Efficient Sign-Based Optimization: Accelerating Convergence via Variance Reduction

Authors: wei  jiang
sifan  yang
wenhao  yang
lijun  zhang
Venue: NeurIPS_cc_2024_Conference
Published: 2024-05-15 12:36:26

Abstract: Sign stochastic gradient descent (signSGD) is a communication-efficient method that transmits only the sign of stochastic gradients for parameter updating. Existing literature has demonstrated that signSGD can achieve a convergence rate of $\mathcal{O}(d^{1/2}T^{-1/4})$, where $d$ represents the dimension and $T$ is the iteration number. In this paper, we improve this convergence rate to $\mathcal{O}(d^{1/2}T^{-1/3})$ by introducing the Sign-based Stochastic Variance Reduction (SSVR) method, which employs variance reduction estimators to track gradients and leverages their signs to update. For finite-sum problems, our method can be further enhanced to achieve a convergence rate of $\mathcal{O}(m^{1/4}d^{1/2}T^{-1/2})$, where $m$ denotes the number of component functions. Furthermore, we investigate the heterogeneous majority vote in distributed settings and introduce two novel algorithms that attain improved convergence rates of $\mathcal{O}(d^{1/2}T^{-1/2} + dn^{-1/2})$ and $\mathcal{O}(d^{1/4}T^{-1/4})$ respectively, outperforming the previous results of $\mathcal{O}(dT^{-1/4} + dn^{-1/2})$ and $\mathcal{O}(d^{3/8}T^{-1/8})$, where $n$ represents the number of nodes. Numerical experiments across different tasks validate the effectiveness of our proposed methods.

Main paper: 
 introduction: 
 This paper investigates the stochastic optimization problemEQUATIONwhere f : R d → R is a smooth and non-convex function. We assume that only noisy estimations of the gradient ∇f (x) can be accessed, represented as ∇f (x; ξ), where ξ is a random sample drawn from a stochastic oracle such that E[∇f (x; ξ)] = ∇f (x).The most well-known method for problem (1) is stochastic gradient descent (SGD), which performs x t+1 = x t -η∇f (x t ; ξ t ) for each iteration, where ξ t is the sample used in the t-th iteration, and η is the learning rate. It has been proved that the SGD method can obtain a convergence rate of O(T -1/4 ) [Ghadimi and Lan, 2013] , where T is the iteration number. Recently, sign stochastic gradient descent (signSGD) method [Seide et al., 2014 , Bernstein et al., 2018] has become popular in the machine learning community, which uses the sign of the stochastic gradient to update, i.e.,x t+1 = x t -η Sign(∇f (x t ; ξ t )). This method can largely reduce the communication overhead in distributed environments, and prior research [Bernstein et al., 2018 [Bernstein et al., , 2019] ] has established that signSGD can achieve a convergence rate of O(d 1/2 T -1/4 ) measured in terms of the l 1 -norm. Since the O(T -1/4 ) rate is already optimal for SGD methods when measured in the l 2 -norm [Arjevani et al., 2023] , we can not further improve the dependence on T for signSGD method, considering that ∥x∥ 2 ≤ ∥x∥ 1 for any x. 2 However, it is also known that variance reduction techniques can further enhance the convergence rate to O(T -1/3 ), under a slightly stronger assumption of average smoothness [Fang et al., 2018 , Wang et al., 2019 , Cutkosky and Orabona, 2019] . This leads to a natural question: Can the convergence of sign-based methods be further improved by employing variance reduction techniques along with the average smoothness assumption? We respond affirmatively by introducing the Sign-based Stochastic Variance Reduction (SSVR) method. By integrating variance reduction technique [Cutkosky and Orabona, 2019] with sign operations, we achieve an improved convergence rate of O(d 1/2 T -1/3 ) measured in the l 1 -norm, matching the optimal rates in terms of T for stochastic variance reduction methods [Fang et al., 2018 , Li et al., 2021 , Arjevani et al., 