . },• Sampling: The server randomly samples a subset of clients A ⊂ [N ] and distributes the global model parameters xt to the selected clients.• Local Model Training: The clients i ∈ A performs K-times local updates via x(i) t,k = x (i) t,k-1 - η∇F i x (i) t,k-1 ⊙ M (i) t ⊙ M (i) t ,where k ∈ {1, . . . , K} and x(i) t,0 = xt . It is noted that M (i) trepresents a binary mask for client i at t-th round, which can be either predefined [2, 13, 25, 33] or determined by the client [21, 45, 76] .• Global Model Aggregation: The clients collect the model updates from the participants A and perform the global model aggregation viaxt+1 = xt -η s Agg i∈A x (i) t,K -xt . Given a set of d-dimension vectors v 0 , . . . , v |A|-1 ∈ R d , Agg i∈A (v) is defined as: (i) For the j-th index, Agg i∈A (v j ) = i∈A v i j / i∈A 1{v i j ̸ = 0} ; (ii) Agg i∈A (v) = ∪ j∈[d]Agg i∈A (v j ). Limitations. The above solution adopts a consistent mask during local model training, which cannot obtain the optimal mask M for various clients as Equation ( 1) expects. Some recent works (e.g., Flado [45] and pFedGate [8] ) have proposed to capture the importance levels of each model parameter in achieving the objective of Equation (1) , where a submodel consists of the most important parameters up to the maximum capacity of a client. In these works, the clients hold the importance scores for each model parameter and extract a submodel accordingly. After the local training of the submodel, the clients take an additional step to optimize the importance scores. Despite the effectiveness of these approaches, their feasibility is compromised due to the massive costs related to storage and computation. These approaches entail a minimum training memory and storage of O(d) and O(d) on client i ∈ [N ], respectively, while HeteroFL [13] ensures the training memory within O(γ i d) and does not require additional storage. Moreover, it is time-consuming to separate model parameters update and mask optimization into two steps. A work [81] attempts to simplify the process by means of greedy pruning, where a submodel consists of the parameters selected from the largest to the smallest absolute values. Apparently, this approach avoids mask optimization while evolving the submodel architectures since they are associated with model parameters. However, this work keeps the mask consistent during local model training, which does not make sense because an update of model parameters should lead to a different mask.Question: Does the paper discuss the limitations of the work performed by the authors?Answer: [Yes]Justification: We mention our limitations and point out a future research direction in the Conclusion (Section 7).Guidelines:• The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.• The authors are encouraged to create a separate "Limitations" section in their paper.• The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.• The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.• The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.• The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.• If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.• While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations. 
theory assumptions and proofs: 
 Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?Answer: [Yes]Justification: We provide the theoretical conclusion in the Convergence Analysis (Section 5), and the detailed proof is provided in Appendix C.Guidelines:• The answer NA means that the paper does not include theoretical results.• All the theorems, formulas, and proofs in the paper should be numbered and crossreferenced.• All assumptions should be clearly stated or referenced in the statement of any theorems.• The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition.• Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.• Theorems and Lemmas that the proof relies upon should be properly referenced. 
fiarse: 
 Solution Overview. In this work, we explore the correlation between the value of model parameters and their importance levels, leveraging the insights from previous research [29, 54] . These studies reveal that the magnitude of model parameters can act as an indicator of their importance levels. This discovery offers an opportunity to simplify Equation (1) . Given that our objective is to extract important parameters for submodel construction, we can approximate this goal by selecting larger parameters to build the submodel, rather than explicitly maintaining an importance score for each parameter. Though simplified, the problem is still challenging since the model parameters themselves are also variables to be optimized. To address this challenge, in Section 4.1, we will present a novel submodel construction method that can jointly select and optimize model parameters.Finally, Section 4.2 introduces our proposed FL algorithm FIARSE that seamlessly integrates the submodel construction method and optimizes the global model. In detail, the clients optimize the model parameters x by leveraging the submodel construction method. The server subsequently aggregates these optimized models from the clients and initiates a new training round. Given that the collected model parameters inherently reflect their importance levels, the aggregated global model parameters also effectively capture their importance from a global perspective. Algorithm 1 concisely presents the pseudocode of FIARSE.As highlighted in the overview, the insight that the values of model parameters are correlated with their importance allows us to reframe the problem of submodel construction. Intuitively, by controlling the number of parameters included in the submodel, we can ensure the computation and/or storage costs of the submodel not exceed the budget of the clients. To achieve this, we establish a threshold for the model parameters based on each client's capacity. Only those parameters whose values exceed this threshold are included in the respective client's submodel. Thanks to the correlation between a parameter's value and its importance level, this approach ensures that the parameters included in the submodel are of greater importance than those that are excluded. This idea can be implemented by converting the mask variable in Equation (1) into a function of the parameter values:EQUATIONwhere Now the question is how to determine the threshold for Equation (2) . In general, the threshold abides by the clients' local computation/storage capacity. Towards this end, we determine the value using TopK γ (•) operation, which selects the top γ values of the given vector. For simplicity, we discuss model-wise threshold selection strategies in this section, where the threshold θ i is set for TopK γi (|x|).M (i) (•)Our proposed method is applicable for settings where different thresholds are assigned to different model parameters. Additional threshold selection strategies will be explored in Appendix D.1.Threshold-Controlled Biased Gradient Descent (TCB-GD). We enhance Equation (2) by integrating straight-through estimation (STE) [4, 49] , where we assume the mask is labeled with 1 with a probability determined by clip |xj |-θi |xj |+θi , 0, 1 , where xj means j-th element of a d-dimension model parameter x. Therefore, the gradient calculated in the backward propagation on client i ∈ [N ] process adheres to:EQUATIONA detailed derivation of the above equality is provided in Appendix B. There are two key differences in comparison with the gradient computation used by local training of partial averaging, i.e., ∇F i x ⊙ M (i) ⊙ M (i) . First, the mask shifts with the model parameters changing. Second, the backward propagation considers the importance levels of model parameters and forms a biased gradient descent. This means the second term tries to make a clear border between the important and non-important parameters.Effectiveness. We analyze our proposed approach based on its two features, namely, thresholdcontrolled and biased gradient computations:• Threshold-controlled: By comparing Equation (3) with Equation (2), we notice that the parameters no less than the designated threshold will be updated. In other words, this gradient descent method only updates the parameters that are greater or equal to the given threshold, and those parameters that are initially smaller than the threshold never get updated. Obviously, the computation cost at each iteration remains constant or even smaller than the cost at the previous iterations.FL clients usually update the model for multiple iterations. According to the description above, the trained submodel is shrinking because some parameters may drop behind the threshold, while no new parameters are introduced to the submodel. Therefore, the proposed gradient descent method keeps the computation cost constant or even smaller than our expectation.• Biased: Biasedness accelerates the update of the parameters near the threshold to distinguish their importance. In other words, less important parameters drop below the threshold and roll out, while the important ones continue to increase until stable. This feature guarantees the model parameters reflect their importance value by minimizing the existence of ambiguous parameters close to the threshold. In other words, the clients can easily identify the unimportant model parameters, facilitating the extraction of a submodel based on parameter values ranging from large to small until it aligns with a client's maximum computation capacity.We further integrate this submodel construction method into our proposed FL algorithm FIARSE and comprehensively discuss how threshold-controlled biased gradient descent benefits the modelheterogeneity FL in the next section.In FIARSE, a global model is initialized with arbitrary parameters x0 ∈ R d (a pretrained model is allowed, which can be regarded as a special case of an arbitrary model).Partial client participation is one of the features of FL algorithms because the server is unlikely to handle all the communications from all clients, especially when the number of clients is considerably large [10, 32, 44, 71, 74] . Therefore, we present our algorithm FIARSE in support of partial client participation: At the beginning of each communication round t ∈ {0, 1, . . . }, the server uniformly samples a group of clients from [N ] without replacement, denoted by A, which consists of A clients.Subsequently, the server broadcasts the submodels to the selected clients and collects and merges their updates into the global model. In the rest of the section, we comprehensively discuss the details of these steps and exemplify them with t-th round. for i ∈ A in parallel doInitializex (i) t,0 = xt ⊙ M (i) t (x t ) 6:for k = 0, . . . , K -1 do 7:g (i) t,k+1 = ∇ x (i) t,k F i x (i) t,k ⊙ M (i) t x (i) t,k 8:x(i) t,k+1 = x (i) t,k -η l • g (i) t,k+1 9:end for 10:∆x (i) t = xt -x (i) t,K 11:Send ∆x(i)t to the server 12:end for13: xt+1 = xt -η s • Agg i∈A ∆x (i) t14: end for Submodel Extraction on Server.Based on the set of participants A, the server learns their computation capacities {γ i } i∈A . Then, the server follows the threshold selection strategies described in Section 4.1 and extracts the submodel for all participants according to their computation capacities. Take the model-wise threshold selection as an example and select a submodel for participant i ∈ A. The threshold is set forθ i = TopK γi (|x|).Then, the server extracts a submodel encompassing the parameters |x t | ≥ θ i and sends it to the participant. This procedure is equivalent to the expression in Line 3 of Algorithm 1, i.e., xt ⊙ M(i) t (x t ). Local Training on Clients i ∈ A.After receiving the submodel from the server, we thereby initialize the local masking function M (i) t (•) for training, which implicitly includes the threshold θ i . The threshold will remain constant during the local training. As outlined in Line 7 -8, the client utilizes the TCB-GD to optimize the local model for K iterations. In each iteration k ∈ {0, . . . , K -1}, the client utilizes the masking function to sort out the parameters that drop behind the threshold. Then, the client computes the gradient ∇ x (i)t,k F i x (i) t,k ⊙ M (i) t x (i) t,kusing Equation (3) (Line 7) and updates the local model via x(i) t,k+1 = x (i) t,k -η l • ∇ x (i) t,k F i x (i) t,k ⊙ M (i) t x (i) t,k(Line 8). As discussed in Section 4.1, the training memory is bounded by O(γ i d), and there is no additional storage requirement. In contrast to other importance-aware works such as Flado, the proposed FIARSE is more feasible in practice.Aggregation on Server. After the selected clients finish their local updates, the server aggregates the updates from the clients (Line 10 -11). Similar to the global model aggregation of partial averaging described in Section 3, FIARSE updates the global model viaxt+1 = xt -η s • Agg i∈A ∆x (i) t (Line 13). A detailed description of the recursive function is placed in Appendix C.1.Given that the global model starts with randomly initialized parameters, this aggregation not only updates the global model parameters but also aligns their values with their importance levels. In the next training round t + 1, the FIARSE will return to Line 3, regenerating a submodel from the updated parameters, which is then sent back to the client. Since the values of the parameters represent better their importance levels than in the last training round, the newly generated submodel will contain more important parameters. The whole algorithm will then be repeated once again, resulting in a newly aggregated global model. This iterative process will progressively select important parameters and exclude unsignificant ones, steering the global model towards a state of convergence in which the importance of parameters is accurately represented. In the coming section, we theoretically analyze the convergence rate of the proposed FIARSE.Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?Answer: [Yes]Justification: We provide the experiment setup in Section 6.1 and Appendix D.2. Also, we release our code at https://github.com/HarliWu/FIARSE.Guidelines:• The answer NA means that the paper does not include experiments.• If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.• If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.• Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.• While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example (a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.• The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.• The full details can be provided either with the code, in appendix, or as supplemental material. 
