Communication-Efficient Federated Group Distributionally Robust Optimization

Authors: zhishuai  guo
tianbao  yang
Venue: NeurIPS_cc_2024_Conference
Published: 2024-05-14 20:57:31

Abstract: Federated learning faces challenges due to the heterogeneity in data volumes and distributions at different clients, which can compromise model generalization ability to various distributions. 
Existing approaches to address this issue based on group distributionally robust optimization (GDRO) often lead to high communication and sample complexity.
To this end, this work introduces algorithms tailored for communication-efficient Federated Group Distributionally Robust Optimization (FGDRO). Our contributions are threefold: Firstly, we introduce the FGDRO-CVaR algorithm, which optimizes the average top-K losses while reducing communication complexity to $O(1/\epsilon^4)$, where $\epsilon$ denotes the desired precision level. Secondly, our FGDRO-KL algorithm is crafted to optimize KL regularized FGDRO, cutting communication complexity to $O(1/\epsilon^3)$. Lastly, we propose FGDRO-KL-Adam to utilize Adam-type local updates in FGDRO-KL, which not only maintains a communication cost of $O(1/\epsilon^3)$ but also shows potential to surpass SGD-type local steps in practical applications.
The effectiveness of our algorithms has been demonstrated on a variety of real-world tasks, including natural language processing and computer vision.