2023] .Furthermore, we investigate a special case of problem (1), in which the objective function exhibits a finite-sum structure:EQUATIONwhere each f i (•) is smooth and non-convex. This problem has been extensively studied in stochastic optimization [Zhang et al., 2013 , Defazio et al., 2014 , Fang et al., 2018] , but is less explored with sign-based methods. Previous literature proposes signSVRG [Chzhen and Schechtman, 2023 ] method to deal with the finite-sum problem, which achieves a convergence rate of O(m 1/2 d 1/2 T -1/2 ). However, its dependence on m is sub-optimal, failing to match the O(m 1/4 T -1/2 ) lower bound [Fang et al., 2018 , Li et al., 2021] for problem (2). To address this gap, we propose the SSVR-FS algorithm, which periodically computes the exact gradient [Zhang et al., 2013, Johnson and Zhang, 2013] and incorporates it into the variance reduction estimator. In this way, we can achieve an improved convergence rate of O(m 1/4 d 1/2 T -1/2 ) for finite-sum problems.Finally, sign-based methods are especially favorable in distributed settings, where the parameter server aggregates gradient signs from each worker through majority vote [Bernstein et al., 2018] , allowing 1-bit compression of communication in both directions. Existing literature [Bernstein et al., 2018 [Bernstein et al., , 2019] ] has proved that signSGD can obtain a convergence rate of O(d 1/2 T -1/4 ) for majority vote in homogeneous settings, where the data across nodes is uniformly distributed or identical. For the more challenging heterogeneous setting, in which data distribution can vary significantly across nodes, existing methods can only achieve convergence rates of O(dT -1/4 + dn -1/2 ) [Sun et al., 2023] and O(d 3/8 T -1/8 ) 3 [Jin et al., 2023] , where n denotes the number of nodes. Note that the first rate indicates that the gradient does not converge to zero as T approaches infinity, and the second one suffers from a high sample complexity. To address these limitations, we first introduce our basic SSVR-MV method, which employs variance reduction estimators to track gradients and replaces the sign operation in each worker as a stochastic unbiased sign operation. This practice ensures 1-bit compression and unbiased estimation at the same time, and the newly proposed method can obtain an improved convergence rate of O(d 1/2 T -1/2 + dn -1/2 ). By further substituting the sign operation in the parameter server with another stochastic unbiased sign operation, our method can further achieve a convergence rate of O(d 1/4 T -1/4 ), which converges to zero as T increases.In summary, compared with existing methods, this paper makes the following contributions:• For stochastic non-convex functions, we develop a sign-based variance reduction algorithm to achieve an improved convergence rate of O(d 1/2 T -1/3 ), surpassing the O(d 1/2 T -1/4 ) rate for signSGD methods. • For non-convex finite-sum optimization, we further improve the our proposed method to obtain an enhanced convergence rate of O(m 1/4 d 1/2 T -1/2 ), which is better than the O(m 1/2 d 1/2 T -1/2 ) convergence rate for SignSVRG method.Table 1: Summary of results for sign-based algorithms. Here, stochastic indicates problem (1), finite-sum represents problem (2), N is the number of stochastic gradient calls, and m is the number of component functions. Note that some rates are measured under squared l 1 -or l 2 -norm, and we convert them to l 1 -or l 2 -norm for a fair comparison.Method Setting Measure Convergence rate signSGD [Bernstein et al., 2018 ] stochastic l1-normO d 1/2 N 1/4EF-signSGD [Karimireddy et al., 2019 ] stochastic l2-norm O 1 N 1/4 signSGD-SIM [Sun et al., 2023] stochastic l1-norm O EF-signSGD method, which integrates error feedback into signSGD to correct errors introduced by the sign operation. Instead of requiring