convergence analysis: 
 Existing convergence analyses of FL algorithms predominantly rely on model-homogeneous settings [65] . However, the exploration of model-heterogeneity FL remains inadequately addressed, with some studies [58, 69, 76, 81] in this domain being recently introduced but relying on full client participation. This section aims to present a thorough convergence analysis of the proposed FIARSE under nonconvex objectives. Specially, our analysis is established under the scenarios of model-heterogeneity FL where not all clients actively participate in the round-by-round training process.Before showing the convergence result, we make the following three assumptions:Assumption 5.1 (Masked-L-smoothness). For all i ∈ [N ], the local objectives F i are L-Lipschitz smooth with a differentiable mask function M (i) : For all w, v ∈ R d , ∇ w F i w ⊙ M (i) (w) -∇ v F i v ⊙ M (i) (v) 2 ≤ L ∥w -v∥ 2 .Assumption 5.2 (Bounded Global Variance). For all i ∈ [N ], the variance between local gradient ∇F i (•) and global gradient ∇F (•) is bounded under the same model parameters: For all w ∈ R d , there exists a constant σ j ≥ 0 for all j ∈ [n] such thati∈N γ ′ j 1 N γ ′ j ∇F [γ ′ j-1 :γ ′ j ] i (w) -∇F [γ ′ j-1 :γ ′ j ] (w) 2 2 ≤ σ 2 j .We further annotate σ 2 = n j=0 σ 2 j . Assumption 5.3 (Masked Reduction). For all i ∈ [N ], the mask-incurred error is bounded with respect to the model parameter xt , t = 0, 1, . . . : There exists a scalar δ 2 t ∈ [0, 1) at round t such that∇F i (x t ) ⊙ M (i) (x t ) -∇ xt F i xt ⊙ M (i) (x t ) 2 2 ≤ δ 2 t ∥x t ∥ 2 2Lipschitz-smooth assumption has gained widespread acceptance in machine learning research, as evidenced by its incorporation in various studies such as [1, 6, 51, 70, 74, 75, 81] . Assumption 5.1 extends this assumption to a scenario where a binary mask is calculated based on the model parameters and subsequently applied to the model. The second assumption, widely made in the previous FL studies [74, 75] , establishes bounds between the local objectives and the global objective due to the occurrence of non-i.i.d. data. Assumption 5.3 draws inspiration from [51, 81] and characterizes the masking performance by comparing gradients with and without the application of the mask. Notably, ifM (i) t = 1 d (i.e., setting the threshold for θ i = 0), then δ 2 t = 0. With these three assumptions, we analyze the convergence rate of our proposed algorithm. Under non-convex objectives, our goal is to evaluate if the gradient norm can approach zero with respect to the model parameters x when the communication round t → ∞. Theorem 5.4 presents the convergence result of the proposed FIARSE, and a detailed proof is deferred to Appendix C. Theorem 5.4. Suppose that Assumption 5.1, 5.2 and 5.3 hold. We define F (x)△ = F x, 1 N ×d , and F (x) ≥ F * for all x ∈ R d . Let the local learning rate satisfy η l ≤ min 1 2L K(K + 1) , 1 6L (K + 1)A , η s 9L , 1 12L √ KN , A 32KN Lη s .Denote T as the total communication rounds. Therefore, the convergence rate of FIARSE for non-convex objectives should beEQUATION)The aforementioned theorem assesses the potential convergence of the global model towards a stable solution when employing the proposed algorithm (FIARSE). This implies that the submodels utilized in the analysis may differ from those employed in updates at each iteration. Consequently, our analysis takes into account clients utilizing the complete model, with the difference relative to their local submodels being constrained by Assumption 5.3. This methodology aligns with the analytical framework advocated by [81] . Next, we set the appropriate learning rates to achieve optimal convergence properties regarding the number of communication rounds, as outlined in the corollary:Corollary 5.5. Suppose that Assumption 5.1, 5.2 and 5.3 hold. We define F (x)△ = F x, 1 N ×d , and F (x) ≥ F * for all x ∈ R d . Let the local learning rate η l = 1 K √T , and the global learning rate η s = 1, where T is the total communication rounds. Then, under non-convex objectives, FIARSE converges to a small neighborhood of a stationary point of standard FL as T is large enough, i.e.,EQUATION)where we treat L as constants.Remark 5.6. Regarding the first term on the right-hand side (RHS) of Equation ( 5), it approaches zero as T tends to infinity. However, an intriguing question arises concerning the potential for the second term to reach zero, given that the norm ∥x t ∥ 2 2 cannot be zero. According to Assumption 5.3, a straightforward case where δ t is always zero occurs when all clients use the full model -that is, when the threshold θ i in equation Equation ( 3) is set to zero in the proposed FIARSE. Therefore, under model-homogeneous federated learning (FL), our theorem aligns with state-of-the-art works [34, 74] in terms of the convergence rate, which focuses solely on the number of communication rounds T , i.e., O 1/T .For model-heterogeneous FL, it is noteworthy that δ t may tend toward zero as t → ∞, leading to1 T T t=1 δ 2 t ∥x t ∥ 2 2approaching zero. As explored by [51, 69, 81] , this occurs because the submodels can replicate the performance of the full model on all clients after a substantial number of communication rounds T . Consequently, our proposed algorithm can achieve a convergence rate of O N σ 2 A √ T , as long as lim t→∞ δ t = 0.Ignoring constant terms, the proposed FIARSE converges at a rate of O σ 2 /T under full-client participation (i.e., A = N ). Recently, some works have reported their convergence rates under model-heterogeneous FL with full-client participation, such as pruning-greedy [81] with a rate of O M σ 2 / √ T and FedDSE [62] with O Kσ 2 / √ T . Evidently, our proposed method exhibits better theoretical performance. 
setup: 
 Datasets and Models. We evaluate the proposed methodology using a combination of two computer vision (CV) datasets and one natural language processing (NLP) dataset. Specifically, we employ CIFAR-10 and CIFAR-100 datasets [37] for image classification, and the AGNews dataset [79] for text classification. For the first two datasets, we conduct training utilizing a ResNet-18 architecture [17] , with modifications made by substituting its batch normalization (BN) layers with static BN counterparts [13] . For the AGNews dataset, we fine-tune a pre-trained RoBERTa-base model [48] .Data Heterogeneity. For CIFAR-10 and CIFAR-100, we follow [22, 30] and partition the datasets into 100 clients based on a Dirichlet distribution setting α = 0.3. As for AGNews, we partition the datasets for 200 clients with Dirichlet distribution as well, but it is with the parameter of α = 1.0. Note that the server or clients do not use any public datasets during the training stage. In the testing phase, we refer to the superset of all clients' test datasets as "global test dataset". System Heterogeneity. To be specific, the parameter γ is quantified as the ratio of a submodel's FLOPs to the total FLOPs of a complete model. The experiments are conducted with four different model sizes represented by γ ′ = {1/64, 1/16, 1/4, 1.0}. The allocation of clients to each level is balanced. It's important to note that our proposed method is flexible and can accommodate varying numbers of complexity levels or client distributions.Implementation. In this setting, we set the participation ratio to 10% by default. We perform 800-round training on the CV tasks while running for 300 rounds on the NLP task. To avoid the randomness of the results, we averaged the results from three different random seeds. In the experiments, we report the results of all the baselines based on the best hyperparameter settings. Due to the space limitation, more experimental results and analysis are deferred to Appendix D. Our code is released at https://github.com/HarliWu/FIARSE. In this setting, we evaluate the submodels' performance on each client's test datasets. To be specific, the local models are extracted from the global models. Figure 2 comprehensively illustrates the number of clients across different test accuracies. Among the four figures presented, all exhibit a left-skewed distribution, with the exception of FedRolex. This outcome aligns with our expectations, as FedRolex employs a rolling-based approach and is unable to concentrate on optimizing submodel performance on the local dataset, while the rest three approaches can spare efforts on a specific (HeteroFL and ScaleFL) or important (FIARSE) part, effectively addressing performance on local datasets. Among these three approaches, we notice that FIARSE stands out with the best results, showcasing superior performance as more clients achieve higher accuracy compared to the other two methods. The averaged results are also reported by Table 1 under the column of CIFAR-10 and "Local" to "Model (1.0)". Specifically, "Model (1/64)" to "Model (1.0)" shows the averaged local performances classified by the model sizes, and the "Local" shows the result by averaging across these four columns. Table 1 also presents the test accuracy of CIFAR-100 and AGNews. The proposed FIARSE achieves at least 2% better than other baselines under these datasets.In this setting, we evaluate the performance of submodels with various sizes on the global test dataset to assess the generalizability of our proposed algorithm. Table 1 presents the results of two CV datasets and one NLP dataset under the column "Global". In conjunction with Figure 3 , our proposed method FIARSE constantly outperforms other baselines in all submodels with different sizes by a substantial margin. Figure 3 dives into the details of training and shows the test accuracy trend throughout the communication rounds. Consider the submodels are expected to surpass a 70% test accuracy threshold. As previously discussed, FIARSE ultimately achieves superior test accuracy compared to other baselines. Across model sizes of {1/64, 1/16, 1/4}, our proposed method requires fewer rounds to reach the targeted accuracy compared to other baselines. While the performance disparity between FIARSE and FedRolex is less discernible under the full model (Model (1.0)), both methods significantly outpace static submodel extraction approaches in achieving 70% accuracy. In summary, the proposed method stands out by attaining the desired submodels with the fewest rounds among the approaches implemented in this section.The above evaluations are conducted on the clients who participated in the training process. However, a more general scenario includes clients who skip the training phase but need models to process newly arrived data. In such cases, our algorithm can enable the server to customize models from the trained global model for them. Same as the expression in Line 3 of Algorithm 1, the server extracts a submodel encompassing the parameters |x t | ≥ θ i and sends it to the client. Note that the unparticipated clients could have capacities different from that of any client involved in the training. In Figure 4 , we employ our algorithm as well as the baselines to extract submodels with different sizes and compare their performance on unparticipated clients. Generally, the performances of all methods increase as the size of extracted submodels grows. Our algorithm consistently outperforms the baseline methods. In contrast, alternative approaches, particularly static ones like HeteroFL and ScaleFL, suffer from more significant drops in performance. For the case of extracting a submodel that is much smaller than the minimum size involved in the training stage, our proposed method demonstrates remarkable superiority, outperforming existing works by at least 10%. 
conclusion: 
 This work introduces an algorithm for model-heterogeneity FL, named FIARSE, that utilizes importance-aware operation to extract various sizes of submodels. In detail, we utilize TCB-GD that is able to optimize the clients' local parameters to reflect their importance levels. Subsequently, we provide a theoretical analysis and highlight that the proposed work can converge to a neighborhood of a stationary point at the rate of O 1/Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?Answer: [Yes] Justification: Figure 3 and 6 display the results as mean and standard deviation. Other experimental results are averaged with three random seeds for different hyperparameter settings. All results are reported under the best hyperparameter setting.Guidelines:• The answer NA means that the paper does not include experiments.• The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.• The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).• The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)• The assumptions made should be given (e.g., Normally distributed errors).• It should be clear whether the error bar is the standard deviation or the standard error of the mean.• It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not verified.• For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).• If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text. 
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 Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?Answer: [Yes] Justification: The details of computation resources are provided in Appendix D.3.Guidelines:• The answer NA means that the paper does not include experiments.• The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.• The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.• The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).• At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file. 
crowdsourcing and research with human subjects: 
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institutional review board (irb) approvals or equivalent for research with human subjects: 
 Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?Answer: [NA] Justification: The paper does not involve crowdsourcing or research with human subjects.Guidelines:• The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.• Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.• We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.• For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review. 
 