Main paper: 
 introduction: 
 Federated learning enables effective model training without the need to share raw data [38, 48] . It is essential in contexts where data privacy and ownership are paramount, such as in inter-hospital collaborations [54] and mobile device networks [20] . However, clients often have data of varying volumes and distinct distributions, which poses notable challenges in maintaining generalization behavior [50, 27] . Generalization here refers to the model's ability to perform consistently across different clients, including those that have not participated in the training [23, 79] .In this study, we tackle the issue using federated group distributionally robust optimization (FGDRO), formulated as follows:EQUATIONHere, w denotes a machine learning model, and N represents the number of clients. For each client i, D i represents its local data distribution, and ℓ i (w) = E z∼Di ℓ(w; z) represents the loss calculated from that local distribution. ∆ N denotes a N -dimensional simplex, which constrains i p i = 1. The vector p = [p 1 , ..., p N ] comprises the weights assigned to each of the N clients. The function ϕ(p) acts as a regularization term, with λ > 0 being an adjustable parameter. This framework aims to assign higher weights to machines with greater losses while discouraging substantial deviations of these weights from a specified distribution.Our study concentrates on two particular forms of the regularization term ϕ, which are well-established regularization techniques [11, 39] , each suited to different tasks and data distributions. Specifically, CVaR is defined as ϕ(p) = I [0,1/K] (p). In this scenario, ϕ(p) is set to 0 if each weight p i falls within the range of [0, 1/K], and is infinite otherwise. FGDRO-CVaR focuses on optimizing for worst-case scenarios or the average of the worst-case losses, making it particularly effective in high-stakes applications like healthcare and finance, where avoiding extreme losses is crucial. However, it can be sensitive to outliers or malicious client attacks. FGDRO-KL, on the other hand, uses Kullback-Leibler (KL) divergence,expressed as ϕ(p) = N i=1 p i log(N p i ). This version of ϕ penalizes deviations of the weight distribution p from a uniform distribution. Fundamentally, when ϕ(p) is strongly convex, as in the case of KL divergence, F (w) can enjoy a smoothness property, while non-strongly convex ϕ(p) would result in non-smooth F (w) [6] . In contrast to CVaR, KL is a softer regularizer to promote smoother and more stable learning. Thus, it can be beneficial in scenarios where robustness to outliers or malicious clients is needed. FGDRO-KL-Adam further enhances FGDRO-KL by incorporating Adam-type updates.Table 1 : Comparison of communication cost and sample complexity on each machine to achieve ϵ-stationary point or near to ϵ-stationary point, where ϵ-stationary point has a (sub-)gradient ∥∂F (w)∥ 2 ≤ ϵ 2 . NDP-SONT denotes the naive deployment of the SONT algorithm [22] O 1 ϵ 16 O 1 ϵ 12 O 1 ϵ 16 [11] DR-DSGD - - O( 1 ϵ 3 ) O( 1 ϵ 6 ) [30] NDP-SONT O 1 ϵ 6 O 1 ϵ 6 O 1 ϵ 6 O 1 ϵ 6 [22] This Work O 1 ϵ 4 O 1 ϵ 8 O 1 ϵ 3 O 1 ϵ 4Previous research addressing these optimization problems in federated learning has struggled with high communication and sample complexity issues, which are basically due to inefficient updates of p on local machines. For example, Deng et al. [11] examined a particular case of the general formula (1) using a CVaR constraint with K = 1, and developed an algorithm for KL regularization as well. They update p only in global communication rounds, while local steps only optimize the local loss function using stochastic gradient descent (SGD). To achieve a ϵ-stationary point or a point near to an ϵ-stationary point, where an ϵ-stationary point has a (sub)gradient ∥∂F (w)∥ 2 ≤ ϵ 2 , their methods required a communication cost of O(1/ϵ 12 ) and a sample complexity of O(1/ϵ 16 ) on each client. For FGDRO with KL regularization, [30] achieves a communication cost of O(1/ϵ 3 ), but it requires the use of large data batches in local update steps in order to get good approximation for the surrogate of p, resulting in a total sample complexity of O(1/ϵ 6 ) per machine.To overcome these limitations, this paper presents specialized algorithms FGDRO-CVaR and FGDRO-KL for FGDRO with CVaR constraint and KL regularization, respectively. Instead of dealing with the constrained primal-dual formulation in (1), we consider their equivalent forms with a compositional structure that get rid of the high-dimensional constrained variable p. We summarize the complexity results in Table 1 .For FGDRO with CVaR constraint, we are the first to consider a constraint-free equivalent form and develop a communication-efficient algorithm for it, significantly reducing communication costs, as shown in Table 1 . In addition to sharing machine learning models, we only introduce an additional scalar threshold to select participants in each round, minimizing additional costs. The equivalent compositional form is a non-smooth two-level compositional function with one auxiliary variable s, which works as a threshold. Only machines whose local losses are greater than s are supposed to contribute to updating the model. In this way, we can simply update the constraint-free scalar variable s locally in each client and average s in communication rounds. However, we do face challenges with non-smooth compositional optimization problems. Our first algorithm FGDRO-CVaR effectively addresses this issues and achieves a communication cost of O(1/ϵ 4 ) and a sample complexity of O(1/ϵ 8 ) on each machine.For FGDRO with KL regularization, while previous literature has explored constraint-free compositional reformulations, they often require large batch sizes on each machine to estimate gradients, making this approach impractical and leading to high sample complexity. In contrast, we utilize moving averages that can work with smaller data batches while still providing accurate gradient estimates, enhancing the efficiency of our method. The equivalent compositional form we consider is a smooth three-level compositional function. In this case, the weights for the clients depend on both local loss functions and global loss functions. We use moving average estimators of these statistics and update the estimators locally. In communication rounds, in addition to averaging the model w, the machines will average the estimator of the global loss function. We have reduced the communication cost and the computation cost compared to the literature as presented in Table 1 .To further enhance our approach, we have developed an adaptive algorithm for solving FGDRO with KL regularization, named FGDRO-KL-Adam. Stochastic adaptive methods apply variable step sizes for each coordinate based on historical gradient information, often yielding better results than non-adaptive techniques, as evidenced by a wealth of research [13, 36, 49, 69] . In federated learning, while Reddi et al. [59] have developed a federated adaptive algorithm and shown its effectiveness in various tasks. However, it limits adaptive steps to global updates on the server, with local updates relying on standard SGD, which may lead to suboptimal results. Moreover, their method is primarily designed for Empirical Risk Minimization (ERM) and is not applicable to address compositional optimization problems. Our FGDRO-KL-Adam allows local updates to use Adam-type updates, which introduces the challenge of handling unbiased gradients, further complicated by the use and updating of the second-order moment. To this end, we update the first-order momentum and secondorder momentum locally and then average them globally during communication rounds. Moreover, our analysis carefully manages the moving estimates of the first and second-order moments, ensuring that the solution provably converges.Our FGDRO-KL-Adam enables local updates with Adam-type methods, which raises the challenge of maintaining unbiased gradients, especially with the adjustment of the second-order moment. Our analysis meticulously handles the moving estimates of both first and second-order moments to guarantee provable convergence. The first-order momentum and second-order momentum are updated locally and then averaged during communication rounds.In summary, our paper contributes in three main areas. First, our FGDRO-CVaR algorithm greatly reduces both communication costs and sample complexity for FGDRO with CvaR constraint problems. Second, our FGDRO-KL algorithm achieves a better sample complexity while maintaining the same communication costs as the existing results. Third, our FGDRO-KL-Adam integrates adaptive step sizes with Adam-type updates, which has the potential to surpass the performance of conventional SGD-based approaches. Extensive testing on diverse real-world datasets has shown that our approach achieves superior performance while substantially reducing communication overhead. 
related work: 