unbiased stochastic gradients in previous literature, Safaryan and Richtarik [2021] assumed that the signs of the stochastic gradient are the same as those of true gradient with a probability greater than 1/2. Under this assumption, they demonstrated that signSGD can obtain a similar convergence rate but does not require large batches anymore. Recently, Sun et al. [2023] proposed the signSGD-SIM method, which incorporates the momentum into the signSGD, achieving a convergence rate of O(dT -1/4 ) with constant batch sizes and an improved convergence of O(d 3/2 T -2/7 ) with second-order smoothness.To deal with the finite-sum problems, Chzhen and Schechtman [2023] developed SignSVRG algorithm, which combines SVRG [Johnson and Zhang, 2013] method with signSGD and achieves a convergence rate of O(d 1/2 m 1/2 T -1/2of O d 1/2 m 1/2 T -1/2 .In distributed settings, sign-based methods with majority vote are also widely investigated. Bernstein et al. [2018 Bernstein et al. [ , 2019] ] first indicated that signSGD and its momentum variant Signum can enable 1-bit compression of worker-server communication, obtaining the O(d 1/2 T -1/4 ) convergence rates in the homogeneous environment. For the more challenging heterogeneous settings, SSDM method [Safaryan and Richtarik, 2021] attains the same O(d 1/2 T -1/4 ) convergence rate, but the information sent back to the server is not a sign information anymore. To remedy this issue, Sto-signSGD algorithm [Jin et al., 2023] is proposed, equipped with a convergence rate of O(d 3/4 T -1/4 ) measured in squared l 2 -norm. More recently, Sun et al. [2023] introduced the MV-signSGD-SIM algorithm and demonstrated a convergence rate of O(dT -1/4 + dn -1/2 ), which could be further enhanced to O(d 3/2 T -2/7 + dn -1/2 ) under second-order smoothness conditions, where n denotes the number of nodes in the distributed system. 
stochastic variance reduction methods: 
 Stochastic variance reduction methods have gained significant attention in the optimization community in recent years. Among the pioneering approaches, the stochastic average gradient (SAG) method [Roux et al., 2012] and Stochastic Dual Coordinate Ascent (SDCA) algorithm [Shalev-Shwartz and Zhang, 2013] utilize a memory of previous gradients to ensure variance reduction, achieving linear convergence for strongly convex functions. To circumvent the need for storing gradients, the stochastic variance reduced gradient (SVRG) [Zhang et al., 2013, Johnson and Zhang, 2013] recalculates the full gradient periodically to enhance the accuracy of gradient estimators, maintaining linear convergence for strongly convex functions. Inspired by SAG and SVRG, Defazio et al. [2014] introduced the SAGA algorithm, which not only provides superior convergence rates but also supports proximal regularization. Subsequently, the stochastic recursive gradient algorithm (SARAH) [Nguyen et al., 2017] employs a simple recursive approach to update gradient estimators, ensuring better convergence for smooth convex functions.For non-convex optimization, inspired by the SVRG algorithm, many methods [Reddi et al., 2016 , Lei et al., 2017 , Zhou et al., 2020] employ variance reduction to design their algorithms and provide the corresponding convergence guarantees. More recent well-known advancements include the SPI-DER [Fang et al., 2018] and SpiderBoost [Wang et al., 2019] methods, which improved the O(T -1/4 ) convergence rate of traditional SGD to O(T -1/3 ) under the average smoothness assumption. The convergence rate can be further improved to O(m 1/4 T -1/2 ) for problems with a finite-sum structure, where m represents the number of component functions. However, these methods typically require a huge batch size to ensure convergence. To avoid this limitation, the stochastic recursive momentum (STORM) method [Cutkosky and