 Appendix: T , where T is the number of communication rounds. Extensive experiments are conducted on ResNet-18 and Roberta-base that demonstrate the significant superiority of our proposed method over the state-of-the-art works.The proposed approach relies on exploiting model sparsity, which is conditionally supported by some hardware. In light of this limitation, one of the future works is to investigate neuron-wise importance-aware submodel extraction, a method that calculates the importance level of neurons without depending on additional information.This work addresses model heterogeneity in federated learning due to varying computational capacities among clients. The proposed method enhances the efficiency of on-device training and reduces computation and energy demands, which is particularly significant for resource-constrained devices like smartphones in real-world applications. Moreover, the method facilitates the practical deployment of federated learning systems in heterogeneous environments, making them more accessible and scalable.Computation Heterogeneity in FL. Computation heterogeneity in FL has become a critical area of research due to the varying computational capacities of participating clients. In traditional FL setups, each client is expected to perform the same volume of computation, such as training on local datasets for a fixed number of iterations, regardless of their hardware capabilities or resource availability [14, 15, 18, 24, 63, 68] . This assumption leads to inefficiencies, particularly when slower devices cause stragglers that delay model aggregation [23, 44, 64] . Recent work has proposed adaptive aggregation strategies to handle computation heterogeneity by assigning different workloads or varying the number of local updates based on client resources, allowing faster clients to contribute more to the global model while slower clients contribute less frequently [43, 50, 53, 57, 64, 72] . However, these works require the clients to train the full model, so they are infeasible when some of the clients cannot load the full model due to limited computation resources. In contrast, our work extracts a submodel for each client that fits within their computational capacity.Model Customization in FL. In addition to submodel extraction, another effective method for facilitating model heterogeneity is model customization [11, 28, 46, 66, 77, 80] . This approach allows clients to construct local models that align with their local computation resources. Since the clients' local models could be very distinct, the existing works implement model aggregation through the technique of knowledge distillation [19, 46, 61, 78] . In detail, each client first locally trains its model on a shared public dataset, and then sends the model's output (i.e., logits) to the server. The server gathers and aggregates clients' outputs into a unified model output and broadcasts it to the clients [60] . Prior to the next round of local training, the clients thereby fine-tune the local models by incorporating the aggregated logits. Although the process effectively transfers knowledge across heterogeneous models, model customization requires a shared public dataset that should have a similar distribution with the clients' training data, which is unattainable in some cases [3, 59] . In our work, we do not require a public dataset, rendering it applicable to a broader range of applications.Model Sparsification in FL. Model sparsification, also known as model pruning, intrigues increasing research focuses with the introduction of lottery ticket hypothesis [16] . This hypothesis addresses the existence of a sparse submodel within a large model, capable of direct training to yield improved performance. Early works [9, 47, 82] have explored sparse models to mitigate the computation overhead. Notably, the industry has recently achieved remarkable advancements in the development of hardware that facilitates the training and inference of sparse models [20, 26, 38, 39] .In the context of FL, two distinct strategies in search of an optimal sparse submodel: dense-to-sparse [27, 31, 41] and sparse-to-sparse [5, 12, 40, 55, 56] . These strategies are distinguished based on whether the global model starts at dense. In both approaches, clients initialize with a global model and iteratively train towards a sparse model to mitigate over-parameterization. Nonetheless, these methods face challenges when client computation capacities differ, particularly when loading the global model exceeds the maximum computation capacities of some clients. Consequently, there arises a need for algorithms capable of tailoring submodels to accommodate various client computation capacities.An existing work, Flado [45] , realizes that the server tailors the submodels to align with various computation resources among clients. It achieves the performance trade-off between the global model and the clients' submodels by differentiating model parameters based on their importance levels. However, this advantage comes at the cost of requiring clients to explicitly maintain the importance score of each model parameter. This requirement, consequently, introduces a considerable overhead on each client in terms of both storage and computation, which imposes great challenges on resource-constrained devices. Contrasting with Flado, our FIARSE explores the correlation between a model parameter's importance level and its value. It implicitly represents the importance level through the parameter's value, thereby eliminating the need to explicitly maintain separate importance scores for each model parameter.Before deriving Equation (3), we should find a formula for ∂M (i) ∂ x :EQUATIONAs defined in the paper, M (i) = ∪ j 1 |xj |≥θi . Therefore, we can obtain Equation (3) viaEQUATIONEQUATIONEQUATIONwhere ⊙ is an element-wise product.C Proof of Theorem 5.5Prior to giving detailed proofs of the theorems, we cover some technical lemmas in this section, and all of them are valid in general cases.Comprehensive description for the aggregation on the server. Since FIARSE generates the submodel based on the parameter importance reflected by the magnitude of the model parameters, a submodel is nested within other submodels that are with a larger size. Given a set of model sizes{γ i } i∈[N ] , we define a sorted set γ ′ = ∪ i∈[N ] {γ i }, where 0 < γ ′ 0 < • • • < γ ′ n-1 ≤ 1 and n = |γ ′ | ≤ N . Let us partition the global model xt into the disjoint submodels xt,[0:γ ′ 0 ] , xt,[γ ′ 0 :γ ′ 1 ] , . . . , xt,[γ ′ n-2 :γ ′ n-1 ] , xt,[γ ′ n-1 :1]. These models are separately held by a set of clientsN γ ′ 0 ⊃ N γ ′ 1 ⊃ • • • ⊃ N γ ′ n-1 ⊃ ∅.In round t, the server samples a set of clients A to train the model, and these models are therefore held byA γ ′ 0 ⊇ A γ ′ 1 ⊇ • • • ⊇ A γ ′ n-1 ⊇ ∅.Notably, the smallest submodel should be held by all clients, i.e.,EQUATIONTherefore, the global update follows xt+1 = xt -η s Agg i∈A xtx (i) t,K . Upon the description, we have the following lemma:Lemma C.1. For a vector v ∈ R d , we partition it into n sets, where we present it in Table 2 . Suppose we select an A-client set out of a total of N clients [N ]. Let us define thatAgg i∈[N ] (v) = j∈[n] 1 N γ ′ j i∈N γ ′ j v (i) [γ ′ j-1 :γ ′ j ] ; Agg i∈A (v) = j∈[n] 1 A γ ′ j i∈A γ ′ j v (i) [γ ′ j-1 :γ ′ j ].Table 2 : Vector Partition across Different Sizes. These notations are the same as the definition in Equation (12) .Submodel v [0:γ ′ 0 ] v [γ ′ 0 :γ ′ 1 ] • • • v [γ ′ n-2 :γ ′ n-1 ] v [γ ′ n-1 :1] Clients set N γ ′ 0 N γ ′ 1 . . . N γ ′ n-1 ∅ Participation set A γ ′ 0 A γ ′ 1 . . . A γ ′ n-1∅ Therefore, we haveEAgg i∈A (v) = j∈[n]1EQUATIONSpecially, if A > N -N γ ′ j , we define N -N γ ′ j A = 0.Proof. For a given submodel v [γ ′ j-1 :γ ′ j ] , there are N γ ′ j clients holding the submodel. In otherwords, N -N γ ′ j clients do not hold any parameters of v [γ ′ j-1 :1] . If a selected client i ∈ A holds the submodel v [γ ′ j-1 :γ ′ j ], it will be divided by an integer k ∈ {1, . . . , A} (i.e., A γ ′ j = k) with a probability of (N γ ′ j -1 k-1 )•( N -N γ ′ j A-k ) ( N A ). Therefore, this part is expected to obtain the following result after aggregation:EQUATIONEQUATIONEQUATIONBy concatenating all j ∈ [n], we can attain the desired results.In addition to the above lemma, we provide a bound for( N A )-( N -c A ) ( N A )and( N A )-( N -c A ) c•( N A ), where c is an integer of {1, . . . , N -A}.Lemma C.2. Given N and A are two integers where A ≤ N , and c is an integer of {1, . . . , N -A}, the following two inequalities are always true:EQUATIONProof. For the first inequality, the lower bound holds when c = 1, while the upper bound constantly satisfies.For the second inequality, we definef (c) = ( N A )-( N -c A ) c•( N A ), and we aim to show that f (c) is monotonically decreasing.Toward the goal, we reduce and prove the following inequality holds:EQUATIONNext, we show the establishment of Equation ( 23) by mathematical induction:• When c = 1, we have LHS = 1 + A N -A , while RHS = 1 + A N -A .Apparently, the inequality holds.