 Federated learning has gained significant attention due to its potential to train machine learning models using data from various sources while ensuring data privacy [38, 48, 54] . Two central challenges to this field are communication cost and client heterogeneity, which have been extensively explored in the literature [63, 76, 77, 73, 62, 34, 64, 2, 31, 68, 4, 33, 35, 71, 70, 34, 18] . This section will dive into the body of literature that focuses on these specific challenges.Non-IID Clients in Federated Learning (FL) One of the key challenges in Federated Learning (FL) is managing client heterogeneity, particularly the issue of non-IID (nonindependently and identically distributed) data across client networks. Efforts to overcome the negative implications of data diversity have led to the development of model personalization techniques [47, 10, 44, 82, 45, 43, 75, 19, 41] . However, these approaches face challenges when dealing with data from unseen or unidentifiable groups. For a comprehensive examination of the challenges and strategies concerning non-IID clients in federated learning, the readers are directed to [27] .Federated Group Distributionally Robust Optimization Since Group Distributionally Robust Optimization has shown effectivenss in addressing non-iid data in centralized setting [14, 51, 12, 56] ,previous research has investigated Federated Group Distributionally Robust Optimization (FGDRO) to address the challenges posed by non-IID clients in federated settings [50, 11] . The DRFA algorithm [11] focuses on a specific instance of (2), applying a CVaR constraint on p with K = 1. It samples machines based on updated probabilities to allow local updates, reducing the need for communication, with these probabilities managed by a central server. However, this approach results in significant communication costs of O(1/ϵ 12 ) and sample complexity of O(1/ϵ 16 ) per machine to achieve an ϵ-stationary point. Recent developments in [22] introduced algorithms for handling Group DRO with a CVaR constraint in centralized settings, but adapting these to federated learning entails substantial communication overheads of O(1/ϵ 6 ). Moreover, [78] introduced the SCAFF-PD algorithm, which is only applicable in convex scenarios and requires the use of the complete dataset in each training round.Regarding KL regularization, [30] achieved a communication cost of O(1/ϵ 3 ), but required the use of large data batches, resulting in a total sample complexity of O(1/ϵ 6 ) per machine. [80] has studied FGDRO with KL regularization in a decentralized setting, which would incur a communication cost of O(1/ϵ 4 ) if directly applied to a centralized federated learning setting.Federated Adaptive Algorithm Stochastic adaptive methods for minimization in non-convex stochastic optimization have garnered significant interest in recent years [13, 36, 49, 69, 42, 84, 65, 7, 46, 25, 16, 81] . These methods, known for assigning unique step sizes to each coordinate, often outperform their non-adaptive counterparts. In federated learning, Reddi et al. [59] have advanced the field with an adaptive algorithm. However, their methodology predominantly applies adaptive steps at the global server level, with local updates still dependent on SGD. This could lead to suboptimal performance. Furthermore, their approach was tailored for Empirical Risk Minimization (ERM) problems and could not be applied for problems considered in this work.3 PreliminariesA function f is C-Lipschitz if f (x) -f (y) ≤ C∥x -y∥. A differentiable function f is L-smooth if ∥∇f (x) -∇f (y)∥ ≤ L∥x -y∥,where ∇f (x) denotes the gradient. For a non-differentiable function f , its subdifferential ∂f (x) is defined as a set of all subgradients as ∂f (x) = {v|f (y) ≥ f (x) + ⟨v, y -x⟩ + o(∥y -x∥)} as y → x. When the context is clear, we also overload the notation ∂f (x) to denote one subgradient from the subdifferential set. We use ∇f (x; z) or ∂f (x; z) to represent an unbiased estimator of gradient or subgradient with a randomly drawn sample z. Additionally, a function f is ρ-weakly convex if f (x) ≥ f (y) + ⟨∂f (y), y -x⟩ -ρ 2 ∥y -x∥ 2 . For a smooth function f (x), x is an ϵ-stationary point if ∥∇f (x)∥ 2 ≤ ϵ 2 . For non-smooth functions, x is an ϵ-stationary point if ∥dist(0, ∂f (x))∥ 2 ≤ ϵ 2 , where dist(x, S) = min x ′ ∈S ∥x -x ′ ∥ 2 measures the distance between a point x and a set S. For non-smooth functions, since it is usually difficult or even impossible to find an ϵ-stationary point, we instead seek an ϵ-near stationary point. Definition 3.1. x is an ϵ-near stationary point of f (•) if ∃x ′ such that ∥x -x ′ ∥ 2 ≤ ϵ and dist(0, ∂f (x ′ )) ≤ ϵ. 
fgdro-cvar: 
 In this section, we present our algorithm designed to tackle Federated Group Distributionally Robust Optimization (FGDRO) with a CVaR constraint. This problem poses substantial challenges due to the complexity of both CVaR and simplex constraints. Typically, during local updates, individual machines do not have access to adequate information to appropriately adjust the weight vector p. Prior approaches, such as the one proposed by [11] , mitigate this issue by updating p during global communication rounds, but results in slower convergence rates. To this end, we reformulate the problem into an equivalent two-level compositional optimization problem without constraints:EQUATIONwhere f (•) = (•) + and g i (w) = E z∼Di ℓ(w, z) is a local loss and s is intended to serve as a threshold value. With s = arg min s ′ 1 N N i=1 f (g i (w) -s ′ ) + K N s ′ , only the K clients with the highest losses will have losses greater than s [53, 83] . During the training phase, K clients with the highest losses are expected to predominantly influence the optimization process.The formulation (2) replaces the constrained high-dimensional vector p with a single unconstrained scalar variable s. However, this adjustment introduces new challenges due to the compositional structure and the non-smooth nature of the outer function f . As a result, it is biased to estimate subgradient ∂f (g i (w) -s)∇g i (w) using a batch of samples. To address this, it is common practice to create an accurate estimate of g i (w) [67, 24, 22, 66, 32, 55, 57] . Specifically, we employ a moving average u as our accurate estimate:EQUATIONAnd then the estimators for sub-gradient of w and s, namely m, v are computed using u. It is notable that for s, it is updated locally using local data and averaged between clients in communication rounds. It will converge alongside w to an ϵ-near stationaty point. This is a fundamental reason why our method achieves a lower communication complexity compared to [11] , as the latter can only update the weight variables at the server node in the communication rounds.Algorithm 1 FGDRO-CVaR 1: Initialization: w1 , s1 = 0, u 0 i,t = 0, 2: for r = 1, ..., R do 3:w r i,0 = wr , s r i,0 = sr , u r i,0 = u r-1 0,Ifor t = 1, ..., I doEach machine samples data z r i,t 6:u r i,t = (1 -β 1 )u r i,t-1 + β 1 ℓ(w r i,t-1 ; z r i,t ) 7: v r i,t = -∂f (u r i,t -s r i,t-1 ) + K N , and s r i,t = s r i,t-1 -η 2 v r i,t 8: m r i,t = ∂f (u r i,t -s r i,t-1 )∇ℓ(w r i,t-1 , z r i,t ), and w r i,t = w r i,t-1 -η 1 m r i,t 9:end for We present the formalization of our FGDRO-CVaR method in Algorithm 1. Next, we show the convergence results of FGDRO-CVaR . We make the following assumptions regarding problem (2). Assumption 4.1.(1) ∀i and ∀z ∈ D i , ℓ(•, z) is C g -Lipschitz and L g -smooth.(2)E z∈Di ∥∇ℓ(w; z) - ∇g i (w)∥ 2 ≤ σ 2 , E z∈Di ∥ℓ(w; z) -g i (w)∥ 2 ≤ σ 2 .Remark. The first assumption about Lipschitz continuity and smoothness of g i is standard in compositional optimization [22, 66, 67] . The second assumption of bounded variance is also common. Assumption 4.1 leads to F (w, s) being weakly convex, which is a key step in the analysis as shown in Appendix A.2The behavior of the estimator u is examined through the following lemma. Lemma 4.2. Under Assumption 4.1, by settingη = O 1/(R) 3/2 , β 1 = O (1/R), I = O(R), Algorithm 1 ensures that E∥u r i,t -g i ( wr t )∥ 2 ≤ (1 -β 1 )E∥u r i,t-1 -g i ( wr t-1 )∥ 2 + 2β 2 1 σ 2 + 3β 2 η 2 I 2 C 2 g + 5 β 1 C 2 g ∥ wr t -wr t-1 ∥ 2 .The convergence result of FGDRO-CVaR is given in the following theorem. η = O 1/(R) 3/2 , β 1 = O (1/R), I = O(R)and ρ = 2L g , the Algorithm 1 ensures that for the output ( w, s), there exists (w ′ , s ′ ) thatEQUATION)Remark. The analysis has utilized Moreau envelop to address the nonsmooth issue [52, 9] . To achieve an ϵ-near stationary point of F (•), we need to set R = O(1/ϵ 4 ) and I = O(1/ϵ 4 ), and thus the sample complexity on each machine isRI = O(1/ϵ 8 ). The total sample complexity of O(n/ϵ 6 )by [22] is achieved by the STORM estimator [8] which incurs additional memory and computational costs due to the requirement of computing gradients using two models at each iteration. Without the STORM estimator, [22] would exhibit a total sample complexity of O(n/ϵ 8 ). When deployed in a federated setting, the complexity for each machine would be O(1/ϵ 8 ), aligning with our results and demonstrating that our approach achieves a linear speed-up in terms of number of machines.Additionally, although we aggregate certain scalar variables (in FGDRO-CVaR, the scalar variable s; in FGDRO-KL and FGDRO-KL-Adam to be presented later, the scalar variable v), similar to the technique in the Remark 3.1 of [61] , we can aggregate these variables using homomorphic encryption, ensuring that their exact values remain confidential. 