Orabona, 2019 ] introduces a momentum-based updating and an adaptive learning rate based on the stochastic gradients, achieving a convergence rate of Õ(T -1/3 ) without necessitating large batches. More recently, variance reduction techniques are widely employed in more complex problems to improve the existing convergence rates, such as compositional optimization [Wang et al., 2017a ,b, Yuan et al., 2019 , Jiang et al., 2022a , 2023] , multi-level optimization [Chen et al., 2021 , Zhang and Xiao, 2021 , Jiang et al., 2022b , 2024b] , adaptive algorithms [Kavis et al., 2022 , Jiang et al., 2024a] , and distributionally robust optimization [Yu et al., 2024] .Algorithm 1 SSVR 1: Input: time step T , initial point x 1 2: for time step t = 1 to T do 3:Draw a batch of samples{ξ 1 t , • • • , ξ B1 t } 4: Compute v t = 1 B1 B1 k=1 ∇f (x t ; ξ k t ) + (1 -β) v t-1 -1 B1 B1 k=1 ∇f (x t-1 ; ξ k t ) 5:Update the decision variable: x t+1 = x t -η Sign (v t ) 6: end for 7: Select τ uniformly at random from {1, . . . , T } 8: Return x τ 
the proposed methods: 
 In this section, we present the proposed methods for the expectation case, i.e., problem (1), and the finite-sum structure, i.e., problem (2), respectively, along with corresponding theoretical guarantees.In this subsection, we introduce our Sign-based Stochastic Variance Reduction (SSVR) method for problem (1). One crucial step in stochastic optimization is to track the gradient of the objective function. Here, we use a variance reduction gradient estimator v t to evaluate the overall gradient ∇f (x t ). In the first iteration (t = 1), the estimator is defined asv 1 = 1 B0 B0 k=1 ∇f (x 1 ; ξ k 1 ), where B 0 is the batch size used in the first iteration. For subsequent iterations (t ≥ 2), v t is updated in the style of STORM [Cutkosky and Orabona, 2019] , i.e.,v t = 1 B 1 B1 k=1 ∇f (x t ; ξ k t ) + (1 -β) v t-1 - 1 B 1 B1 k=1 ∇f (x t-1 ; ξ k t ) ,where β represents the momentum parameter and B 1 is the batch size. This method ensures that the expectation of the estimation error E[∥v t -∇f (x t )∥ 2 ] would be reduced gradually. After obtaining the gradient estimator v t , we update the decision variable using the sign of v t :x t+1 = x t -η Sign (v t ) . The whole algorithm is outlined in Algorithm 1. Next, we introduce the following assumptions for our SSVR method, which are standard and commonly adopted in the analysis of variance reduction methods and stochastic non-convex optimization [Fang et al., 2018 , Wang et al., 2019 , Cutkosky and Orabona, 2019 , Li et al., 2021] .Assumption 1 (Average smoothness)E ξ ∥∇f (x; ξ) -∇f (y; ξ)∥ 2 ≤ L 2 ∥x -y∥ 2 .Assumption 2 (Bounded variance)E ξ ∥∇f (x; ξ) -∇f (x)∥ 2 ≤ σ 2 .With the above assumptions, we can obtain the theoretical guarantee for our method as stated below.Theorem 1 Under Assumptions 1 and 2, by settingβ = O( 1 T 2/3 ), η = O( 1 d 1/2 T 2/3 ), B 0 = O(T 1/3), and B 1 = O(1), our SSVR method ensures:E [∥∇f (x τ )∥ 1 ] ≤ O d 1/2 T 1/3 .Remark: This convergence rate surpasses the O(d 1/2 T -1/4 ) rate achieved by previous sign-based methods [Bernstein et al., 2018 [Bernstein et al., , 2019]] , and it also outperforms the O(d 3/2 T -2/7 ) convergence rate under the second-order smoothness condition [Sun et al., 2023] . Specifically, to ensure that Update the decision variable: x t+1 = x t -ηSign (v t ) 9: end for 10: Select φ uniformly at random from {1, . . . , T } 11: Return x φ 3.2 Sign-based stochastic variance reduction for finite-sum structureE[∥∇f (x τ )∥ 1 ] ≤ ϵ,We now extend our SSVR method to deal with the finite-sum structure in problem (2). In this context, we introduce the following assumption for each component function, which is standard and widely adopted in existing literature [Fang et al., 2018 , Wang et al., 2019 , Li et al., 2021] .Assumption 3 (Smoothness) For