• Let us assume that, when c = k, the conclusion holds, i.e., 1EQUATIONEQUATIONEQUATIONThis shows that the inequality (i.e., Equation ( 23)) still holds, indicating that f (c) is monotonically decreasing. As c is an integer of {1, . . . , N -A}, we haveEQUATIONApparently, we can attain the desired results as mentioned in Equation (18) .Lemma C.3. Suppose that Assumption 5.1, 5.2 and 5.3 hold. Let the local learning rate satisfyη l ≤ 1 2L √ K(K+1). With FIARSE, we have the following conclusion:EQUATIONProof. First, we establish a recursive relationship between E xEQUATION(30)EQUATION(32)EQUATION(33)EQUATION(34)EQUATION)where the second term in Equation ( 30) is expanded to four parts: Equation (32), Equation (33) , Equation (34) , and Equation (35) . Furthermore, applying Assumption 5.1 to Equation ( 32) and Assumption 5.3 to Equation ( 33), along with the equalities∇ xt F i (x t ) ⊙ M (i) t (x t ) = ∇F [0:γi] i (x t ) and ∇F (x t ) ⊙ M (i) t (x t ) = ∇F [0:γi] (x t ), we simplify the above inequality forEQUATION)EQUATIONLet C beEQUATION)AsEQUATIONEquation ( 42) holds because 1 + 2 K k ≤ 9 for all k ∈ N + . Therefore, for all k ∈ [K], we haveEQUATION(43)EQUATION)With this conclusion, we can bound the left-hand side of Equation ( 28), which isEQUATION)EQUATIONFinally, by applying Assumption 5.2 to Equation ( 46), we achieve the desired conclusion.Lemma C.4. Suppose that Assumption 5.1, 5.2 and 5.3 hold. Let the local learning rate satisfyEQUATIONWith FIARSE, we have the following conclusion:n j=0 N A - N -N γ ′ j A N A • E v [γ ′ j-1 :γ ′ j ] t 2 2 ≤ 4Aδ 2 t ∥x t ∥ 2 2 + 72KAη 2 l L 2 σ 2 + ∥∇F (x t )∥ 2 2(48)Proof. In the beginning, let us defineEQUATIONEQUATION)According to Equation ( 47EQUATIONTo find a bound for Equation (48) , we try to bound E ∥v∥ 2 2 using generalized mean inequality and Cauchy-Shwarz inequality:EQUATION≤ 2 1EQUATIONWith the inequality above, the LHS of Equation ( 48) is bounded byEQUATIONAccording to Lemma C.2, we haveEQUATIONTherefore, we further bound Equation (55) forEQUATIONwhere the last equation holds because the second norm is non-zero when the client i ∈ [N ] owns the part [γ ′ j-1 : γ ′ j ], j ∈ {0, . . . , n}, and different parts (i.e., [γ ′ j-1 : γ ′ j ] for different js) are independent with each other when calculating the second norm. With Assumption 5.3 and Assumption 5.1, we separately bound two terms in Equation (59) forEQUATIONTherefore, With the inequality above, the LHS of Equation ( 48) is bounded byEQUATION)EQUATIONSince the learning rate meets the constraint of Lemma C.3, Equation ( 63) is obtained when Lemma C.3 applies. Furthermore, with the defined learning rate, we can attain the desired conclusion of Equation ( 48).Lemma C.5. Suppose that Assumption 5.1, 5.2 and 5.3 hold. Let the local learning rate satisfyη l ≤ min 1 2L √ K(K+1) , 1 6L √ (K+1)A. With FIARSE, we have the following conclusion:EQUATIONProof. The recursive function of the global updates of FIARSE follows thatEQUATIONLet us define gEQUATIONTherefore,EQUATIONThe bound for E ∥x t+1 -xt ∥ 2 2 is formulated and simplified as follows:EQUATIONEquation ( 69) is obtained based on Cauchy-Shwarz inequality, and Equation ( 70) holds according to Lemma C.1.EQUATIONEQUATIONSinceEQUATIONwhereg[γ ′ j-1 :γ ′ j ](i) t,kis defined in Equation ( 50), we bound g[γ ′ j-1 :γ ′ j ](i) t,k 2 2by splitting it into four terms:EQUATIONTherefore, to bound Equation ( 70), we should analyze the following inequality, i.e.,EQUATIONEQUATIONEQUATION)There are four terms in the above inequality, i.e., Equation (77), Equation (78) , Equation ( 79), and Equation ( 80). Subsequently, we analyze these four terms one by one.• For Equation (77), we apply Lemma C.2 and obtain thatEQUATIONIn Lemma C.4, we mention thatEQUATION; g[0:γi](i)EQUATIONTherefore, Equation ( 77) is further bounded byEQUATION• Similarly, Equation ( 78) is bounded forEQUATIONIn view that the non-zero part of ∇F[0:γi] i (x t ) is equivalent to M (i)t (x t ), we apply Assumption 5.3 and attain thatEQUATIONTherefore, the bound of Equation ( 78) isEQUATION• For Equation (79), we apply( N A )-( N -N γ ′ j A ) ( N A )≤ 1 and Assumption 5.2 and obtainEQUATION• With 80) is bounded and simplified for( N A )-( N -N γ ′ j A ) ( N A ) ≤ 1, Equation (n j=0 N A - N -N γ ′ j A N A • E ∇F [γ ′ j-1 :γ ′ j ] (x t ) 2 2 ≤ n j=0 E ∇F [γ ′ j-1 :γ ′ j ] (x t ) 2 2 = ∥∇F (x t )∥ 2 2 (88)In conclusion, Equation ( 70) is bounded byEQUATION(90)By applying Lemma C.3, we further simplify the inequality forEQUATION(92)By applying the above learning rate, we can attain the desired conclusion.C.3 Main Proof of Theorem 5.4As we set F (x) with the mask M = 1 N ×d , Assumption 5.1 is reduced to the statement that for allv, v ∈ R d , ∥∇F i (v) -∇F i (v)∥ 2 ≤ L∥v -v∥ 2 , ∀i ∈ [M ].Therefore, the global objective function F (•) is a L-smooth function. As a result, we haveEQUATIONThe iteration function in FIARSE follows that:• Local updates:EQUATION• Global update:EQUATIONSimilar to Lemma C.5, we define gEQUATIONTherefore,EQUATIONIn the proposed algorithm, the parameters are updated only when the submodels hold the counterpart. In this means, the definition forEQUATIONTherefore, by applying Lemma C.1, we haveE ⟨∇F (x t ) , xt+1 -xt ⟩ = n j=0 E ∇F [γ ′ j-1 :γ ′ j ] (x t ) , x[γ ′ j-1 :γ ′ j ] t+1 - x[γ ′ j-1 :γ ′ j ] t (99) = n j=0 E ∇F [γ ′ j-1 :γ ′ j ] (x t ) , -η s η l • 1 N γ ′ j • N A - N -N γ ′ j A N A i∈N γ ′ j K-1 k=0 g [γ ′ j-1 :γ ′ j ](i) t,k (100) = -η s η l K n j=0 N A - N -N γ ′ j A N A E ∇F [γ ′ j-1 :γ ′ j ] (x t ) , 1 N γ ′ j K i∈N γ ′ j K-1 k=0 g [γ ′ j-1 :γ ′ j ](i) t,k (101) ≤ - η s η l K 2 n j=0 N A - N -N γ ′ j A N A ∇F [γ ′ j-1 :γ ′ j ] (x t ) 2 2 (102) + η s η l K 2 n j=0 N A - N -N γ ′ j A N A • E ∇F [γ ′ j-1 :γ ′ j ] (x t ) - 1 N γ ′ j K i∈N γ ′ j K-1 k=0 g [γ ′ j-1 :γ ′ j ](i) t,k 2 2 (103)where the last equation is built upon ⟨a, b⟩ = -1 2 ∥a∥EQUATIONand we simplify Equation ( 103) with the conclusion from Lemma C.4:EQUATION(106) By applying the above local learning rate, we can further simplify the equation forEQUATIONPlugging Equation (107) and Lemma C.5 back to Equation (93), we haveEQUATIONWith the above learning rate, we reorder the formula of Equation (108) forEQUATIONBy summing Equation (109) for all t ∈ {0, . . . , T -1}, we have:F * -F (x 0 ) ≤ EF (x T +1 ) -F (x 0 ) = T t=0 (EF (x t+1 ) -F (x t )) (110) ≤ - η s η l KA 8N T -1 t=0 ∥∇F (x t )∥ 2 2 + 4η s η l KA T -1 t=0 δ 2 t ∥x t ∥ 2 2 + 4η s η 2 l K η s KL + 9Kη l L 2 σ 2 T (111)By reorganizing the inequality above, we have: where L is the number of layers of the global model. Therefore, the threshold for each layer is θ i,l = TopK γi (|x l |), and1 T T -1 t=0 ∥∇F (x t )∥ 2 2 ≤ 8 (F (x 0 ) -F * ) N η s η l KAT + 64N A η s η l KLσ 2 + 32N T t∈[T ] δ 2 t ∥x t ∥θ i = {θ i,l } l∈[L] .• Sharding-wise threshold is a way in the middle that partitions the model into several shardings, and each sharding encompasses a couple of consecutive layers. This is designed for the case when the parameters have distinct distributions across the model. In this case, we partition the layers [L] into multiple group, i.e., L = {[l j : l j+1 ]}. Therefore, the threshold for each sharding is θ i,[lj :lj+1] = TopK γi (|x [lj :lj+1] |).Table 3 lists the hyperparameters that we use in the experiments. For CV datasets, we adopt vanilla SGD or momentum SGD (with the setting of 0.9) as an optimizer. For the NLP dataset, we fine-tune the pretrained model with AdamW and set the parameters (β 1 , β 2 ) for (0.9, 0.95). In our experiments, we keep the learning rate constant. There is no weight decay during our training.We train the neural network and run the program on a server with 8 NVIDIA A6000 GPUs, an Intel Xeon Gold 6254 CPU, and 256GB RAM. Our codes are running with Python 3.7 and Pytorch 1.8.1.Participation rates of 20%. Justification: We highlight our contributions in the Introduction (Section 1) and clearly present the idea of the proposed FIARSE.Guidelines:• The answer NA means that the abstract and introduction do not include the claims made in the paper.• The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.• The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.• It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. 