fgdro-kl: 
 In this section, we present our FGDRO-KL algorithm for solving problem (1) with a KL regularization.Unlike the CVaR constraints that focus on the top K clients, KL regularization takes into account all clients, assigning them varying weights. Additionally, FGDRO with KL regularization is smooth, and strongly concave with respect to p. Nevertheless, it is subject to the simplex constraint on p. To address this, we use an equivalent form derived from the KKT conditions, as referenced in [56, 30] :EQUATIONThis formulation eliminates the constrained vector p, and F (w) is smooth since KL regularization is strongly concave [6] . However, this formulation has a three-level composition structure and thus, using a batch of data in a three-level composition can result in biased gradient estimation. Furthermore, the gradients on one machine are depend on other machines. Specifically, we denote g i (w) = exp(E z∈Di ℓ(w; z)/λ) and g(w) = 1 N N i=1 g i (w), with ℓ(w; D i ) =E z∈Di ℓ(w; z), then, the gradient of F (w) in ( 5) is given by:EQUATIONIt is crucial to recognize that the gradient for machine i, i.e., ∇ℓ(w; D i ), is scaled by g i (w)/g(w). This scaling indicates that machines experiencing larger loss functions exert more influence over the training process. To mitigate the biased gradient estimation, we approximate g i (w) and g(w) based on moving average estimators u and v. On each machine, u serves as a moving average estimator for the local loss function ℓ(w; D i ), with exp(u r i,t /λ) providing a local approximation of g i (w). u is updated and maintained locally without need for averaging during communication rounds. v estimates the global statistic g(w), and is updated locally but averaged during global communication rounds. Subsequently, a moving average estimator of the gradient, denoted as m is constructed using u and v. For specific update rules, please refer to Algorithm 2.For analysis, we make the following assumptions regarding problem (1) with a KL regularization:Assumption 5.1. (1) ∀i and ∀z ∈ D i , g i (•) is C g -Lipschitz and L g -smooth. (2) E z∈Di ∥∇ℓ(w; z) - ∇ℓ(w; D i )∥ 2 ≤ σ 2 , E z∈Di ∥ℓ(w; z) -ℓ(w; D i )∥ 2 ≤ σ 2 . (3) f is C f -Lipschitz and L f -smooth. (4) ∀i and ∀z ∈ D i , 0 ≤ ℓ(•; z) ≤ C 0 , ℓ(•) is C ℓ -Lipschitz and L ℓ -smooth.The behavior of the u and v estimators can be bounded similar to the previous section and are shown in Appendix B. The estimator m for gradient can be bounded as η = O 1 √ RI , β 1 = O 1 √ RI , the Algorithm 2 ensures that ∥ mr t -∇F ( wr t )∥ 2 ≤ (1 - β 3 2 )∥ mr t-1 -∇F ( wr t-1 )∥ 2 + β 3 C 2 ℓ C 2 1 1 N N i=1 ∥u r i,t-1 -ℓ( wr t-1 ; D i )∥ 2 + β 3 C∥v r t -g( wr t )∥ 2 + 3η∥∇F ( wr t-1 )∥ 2 + β 2 3 C 2 1 σ 2 N + 2β 3 C 2 1 L 2 ℓ η 2 I 2 G 2 .Algorithm 2 FGDRO-KL 1: Initialization: w1 , u 0 i,I , v1 , m1 2: for r = 1, ..., R do 3:w r i,0 = wr , m r i,0 = mr , u r i,0 = u r-1 i,I , v r i,0 = vr 4:for t = 1, ..., I do 5:Each machine samples data z r i,t 6:u r i,t = (1 -β 1 )u r i,t-1 + β 1 ℓ(w r i,t-1 ; z r i,t ), and v r i,t = (1 -β 2 )v r i,t-1 + β 2 exp(u r i,t /λ) 7: h r i,t = exp(u r i,t ) v r i,t ∇ℓ(w r i,t-1 ; z r i,t ), and m r i,t = (1-β 3 )m r i,t-1 + β 3 h r i,t 8: w r i,t = w r i,t-1 -ηm r i,t 9:end for 10: w r i,0 = wr , m r i,0 = mr , q r i,0 = qr , u r i,0 = u r-1 i,I , and v r i,0 = vr 4:wr+1 = 1 N N i=1 w r i,I , vr+1 = 1 N N i=1 v r i,for t = 1, ..., I doEach machine samples data z r i,t 6: The precisions of the u, v, m estimators depend on each other. The idea is to get E∥u r i,t -ℓ( wr t ; D i )∥ 2 , E∥v r t -g( wr t )∥ 2 , ∥ mr t -∇F ( wr t )∥ 2 and E∥∇F ( w)∥ 2 jointly converge, and then we finally have the following theorem to guarantee the convergence:u r i,t = (1 -β 1 )u r i,t-1 + β 1 ℓ(w r i,t-1 ; z r i,t ), and v r i,t = (1 -β 2 )v r i,t-1 + β 2 exp(u r i,t /λ) 7: h r i,t = exp(u r i,t ) v r i,t ∇ℓ(w r i,t-1 ; z r i,t ) 8: m r i,t = (1-β 3 )m r i,t-1 + β 3 h r i,t , and q r i,t = (1-β 4 )q r i,t-1 + β 4 (h r i,t ) 2 9: w r i,t = w r i,t-1 -η m r i,t √ q r i,EQUATIONRemark. To achieve an ϵ-stationary point, i.e., ∥∇F ( w)∥ 2 ≤ ϵ 2 , we need to setR = O(1/ϵ 3 ), I = O(1/ϵ), η = O(ϵ 2 ) and β 1 = O(ϵ 2 ).Compared to [30] , our approach maintains a communication complexity of O(1/ϵ 3 ), but significantly reduces the sample complexity on each machine from O(1/ϵ 6 ) from O(1/ϵ 4 ), requiring only a batch size of O(1) rather than a large batch size of O(1/ϵ 2 ).Our results match the communication and sample complexity in [17] , which tackles a simpler twolevel compositional problem and achieved sample complexity of O(1/ϵ 4 ) per machine. Considering that the sample complexity for a two-level compositional problem in a centralized setting would be O(n/ϵ 4 ) [66] , our approach realizes a linear speed-up proportional to the number of machines. 
fgdro-kl-adam: 
 In this section, we introduce an adaptive algorithm, FGDRO-KL-Adam, to address the problem (5) with KL regularization, as detailed in Algorithm 3. This algorithm incorporates Adam-type updates at local steps, which have been shown to outperform SGD in centralized settings. While previous studies [59] in federated settings have implemented Adam-type updates at the global step but retained SGD for local updates, which may be sub-optimal.Similar to Algorithm 2, u and v are used to estimate the local loss and the global function g(w), respectively. The variables h and m are updated in a manner consistent with Algorithm 2. Under Assumption 5.1, the behavior of the u, v, m estimators is addressed as previously discussed.The primary distinction in Algorithm 3 lies in its adaptive updates for the local model w r i,t . Here, m serves a role akin to the first-order momentum in Adam, and we introduce q r i,t to estimate the second-order momentum:EQUATIONSubsequently, the local models are updated adaptively using the formula:EQUATIONwhere both the square root and division are performed element-wise.A key step in the analysis is to address the coordinate-wise update, as in the following lemma.Lemma 6.1. Using L-smooth of F , for some proper constants C and G, by settingEQUATIONFinally, we show that FGDRO-KL-Adam has same convergence rate as FGDRO-KL. EQUATION)Remark. To achieve an ϵ-stationary point, i.e., ∥∇F ( w)∥ 2 ≤ ϵ 2 , we just need to set R = O(1/ϵ 3 ),I = O(1/ϵ), η = O(ϵ 2 ) and β 1 = O(ϵ 2). The communication and sample complexities are the same as in Theorem 5.3. Our analysis, following the framework in [16] , requires q r i,t + τ to be both upper and lower bounded. It is achieved by the upper bound assumption and choice of τ , ensuring τ ≤ q r i,t + τ ≤ G + τ , which is utilized similarly in [59] . However, Guo et al. [16] did not cover the federated learning scenario or the compositional problems. It is important to note that we have not developed an Adam-type variant for FGDRO-CVaR. This is due to the need for accurate gradient estimation in the analysis of Adam-type updates, which is achieved using the moving estimator m. But in CVaR variant, the nonsmooth nature renders a moving average for subgradient ∂F (w) not provably accurate. 
experiments: 
 Datasets and Neural Networks We use Pile [15] , CivilComments [5] , Camelyon17 [1] , iWild-Cam2020 [3] , and Poverty [74] . For Pile, we preprocess it as [72] , for the others, we use the preprocessed version by [37] . We utilized natural data splits where data from the same hospital, web source, location, or demographic group were placed on the same client machine. These experiments have involved with highly imbalanced number of data on clients. Data statistics are presented in the Appendix E. Additiona experiments on Cifar 10 with Dirichlet distributions over 100 clients are reported in Appendix G.The Pile data set is a large language data set. We use the uncopyrighted version [29] which has 17 domains, and each domain is allocated to one machines. We use the GPT2 model [58] as implemented by [28] with 12 hidden layers, 12 attention heads, and 768 embeddings and hidden states. We measure the performance using worst log-perplexity and average log-perplexity of testing groups. CivilComments is a toxicity classification of the online comment task in diversified demographic identities. We train on four groups based on the presence of 'Black' and toxicity labels, deploying each on a separate machine, and use the DistilBERT base-uncased model [60] to predict toxicity. We measure the performance using worst group accuracy and average accuracy of testing groups. Camelyon17 focuses on tumor detection from lymph node images [1] , with data from five hospitals split into training (3), validation (1), and testing (1) sets. Training uses three machines, each processing data from one hospital, using DenseNet-121 [26] . The iWildCam2020 dataset consists of wildlife images from various camera traps [3] , the dataset is split into training, validation, and testing