each i ∈ {1, 2, • • • , m}, the gradient functions satisfy:∥∇f i (x) -∇f i (y)∥ ≤ L∥x -y∥.To handle the finite-sum problems, we retain the core structure of our SSVR method while incorporating elements from the SVRG [Zhang et al., 2013, Johnson and Zhang, 2013] approach. Specifically, we compute a full batch gradient at the first step and every I iteration, i.e.,∇f (x τ ) = 1 m m i=1 ∇f i (x τ ).For other iterations, we randomly select an index i t from the set {1, 2, • • • , m} and construct a variance reduction gradient estimator v t as follows:v t = ∇f it (x t ) + (1 -β)(v t-1 -∇f it (x t-1 )) STORM estimator -β (∇f it (x τ ) -∇f (x τ ))error correction.(3)The first two terms of v t align with the STORM estimator, and the last term measures the difference of past gradients between the selected component function ∇f it (x τ ) and the overall objective ∇f (x τ ).Note that the STORM estimator employs the component gradient ∇f it (x t ) to track the overall gradient ∇f (x t ), which leads to an estimation error due to the gap between the component function and the overall objective. This gap can be effectively mitigated by the error correction term we introduced in equation ( 3). With such a design, we can obtain a better gradient estimation of the overall gradient, and ensure that the estimation error E[∥∇f (x t ) -v t ∥ 2 ] can be reduced gradually. After computing v t , we utilize its sign information to update the decision variable. The detailed procedure is outlined in Algorithm 2. Next, we present the theoretical convergence for this method.Theorem 2 Under Assumption 3, by settingβ = O( 1 m ), I = m, and η = O( 1 m 1/4 d 1/2 T 1/2 ), our algorithm ensures: E[∥∇F (x φ )∥ 1 ] ≤ O m 1/4 d 1/2 T 1/2 . Remark: To ensure E[∥∇F (x φ )∥ 1 ] ≤ ϵ, the sample complexity is O(m + d √ mϵ 2 ), which improves over the O( dm ϵ 2 ) complexity of the previous SignSVRG method [Chzhen and Schechtman, 2023] . 
sign-based stochastic variance reduction with majority vote: 
 Sign-based methods are advantageous in distributed settings for their low communication overhead, as they can only transmit sign information between nodes via majority vote. This section explores signbased stochastic methods with majority vote, a typical example of distributed learning extensively Draw sample ξ j t and compute v j t = ∇f j (x t ; ξ j t ) + (1 -β) v t-1 -∇f j (x t-1 ; ξ j t )5:Option 1: Send S R (v j t ) to the parameter server, where R = 4G6:Option 2: Send S G (v j t ) to the parameter server, wherevj t = Π G [v j t ] 7:On parameter server:8:Option 1: Update the decision variable x t+1 = x t -ηv t 12: end for 13: Select τ uniformly at random from {1, . . . , T } 14: Return x τ studied in previous sign-based algorithms [Bernstein et al., 2018 , 2019 , Safaryan and Richtarik, 2021 , Sun et al., 2023] . To begin with, we investigate the following distributed learning task:Send v t = Sign 1 n n j=1 S R v j t toEQUATIONwhere D j represents the data distribution for node j, and f j (x) is the corresponding loss function. Some previous studies [Bernstein et al., 2018 [Bernstein et al., , 2019] ] investigate the homogeneous setting, which assumes the data across each node is uniformly distributed or identical, ensuring thatE[f i (x)] = f (x).In contrast, this paper considers the more challenging heterogeneous setting [Jin et al., 2023 , Sun et al., 2023] , where data distributions can vary significantly across nodes.For sign-based methods in distributed settings, each node j computes a gradient estimator v j t and transmits its sign, i.e., Sign(v j t ), to the parameter server. Note that the server can not directly send the aggregate information n j=1 Sign(v j t ) back to each node, since it loses binary characteristic after summation. A natural solution is to apply another sign operation to update the decision variable