Decision: The paper is well-written and supported by thorough theoretical analysis and strong experimental results. All reviewers have expressed positive feedback and recommend its acceptance.

Reviews: review: paper_summary: paper_summary: This paper tackles the problem of model heterogeneity in federated learning, where clients have different computational capabilities. The authors propose FIARSE, a method that extracts submodels of varying sizes from a global model based on the importance of parameters. The key idea is using a threshold-controlled biased gradient descent to jointly optimize model parameters and determine their importance. They provide theoretical convergence analysis and empirical results showing FIARSE outperforms existing methods.
strength_weakness: strength_weakness: - Because importance is determined by the size of model parameters, smaller parameters may never be selected, which is equivalent to them being 'deactivated.' This seems to have some impact on model performance.
- Similarly, the paper doesn't explore how FIARSE might impact model fairness across different client groups. Could the importance-based extraction inadvertently amplify biases present in the data?
- How sensitive is FIARSE to the choice of threshold? Some analysis on the impact of different thresholding strategies could be valuable.
- Figure 3 seems to only show one set of experimental results. Due to the curves being quite close, they may be influenced by random factors. Can we conduct multiple experiments and display the results as mean + std?
questions: questions: see Weaknesses
limitations: limitations: yes
paper_summary: paper_summary: The authors proposed FIARSE, a model-heterogeneous federated learning framework that addresses the limitations of existing static and dynamic sub-model extraction methods. They introduced an importance aware sub-model extraction technique to extract heterogeneous sub-model from a shared global model. This approach has demonstrated significantly better performance compared to other model-heterogeneous federated learning methods, such as FedRolex and HeteroFL.
strength_weakness: strength_weakness: Weakness:

1. The performance comparison under different data heterogeneity settings is missing. 
2. The authors should discuss the communication cost and computation overhead to reach the target accuracy. 
3. The performance evaluation on different ratios of client model heterogeneity distribution is missing.
questions: questions: 1. Can the proposed technique generalize to various data heterogeneity scenarios? 
2. Is equal weighting used when aggregating local updates? 
3. What will be the impact of client model heterogeneity distribution on global model accuracy??
limitations: limitations: Refer to the weakness.
paper_summary: paper_summary: The authors address the low device capacities in federated learning. The authors notice that existing submodel extraction methods lack awareness of parameter importance when reducing the model, which may limit the model's performance. They propose an importance-aware dynamic submodel extraction method without the need for additional information beyond the model parameters. Both theoretical and empirical results confirm the effectiveness of the proposed method.
strength_weakness: strength_weakness: 1) The authors directly masks unimportant parameters during model training, which may not be as efficient as structured-pruning methods (e.g., fedrolex, heterofl). It would be better to take a full comparison on capacity issues like model size, peak memory and FLOPs of different methods.

2) Since only a fraction of model parameters will be assigned with large importance, I'm wondering whether the proposed method will degrade to static submodel extraction where fixed parameters may be always considered important and thus lead to limited model representability. It would be better to contain more analysis of the parameters' importance distributions.