segments. We use ResNet50 [21] across all datasets and measure performance via Macro F1 score. The Poverty dataset contains geographic data aimed at predicting regional poverty levels [74] . We use ResNet50 [21] for our models and evaluate performance using both the Pearson correlation on the worst-performing region and the average across regions. Baselines We compare our algorithms FGDRO-CVaR, FGDRO-KL, and FGDRO-KL-Adam with four baselines: FedAvg [48] , SCAFFOLD [34] , FedProx [40] , FedAdam [59] , DRFA [11] , and DR-DSGD [30] . For each algorithm, we repeat the experiments 3 times with different random seeds and report the averaged performance. Following [72] , our FGDRO algorithms for Pile initially train for 20K iterations to obtain domain weights, which are then fixed during subsequent training phases.We report the experimental results for natural language processing in Table 2 and those for computer vision in Table 3 . We can see that our methods outperform the baselines in most tasks. Our approaches improve worst-case performance without hurting average case performance. Furthermore, FGDRO-KL-Adam has demonstrated superior performance compared to FGDRO-KL in most cases.Ablation Studies Here we present some ablation study to examine some aspects of our algorithm design. First, in Figure 1 (a), we vary the communication interval I in experiments on the Camelyon dataset. We can see that both our FGDRO-CVaR and FGDRO-KL-Adam algorithms can tolerate skipping a large number of communications without degrading the performance.To demonstrate the effect of the local adaptive updates. We develop a LocalAdam algorithm (see Appendix D), which optimizes ERM using our design of using Adam steps in local updates. The results are plotted in Figure 1  
impact statements: 
 This paper is meant to advance the field of federated machine learning. We do not see noticeable negative impact.For non-smooth functions, since it is usually difficult or even impossible to find an ϵ-stationary point, we are interested in finding an ϵ-near stationary point. Following a common technique for finding an ϵ-near stationary point for weakly convex function, we use the Moreau envelope of a general non-smooth ρ-weakly-convex F (x) [9] , defined asψ (1/ ρ) (x) = min x ′ [F (x ′ ) + ρ 2 ∥x ′ -x∥ 2 ].By the properties of Moreau envelop [52, 9] , we have that if ρ = 2ρ, then ψ 1/ ρ(•) is smooth and an ϵ-stationary point of ψ 1/ ρ(•) is an ϵ-near stationary point of F (•).Since F (w, s) is non-smooth, we investigate its Moreau envelop, which isψ (1/ ρ) (w, s) = min w ′ ,s ′ [F (w ′ , s ′ ) + ρ 2 (∥w ′ -w∥ 2 + ∥s ′ -s∥ 2 )].A.1 Proof of Lemma 4.2Proof. By updating rule, we haveEQUATION)whereEQUATIONwhere the last equality uses E t-1 [ℓ( wr t-1 , z r i,t ) -g i ( wr t-1 )] = 0. Since f (•, z) and g(•, z) are Lipschitz and smooth, we know ∥m r i,t ∥ 2 ≤ C 2 g . We also haveEQUATIONTherefore,EQUATIONA.2 Proof of Theorem 4.3Proof. Since f (•) is 1-Lipschitz, convex and monitonically nondecreasing, while ℓ(•, z) is C g -Lipschitz and L g -smooth, we know that F (w, s) is ρ F := L g -weakly convex byEQUATIONwhere the first inequality uses the convexity of f (•), and the second inequality uses that ∂f (•) ≥= 0 and the L g -smoothness of g(•).With ρ = max 2ρ F , 1, we defineEQUATIONandEQUATIONthen we have the following [9] ,EQUATION)ThenEQUATIONwhere we used ∥m r i,t ∥ 2 ≤ C 2 g . Denote mr t = 1 N N i=1 ∂ u f (u r i,EQUATION)Since f is convex, ∂ u f ≥ 0 and g i is ρ g := L g -weakly convex, we haveEQUATIONNotingEQUATION⟨ mr t , ŵr t-1 -wr t-1 ⟩ + ⟨v r t , ŝr t-1 -sr t-1 ⟩ = 1 N N i=1⟨∂ u f (u r i,t , s r i,t-1 )∇g i ( wr t-1 ), ŵr t-1 -wrEQUATIONPutting ( 21) and ( 23) together, we obtainEQUATION)Thus,EQUATIONIt follows thatEQUATION)EQUATIONHence, taking telescoping sum over Lemma 4.2 sum we haveEQUATIONThus,EQUATIONWithEQUATIONWe also have that ∥ ŵ -w∥ 2 + ∥ŝ -s∥ 2 = ρ∥∇ψ 1/ ρ( w, s)∥ 2 , |dist(0, ∂F ( ŵ, ŝ))| 2 ≤ ∥∇ψ 1/ ρ( w, s)∥ 2 [9]. We can conclude by setting parameters as in the theorem.The behavior of u, which is an estimator of ∇g i (w)'s, is given in the following lemma.Lemma B.1. Under Assumption 5.1, with some constant G, by settingEQUATION)The behavior of v, which is an estimator of 1 N i g i (w), is given in the following lemma.Lemma B.2. Under Assumption 5.1, with some constant C 1 , by setting EQUATIONEQUATIONwhere (a) holds due to the fact that E t-1 [ℓ(w r i,t ; z r i,t ) -ℓ(w r i,t ; D i )] = 0. Moreover, we haveEQUATIONandEQUATIONwhich yieldsEQUATIONwith C 1 = exp(C 0 /λ). Thus,EQUATIONwithC 2 = 288(C 2 ℓ C 2 1 (σ 2 + C 2 ℓ )).Proof. We haveEQUATIONwhereC 1 = exp(C 0 /λ).Thus, it follows thatEQUATIONWe address each term as follows. 1 ⃝ can be bounded asEQUATION⃝ can be bounded asEQUATION3 ⃝ can be bounded asEQUATIONwhere (a) follows from (35) . 4 ⃝ can be bounded asEQUATIONas machines are independent conditioned on iteration t -1.⃝ can be bounded asEQUATIONwhere the first inequality uses v r i,t ≥ 1 as ℓ(•) ≥ 0. F (w) is L F := C f L g + C 2 g L f -smooth. With η ≤ β 3 /(3LEQUATIONwhere (a) usesEQUATIONThus,EQUATIONWe conclude the proof by setting the parameters as in the Lemma.Proof of Theorem 5.3. Using Lemma 5.2, with β 2 = O(β 3 ), we haveEQUATION)Using L F -smooth of F , we have (FEQUATIONTherefore,EQUATION)We conclude the proof by setting the parameters as the theorem.Proof of Theorem 6. EQUATIONwhere (a) holds due to the fact that E t-1 [ℓ(w r i,t ; z r i,t ) -ℓ(w r i,t ; D i )] = 0. And we haveEQUATIONwhere C 1 = exp(C 0 /λ).Moreover, we haveEQUATION)Using L F -smooth of F , we have FEQUATION)We conclude the proof by setting the parameters as the theorem.In this section, we present Algorithm 4, which uses Adam type updates in local steps to solve an ERM problem. w r i,0 = wr , m r i,0 = mr , q r i,0 = qr 4:for t = 1, ..., I doEach machine samples data z r i,th r i,t = ∇ℓ(w r i,t-1 ; z r i,t )7:m r i,t = (1-β 3 )m r i,t-1 + β 3 h r i,t , q r i,t = (1-β 4 )q r i,t-1 + β 4 (h r i,t ) 2 8: w r i,t = w r i,t-1 -η m r i,t √ q r i,t +τ 9:end for Running time is reported in Tabel 6. Each algorithm was run on a high performance cluster where each machine uses a NVIDIA A100 GPU. We create an imbalance dataset by reducing the data of 5 classes by 80% and then distributed the data across 100 clients according to two different Dirichlet distributions: Dirichlet (0.3) and Dirichlet (10), using code released by [61] . We use a two layer CNN as the model. Results of algorithms are summarized in Table 7 . Figure 2 and Figure 2 illustrate the communication complexity of each method by comparing the worst-case testing accuracy against the number of local updates and the communicated data sizes. In Table 8 , 9 and 10, we evaluate the statistical significance of our algorithms' advantages over the baselines. In each cell, the three letters (representing FGDRO-CVaR, FGDRO-KL, and FGDRO-KL-Adam, respectively) indicate whether each of our algorithms has outperformed the corresponding baseline in that row with a confidence level greater than 95% (p-value less than 0.05). 'Y' indicates Yes, and 'N' indicates No. 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).Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines?Answer: [Yes] Justification: This paper conform with the NeurIPS Code of Ethics.Guidelines:• The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.• If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics. • The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction). 
broader impacts: 
 Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?Answer: [Yes] Justification: The broader impacts have been discussed in Section 9Guidelines:• The answer NA means that there is no societal impact of the work performed.• If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact. • Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.• The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster. • The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology. • If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML). 
safeguards: 
 Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?Answer: [NA] Justification: This paper does not release any new data or models.Guidelines:• The answer NA means that the paper poses no such risks.• Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters. • Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images. • We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.12. Licenses for existing assets Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?Answer: [Yes] Justification: The assets we use are all publicly available and properly credited.Guidelines:• The answer NA means that the paper does not use existing assets.• The authors should cite the original paper that produced the code package or dataset.• The authors should state which version of the asset is used and, if possible, include a URL. • The name of the license (e.g., CC-BY 4.0) should be included for each asset.• For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided. • If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset. • For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided. 
 