as:x t+1 = x t -η Sign   1 n n j=1 Sign(v j t )   .This process is called majority vote [Bernstein et al., 2018] , as each worker votes on the sign of the gradient, with the server tallying these votes and broadcasting the decision back to the nodes. However, the sign operation introduces bias in the estimation, and employing it twice can significantly amplify this bias, particularly in a heterogeneous environment. Previous analysis [Chen et al., 2020] indicates that signSGD fails to converge in the heterogeneous setting. To deal with this problem, we introduce an unbiased sign operation S R (•), which is defined below.Definition 1 For any vector v with ∥v∥ ∞ ≤ R, define the function mapping S R (v) as:[S R (v)] k =      +1, with probability 1 2 + [v] k 2R , -1, with probability 1 2 -[v] k 2R .(5)Remark: This operation provides an unbiased estimation of v/R, such that E[S R (v)] = v/R. It is worth noting that the function mapping is valid when ∥v∥ ∞ ≤ R, since the probability should always fall within [0, 1]. For this purpose, we need to further assume that the gradient is bounded.Utilizing this unbiased sign operation, we can update the decision variable as:x t+1 = x t -η Sign   1 n n j=1 S R (v j t )   .After applying S R (•), the output is a sign information, which can be transported between nodes efficiently. The complete algorithm, named SSVR with majority vote (SSVR-MV), is described in Algorithm 3 (with Option 1). Note that in Step 4, we set v j 1 = ∇f (x 1 ; ξ j 1 ) when t = 1. Next, we present the convergence guarantee for the proposed algorithm with the following assumption.Assumption 4 For each node j, the stochastic gradient is bounded by G in the infinity norm, such that ∥∇f j (x; ξ)∥ ∞ ≤ G.Theorem 3 Under Assumptions 1, 2 and 4, by setting β = 1 2 and η = O(1 T 1/2 d 1/2), our SSVR-MV method (with Option 1) ensures:E [∥∇f (x τ )∥ 1 ] ≤ O d 1/2 T 1/2 + d n 1/2 .Remark: Our rate is better than the previous result of O(dT -1/4 + dn -1/2 ), and also outperforms the rate of O(d 3/2 T -2/7 + dn -1/2 ) under the second-order smoothness [Sun et al., 2023] .Although the above convergence rate is superior to previous results, we have to note that the gradient does not converge to zero even as T → ∞. To address this issue, we propose replacing another sign operation with the S 1 (•) mapping, as defined in equation ( 5) with R = 1. Additionally, in our prior analysis, we ensured that each v j t is bounded by assuming the stochastic gradient is bounded and using a constant β. Here, we instead suppose that the true gradient is bounded, as detailed below.Assumption 4 ′ For each node j, the gradient is bounded such that ∥∇f j (x)∥ ≤ G.Remark: This assumption is weaker than the one used by Sun et al. [2023] , which assumes all stochastic gradients are bounded, i.e., ∥∇f j (x; ξ)∥ ≤ G.To ensure each gradient estimator is bounded, we employ a projection operation vjt = Π G [v j t ], where Π G denotes the projection onto a ball of radius G. This allows us to utilize an unbiased sign mapping S G (v j t ) before transmission to the parameter server. The revised algorithm is presented in Algorithm 3 (with Option 2), and the modifications lie in Steps 6 and 9. We now present the convergence guarantee for this modified approach below.Theorem 4 Under Assumptions 1, 2 and 4 ′ , by settingβ = O( 1 T 1/2 ) and η = O( 1 d 1/2 T 1/2), our SSVR-MV method (with Option 2) ensures:E [∥∇f (x τ )∥] ≤ O d 1/4 T 1/4 .Remark: This rate converges to zero as T → ∞, and offers a significant improvement over the previous results of O(d 3/8 T -1/8 ) [Jin et al., 2023] . Our result is also better than the O(d 1/2 T -1/4 ) convergence rate obtained by Safaryan and Richtarik [2021] , whose algorithm requires transmitting n j=1 sign(v j t ) back to all nodes, which is actually not sign information anymore. 