3) Fjord  [6] is an important baseline whose results should be empirically compared.
questions: questions: Please see weakness
limitations: limitations: Please see weakness
paper_summary: paper_summary: This works tackles the known problem of model heterogenity in FL, in which each client is allowed to use a different model according to its computational capabilities, while contributing to learn a global model. The proposed approach (FIARSE) samples submodels out of the global model by estimating the importance of its parameters, and pruning out least important components. The work also provides a theoretical foundation for submodel extraction, even if it is not clear how this analisis relates to the ones proposed in other works. The experiments are carried on CIFAR10/100 (CV task) and AGNews (NLP task).
strength_weakness: strength_weakness: 1. **FIARSE induces unstructured sparsity in the model.** This is relevant for claims of the paper, because it influences the efficiency of the proposed approach: not all the hardware supports sparse operations, and in general the performance are lower compared to dense models that are actually smaller by the same factor of the induced sparsity. This particularly outstanding given that previous works produce structural sparsity in the given model [1]. As such, this aspect needs to be more clearly stated and analyzed in support of the work's claims: for examples, it would be appropriate to discuss in more detail the computational performance of the proposed method with respect to the others, varying the submodel dimension.
2. **Insufficient comparison with SOTAs.** The proposed approach misses comparison with recent approaches [1,2] which have the advantage of producing structured sparsity in the subsampled model. The works are correctly cited in the introduction, which raises some concers about the choice of excluding them from the evaluation. i believe that those FjORD and NeFL needs to be considered in the evaluation, and better described in the related work section (see next point).
3. **Writing can be improved.** The works presents a discussion on approaches for model-heterogeneous FL in sections 1-2, that is essentially in the introduction and in one sections where the research problem is stated. I found that this structure makes the discussion of the related works not cohesive, resulting in not clear relationship with previous works. Ideally, one would like to understard which are the similar works and how the proposed approach differs from them. In its current form, it's difficult to have such clear idea.
Also on the theory, the work does not detail what is the improvement and/or generalization with respect to previous works [3].
4. **Quality of images.** This is not really I take into account for assessing the paper, but please note that the figures are low-quality so they get grainy on the printed paper.

Points (1-2) are the most important for my evaluation, I believe this could be a good work but the empirical evaluation need to be more convincing in showing an advantage over existing approaches, not only in term of model quality (e.g. accuracy) but also in terms of resources.

[1] Horvath et al., FjORD: Fair and Accurate Federated Learning under heterogeneous targets with Ordered Dropout, NeurIPS 2021

[2] Kang et al., NeFL: Nested Federated Learning for Heterogeneous Clients

[3] Shulgin et al., Towards a Better Theoretical Understanding of Independent Subnetwork Training, ICML 2024
questions: questions: 1. In the introduction, model-heterogeneous FL approaches are divided into static and dynamic submodel extraction, by looking at the costruction of submodels during the training. Why FjORD is regarded as static? In FjORD at each optimization step of portion of contiguous weights are sampled for training according to the client's computational capabilities, so this would make it dynamic based on the above distinction.
2. What is the improvement and/or generalization on the theory part with respect to previous works?
limitations: limitations: Yes

Comments: title: We look forward to your response as the discussion period ends in one day
We look forward to your acknowledgement
We look forward to your acknowledgement
Thanks for your response
Rebuttal by Authors (cont.)
Rebuttal by Authors
comment: Dear authors, thanks again for additional clarifications.
Yes, debating the category of FjORD (as any other competitor) is indeed minor, it was a doubt and not a concern. The explanations you provided will enhance the clarity of the paper in classifying the approaches, given that there is particular accent on FIARSE being dynamic.

Thanks again for the discussion.
Dear Reviewer inPJ, 

Thank you for your feedback. We are pleased that our rebuttal has addressed your concerns and sincerely appreciate your positive rating.

Best,

Authors
Thanks for the detailed rebuttal.

Some of my concerns have been addressed.

I would raise my score.
Dear Reviewer inPJ,

Thank you once again for your commitment to reviewing our paper and assisting us in improving our work. We would like to remind you that the discussion window will close in less than 24 hours, and we eagerly await your feedback.

We have provided detailed explanations for each of your concerns. We would greatly appreciate it if you could review our responses and let us know if they fully or partially address your concerns. Any additional comments you may have would also be highly appreciated.

Best,

Authors
Dear Reviewer dCzP,

Thank you for your valuable feedback and for highlighting the hardware limitations of unstructured pruning on resource-constrained devices. While this approach appears cost-effective from a conceptual standpoint, we recognize the practical challenges it presents. We will discuss this limitation in a new section titled *Limitations*.

We are pleased that our response has largely addressed your concerns, and we appreciate your positive rating.

Best,

Authors
Thank you for the detailed responses. I think it is still a concern that unstructured pruning might weaken the adaptability of this method in practice since clients' heterogeneity of memory capacity can be one of the most important system constraints in cross-device FL. This response has addressed most of my questions, thus I will raise my rating to 5.
Dear Reviewer PQwa,

Thank you for your valuable feedback. We are pleased that our response and the additional experiments have addressed your concerns, and we appreciate your positive rating.