 Appendix:  


Decision: Authors propose algorithms for federated distributionally robust optimization (DRO). DRO is claimed to be particularly relevant in federated setting because of the potential for non-IID data on client. They argue that existing methods have higher communication and sample complexity and provide 3 algorithms to address CVaR and KL regularized DRO frameworks, and analyze their communication complexities. They show that the algorithms have different theoretical goals and strengths. Finally authors provide multiple experimental results showing the robustness of their algorithms when compared to other federated baselines. Reviewers raised multiple concerns including lack of experimental details, lack of baselines and comparison to past literature, confusion about experimental details, unclear delineation of technical and algorithmic novelty, lack of statistical uncertainty in experiment, and too strong assumptions. Authors were able to address most of these concerns during the rebuttal phase. AC notes that paper lacks any comparison to non federated (centralized) learning methods. This is particularly important to understand the performance gap between centralized and federated setting. AC highly encourage the authors to add this baseline in the next revision.

Reviews: review: paper_summary: paper_summary: This paper aims to improve the efficiency of existing federated group distributionally robust optimization (FGDRO) when considering two specific types of regularization, condition value at risk and KL divergence. To address the first type of problem, the authors propose FGDRO-CVaR that reduces the sample complexity and communication costs simultaneously. For KL conditions, the proposed FGDRO-KL reduces the sample complexity while retaining the same communication costs. Moreover, the authors integrate the notion of Adam into FGDRO-KL, yielding FGDRO-KL-Adam and achieving better convergence speed.
strength_weakness: strength_weakness: 1. The background can be more thoroughly explained. The authors are encouraged to provide additional context to address the following questions, which will greatly enhance the paper's completeness. Why is federated group distributionally robust optimization (FGDRO) an important problem or technique? What are the sources of the additional communication costs? Why is it necessary to consider two different types of regularization? Are these types of regularization relevant to different applications?

2. My major concerns lie in the experiments and their settings. 

    - **Data Splits.** While FGDRO's main advantage appears to be its ability to address non-IID optimization, the experimental setup concerning data splits lacks clarity. An analysis of the non-IID levels, such as those derived from different Dirichlet-distributed data splits with varying $\lambda$ values, is missing. Including more representative baselines, such as SCAFFOLD and FedProx, which are also designed for non-IID optimization, could further enhance the analyses.

    - **Performance.** The proposed method performs similarly to the baselines in most experiments. For example, in Tables 2 and 3, apart from the Adam variant, the proposed method is comparable to the baselines. This would be acceptable if the proposed method demonstrated improved efficiency; however, relevant analyses on this aspect are absent from the experiments.

    - **Communication or Sample Complexity Analysis.** An empirical analysis comparing complexity versus utility would be beneficial and highlight the advantages of the proposed method. For instance, the experiment in Figure 1 can be extended to a comparison among different baselines.
questions: questions: 1. The datasets considered in this paper do not seem common in the existing literature. Could the authors report numbers on datasets like CIFAR-10/100 or EMNIST? Why did the authors choose the datasets in the paper?
limitations: limitations: Yes, the authors have discussed the limitations in the paper.
paper_summary: paper_summary: The paper presents three methods for Federated Learning Group Distributionally Robust Optimization: (i) one tailored to reduce the CVaR which optimizes the top K-losses, (ii) another one tailored to tackle the KL divergence, and finally (iii) one that uses Adam locally. The paper is well written and the ideas are presented. To the best of my knowledge, the proofs are correct. My main concerns are regarding the relevance and importance of the subject, the lack of experiments, and the lack of empirical studies on communication efficiency.
strength_weakness: strength_weakness: [W1] The relevance of the subject is not entirely addressed. See [Q1]

[W2] The experiment section is limited. In particular, the paper does not present any intuition on the problems they are solving. They do not consider the number of samples per server for example. I believe the authors should include a class imbalance problem [AN AGNOSTIC APPROACH TO FEDERATED LEARNING WITH CLASS IMBALANCE - Shen et al, ICLR 22].

[W3] Communication efficiency is not properly addressed by the authors. The authors show the number of communication rounds required, but they do not take into account how much is communicated. The authors claim that this method is more efficient in terms of communication, and they show it theoretically, but in the experiment section, there is no evidence of communication efficiency. I suggest the authors reveal the communication cost associated with each method, measured in the amount of data shared between servers. 

[W4] Privacy is an important subject of Federated Learning, but in this paper, there is no analysis of the privacy aspect. Can the authors elaborate on the privacy aspect of this work? 