Best, 

Authors
Dear Reviewer dCzP, 

We appreciate your high-quality and constructive comments on our work. As we approach the end of the author-reviewer discussion period, we kindly request that you review our response. Feel free to let us know if our response addresses your concerns and if you consider updating the rating. We are more than happy to answer any further questions you may have.

Best, 

Authors
Dear Reviewer inPJ, 

We appreciate your high-quality and constructive comments on our work. As we approach the end of the author-reviewer discussion period, we kindly request that you review our response. Feel free to let us know if our response addresses your concerns and if you consider updating the rating. We are more than happy to answer any further questions you may have.

Best, 

Authors
I thank the authors for providing detailed responses and additional experiments. My questions have been addressed. After reviewing all the feedback and responses to the comments, I'll keep my original rating. Thank you.
Thanks for your feedback. We are happy that our rebuttal has addressed your concerns and appreciate your positive rating.

Regarding your comments on Question 1, we believe that FjORD can be categorized as either a static or dynamic submodel extraction approach. As you know, FjORD employs ordered dropout, allowing clients to randomly train a submodel within their computational limits at each local step. In our paper, static submodel extraction means that the server consistently extracts a fixed part of a full model and transmits it to a client at every **communication round**. It is reasonable to consider FjORD as a *static* submodel extraction approach since a submodel received by a client is consistently extracted from a specific part of a full model at every **communication round**. On the other hand, FjORD can also be viewed as a *dynamic* submodel extraction approach because a client trains a random portion of the received submodel at each **local training round** (a.k.a. local step).

From our perspective, debating the category of FjORD is minor; the real challenge lies in understanding why the proposed FIARSE significantly outperforms FjORD, as shown in the table of **Response [W2]**. FjORD adopts ordered dropout to generate submodels of various sizes, which prevents it from constructing optimal submodels for different sizes. In contrast, FIARSE captures the importance of parameters, exploring and building various submodels until they achieve optimal performance for each size.
I thank the authors for the detailed response. I have read and considered all the reviews and relative rebuttals. I find the authors nicely answered to my concerns, and it is now clearer what is the comparison with the state-of-art in terms of performance on a broader set of metrics, and not just accuracy. As such, I raised my score accordingly, please include the std in the revised manuscript when reporting the results.

More as a perplexity than a concern for evaluation, I still don't get the answer to my Q1 and (static vs dynamic submodel extraction) if this classification helps in navigating the literature. Reading again the text, it seems that "static" means that clients' extract the same submodel at each round (see lines 31-32 and Figure 1), while "dynamic" can explore different parameters at different round. To this respect, FjORD samples the parameters to use at each forward pass, so it seems it should be dynamic. It would be nice to further clarify it in the revision.
**Response to [W4].** Thanks for your suggestion. We have updated the experimental figure in the supplementary materials of the rebuttal. I have taken into account the suggestions from Reviewer inPJ and made the necessary updates accordingly, i.e., we will replace Figure 3 in the manuscript with Figure 2 in the supplementary material.
**Response to [W1].** Thanks for your question. While we acknowledge the limitations of unstructured pruning in the *Conclusion*, we also highlight significant industry progress in Appendix A, supported by notable evidence [1,2]. For instance, Nvidia GPUs support unstructured pruning through the NVIDIA Ampere Architecture and NVIDIA TensorRT.

The rest of the response demonstrates the advantages of FIARSE in terms of computation and communication overhead. We train ResNet-18 using CIFAR-10, which is partitioned into 100 clients using Dirichlet distribution with a parameter 0.3. In each communication round, we sample 10 clients, and these clients locally train the model for 4 epochs with a batch size of 20.

According to the numerical results presented in Table 1 and Figure 3 in the manuscript, all baselines can achieve a global accuracy of 65% or higher, and various submodels can realize a global accuracy of 60%. Therefore, the table below presents the number of communication rounds, overall computation, and communication costs when **the average global accuracy reaches 65% and all submodels should achieve an accuracy higher than 60%**:

|Method|Rounds|Total computation costs (TFLOPs)|Total communication costs (GB)|
|-|-|-|-|
|HeteroFL|560|4089.59|309.86|
|FedRolex|NA(640)|4673.82|354.13|
|ScaleFL|460|5144.41|274.57|
|FIARSE|**380**|**3620.36**|**210.08**|

Notes to the table: (i) we evaluate the result every 20 rounds, so the value of rounds can be divided by 20. (ii) Model 1/64 of FedRolex never achieves an accuracy above 60%. So we present its result of the 640th round, where FedRolex achieves an average global accuracy of 65% for the first time.

The table demonstrates the superiority of the proposed FIARSE: it requires the fewest communication rounds, and the least communication and computation overhead among the selected methods.

**References:**
[1] How Well Do Sparse ImageNet Models Transfer?, CVPR'22
[2] Sparse Fine-tuning for Inference Acceleration of Large Language Models, 2023

***

**Response to [W2].** Thanks for your suggestion. We conduct empirical studies to train a ResNet with CIFAR-10 using 100 clients, while only 10 clients are selected to train the model in each communication round. The tables below make a comparison between NeFL, FjORD, and the proposed FIARSE. 

|Method|Model (1/64)|Model (1/16)|Model (1/4)|Model (1.0)|
|--|--|--|--|--|
|NeFL|65.88|72.08|76.52|79.44|
|FjORD|71.36|71.48|70.04|77.40|
|FIARSE|73.12|77.20|77.24|82.04|

Among these three methods, the proposed FIARSE significantly outperforms another two baselines.

***

**Response to [W3/Q2].** Thanks for your suggestion and for bringing interesting work. Due to the space limit, we moved our related work to Appendix A and discussed the similarities and differences compared to the previous works. For example, this work achieves model sparsification in federated learning, but different from the previous studies [1,2], we consider each client to possess heterogeneous computation capacities. We understand that paper organization is not a common practice, and we will insert a brief summarization of related work between *Introduction* and *Preliminary*.

IST [3] focuses on model parallelism in federated learning, where each node independently trains a submodel, and the server assembles all submodels into a global network. In contrast, our proposed FIARSE explores submodel extractions for different model sizes, which can be efficiently trained to achieve remarkable performance. These two works address different problems. From a theoretical perspective, IST analyzes the convergence while training with various submodels. Similar to the conclusion drawn from Theorem 2 of IST, the proposed FIARSE converges to a neighborhood of a stationary point. IST confirms that the bias cannot be eliminated, indicating that the model keeps fluctuating within a constant area. In contrast, our theoretical analysis demonstrates the possibility that FIARSE can minimize this neighborhood area to zero.

**References:**
[1] Sparse Random Networks for Communication-Efficient Federated Learning, ICLR'23
[2] DisPFL: Towards Communication-Efficient Personalized Federated Learning via Decentralized Sparse Training, ICML'22
[3] Towards a Better Theoretical Understanding of Independent Subnetwork Training, ICML'24

***

**Response to [Q1].** In our opinion, a static model never explores other parameters, while a dynamic model investigates new parts of the model. According to FjORD, the model architecture is fixed at the beginning of training, meaning the submodel never actively explores new parts of the model. Therefore, we consider FjORD to be a static strategy. We will clarify this classification in the Introduction.