[W5] Federated learning is a technique used to train on a set of machines. The idea is that the number of machines that participate is large. It appears to me that the largest number of servers is 17. This seems to me insufficient for a distributed learning problem.
questions: questions: [Q1] Why should be designed solutions that are distributional robust? And, at what cost? If we compare a method that simply maximizes/minimizes the FL problem, what is the overall loss? I believe the overall loss should be smaller, given that being distributionally robust is a particular case, and therefore, the unconstrained problem achieves a smaller minimum.
limitations: limitations: See weaknesses.
paper_summary: paper_summary: This paper addresses the challenge of reducing communication costs and sample complexity in Federated Group Distributionally Robust Optimization (FGDRO). The authors present the FGDRO-CVaR algorithm and the FGDRO-KL algorithm to address different constraints. Subsequently, they conduct extensive experiments across various real-world tasks, including NLP and CV tasks. The corresponding empirical results confirm the effectiveness of their proposed methods.
strength_weakness: strength_weakness: 1. The contributions and novelties of this paper are unclear. It appears that the authors have directly combined existing federated adaptive algorithms with pre-existing federated group DRO methods in this paper.
questions: questions: 1. The introduction's treatment of the concept of generalization appears incomplete. It is evident that there are two levels of generalization in Federated Learning, as delineated in [1] and [2].

2. As highlighted in the aforementioned weaknesses, the authors should provide additional clarification regarding the contributions and novelty of this paper. Overall, it appears that the proposed method is primarily a direct combination of existing methods.

3. Similarly, the authors should delineate the challenges and innovations intrinsic to their theoretical analysis. Specifically, they should underscore the complexities involved in analyzing federated adaptive algorithms when applied in federated group DRO.

4. On line 154 of Page 4, what is the relationship between the "accurate estimate" and the "moving average" in the subsequent sentence?

5. Some minor points to address. The authors might consider offering more empirical results on convergence analysis in the experimental section. Additionally, they should further consider the statistical significance of these convergence analyses.

[1] Hu X, Li S, Liu Y. Generalization bounds for federated learning: Fast rates, unparticipating clients and unbounded losses[C]//The Eleventh International Conference on Learning Representations. 2023.

[2] Yuan H, Morningstar W, Ning L, et al. What do we mean by generalization in federated learning?[J]. arxiv preprint arxiv:2110.14216, 2021.
limitations: limitations: The authors have adequately addressed the limitations and potential negative societal impact of their work.
paper_summary: paper_summary: This work introduces three algorithms for communication-efficient Federated Group Distributionally Robust Optimization. The effectiveness of the proposed algorithms are verified through both theoretical and experimental results.
strength_weakness: strength_weakness: 1) This work proposes three algorithms, including FGDRO-CVaR, FGDRO-KL, and FGDRO-KL-Adam. There lacks a comparison between these algorithms. For example, what are the connections and differences between these algorithms?
2) The analysis for FGDRO-CVaR assumes the loss function to be rho-weakly convex, which is missing from the main context.
questions: questions: 1) Missing reference: How about the comparison with this work [1]?

[1] Communication-Efficient Distributionally Robust Decentralized Learning https://arxiv.org/pdf/2205.15614

2) What are the experimental setups for the number of clients and non-IID?

3) In experimental results (Tables 2 and 3), FGDRO-CVaR seems to have no advantages in both task; why do we need this algorithm? Besides, it is better to highlight the best-performance results in Tables 2 and 3.

4) Intuitively, using Adam optimizer can bring training speedup and is supposed to outperform other algorithms. But why do the results show that sometimes FGDRO-KL is better than FGDRO-KL-Adam?

5) In proof, is an assumption of bounded gradient needed? If I don't misunderstand, Line 550 indicates such an assumption.

6) If the loss function assumes to be convex, analyzing the optimal distance between the loss value the minimum loss should be better in Theorem 6.2.
limitations: limitations: See weakness.

Comments: title: Thank you for raising the score
Regarding your concern about the bounded stochastic gradient assumption
Reponse
Thank you for the response
comment: We are glad to hear that your concerns have been addressed by our response. As suggested, we will include more discussion on the background, differences between methods and new results. Thank you!
I want to thank the authors for their response, which effectively addressed my concerns. Although I still have some reservations about the performance of common baselines like FedProx and SCAFFOLD in the rebuttal, I encourage the authors to provide more context on the background, clearly outline the differences between the proposed methods and the baselines, and include the new results. Given this, I’d like to raise my score to 5.
Thank you for your valuable feedback! After a thorough analysis, we have managed to relax the bounded stochastic gradient assumption ($||\nabla \ell(\mathbf{w};\mathbf{z})||^2\leq G^2$) to more standard assumptions in the literature of federated learning and optimization. Specifically, we now assume a bounded true gradient ($||\nabla \ell(\mathbf{w};\mathcal{D}\_i)||^2\leq C_\ell^2$, where $\ell(\mathbf{w};\mathcal{D}\_i) = \mathbb{E}\_{\mathbf{z}\in\mathcal{D}\_i} \ell(\mathbf{w}; \mathcal{D}\_i)$) and bounded variance ($\mathbb{E}_{\mathbf{z} \in \mathcal{D}\_i}||\ell(\mathbf{w}; \mathbf{z}) - \ell(\mathbf{w}; \mathcal{D}\_i)||^2 \leq \sigma^2$). We have updated the assumption and analysis accordingly, and you can find the revised details in the "updated_analysis.pdf" file at https://anonymous.4open.science/r/NeurIPS_11919-3402. For example, your original concern about the usage of the bounded stochastic gradient assumption in Line 550 has now been addressed with these relaxed assumptions in Lines 550-551.
I appreciate the author's response. I choose to keep my score.
Thanks to the authors for the additional explanations of the other issues that I have mentioned in the review and more experiments, which have solved all of my concerns. I recommend the acceptance of this paper. However, since I am not an expert in this field, I decide to keep my current score.
After reading other reviews and the authors' responses, I would keep my score. My major concern is that the bounded gradient assumption is somehow too strong.