experiments: 
 In this section, we assess the performance of the proposed methods through numerical experiments.We first evaluate the SSVR and SSVR-FS algorithms within the centralized setting, and then assess the performance of SSVR-MV method in the distributed learning environment. All experiments are conducted on NVIDIA 3090 GPUs. To begin with, we conduct numerical experiments on multi-class image classification tasks to validate the effectiveness of our proposed methods. Concretely, we train a ResNet18 model [He et al., 2016] on the CIFAR-10 dataset [Krizhevsky, 2009] . We compare the performance of our SSVR and SSVR-FS methods against signSGD [Bernstein et al., 2018] , signSGD-SIM [Sun et al., 2023] ,and SignSVRG [Chzhen and Schechtman, 2023] . For hyper-parameter tuning, we either follow the recommendations from the original papers or employ a grid search to determine the best settings. Specifically, the momentum parameter β is searched from the set {0.1, 0.5, 0.9, 0.99}, and the learning rate is fine-tuned within the range of {1e-5, 1e-4, 1e-3, 1e-2, 1e-1}.Results. The training loss, gradient norm, and testing accuracy are presented in Figure 1 , with curves averaged over five runs. We observe that all methods exhibit a rapid decrease in training losses, with our methods showing a more pronounced reduction in the gradient norm. In terms of testing accuracy, our SSVR algorithm outperforms other sign-based methods, and our SSVR-FS method achieves superior accuracy in the final epochs.Subsequently, we conduct experiments to evaluate the effectiveness of the SSVR-MV method in the distributed environment. Specifically, we train a ResNet50 model [He et al., 2016] on the CIFAR-100 dataset [Krizhevsky, 2009] with 4 and 8 nodes respectively. We compare the performance of our method against signSGD (with majority vote) [Bernstein et al., 2018] , Signum (with majority vote) [Bernstein et al., 2019] , SSDM [Safaryan and Richtarik, 2021] , Sto-signSGD [Jin et al., 2023] , and MV-signSGD-SIM [Sun et al., 2023] . The hyper-parameter tuning follows the same methodology as in the centralized environment experiment.Results. We plot the training loss and testing accuracy in Figure 2 , with all curves averaged over five runs. The results indicate that the training loss of our SSVR-MV algorithm decreases rapidly, and our method obtains higher testing accuracy compared to other methods, both in experiments with 4 nodes and 8 nodes. 
conclusion: 
 In this paper, we explore sign-based stochastic variance reduction (SSVR) methods, which use only the sign information of variance-reduced estimators to update decision variables. The proposed method achieves an improved convergence rate of O(d 1/2 T -1/3 ), surpassing the O(d 1/2 T -1/4 ) convergence rate of signSGD methods. When applied to finite-sum problems, this rate can be further enhanced to O(m 1/4 d 1/2 T -1/2 ), which is also better than the O(m 1/2 d 1/2 T -1/2 ) convergence rate of SignSVRG. Finally, we investigate the SSVR method in distributed settings and devise novel algorithms to attain convergence rates of O(d 1/2 T -1/2 + dn -1/2 ) and O(d 1/4 T -1/4 ), which improve upon the previous results of O(dT -1/4 + dn -1/2 ) and O(d 3/8 T -1/8 ) respectively. 
safeguards: 
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 Appendix: Firstly, note that Assumption 1 indicates that the objective function f (x) is also L-smooth [Li et al., 2021] . Given this property, we have thatEQUATIONwhere the last inequality is due to the fact thatEQUATIONSumming up and rearranging the equation ( 6), we derive:E 1 T T t=1 ∥∇f (x t )∥ 1 ≤ f (x 1 ) -f (x T +1 ) ηT + 2 √ d • E 1 T T t=1 ∥∇f (x t ) -v t ∥ + ηLd 2 ≤ ∆ f ηT + 2 √ d • E 1 T T t=1 ∥∇f (x t ) -v t ∥ 2 + ηLd 2 (8)where we define ∆ f = f (x 1 ) -f * , and the second inequality is due to Jensen's Inequality.Next, we can bound the termE 1 T T t=1 ∥∇f (x t ) -v t ∥ 2 as follows. E ∥∇f (x t+1 ) -v t+1 ∥ 2 = E   (1 -β)v t + 1 B 1 B1 k=1 ∇f (x t+1 ; ξ k t+1 ) -(1 -β) 1 B 1 B1 k=1 ∇f (x t ; ξ k t+1 ) -∇f (x t+1 ) 2   = E (1 -β)(v t -∇f (x t )) + β 1 B 1 B1 k=1 ∇f (x t ; ξ k t+1 ) -∇f (x t ) + ∇f (x t ) -∇f (x t+1 ) + 1 B 1 B1 k=1 ∇f (x t+1 ; ξ k t+1 ) - 1 B 1 B1 k=1 ∇f (x t ; ξ k t+1 ) 2   ≤ (1 -β) 2 E ∥v t -∇f (x t )∥ 2 + 2β 2 E   1 B 1 B1 k=1 ∇f (x t ; ξ k t+1 ) -∇f (x t ) 2   + 2E   1 B 1 B1 k=1 ∇f (x t+1 ; ξ k t+1 ) -∇f (x t ; ξ k t+1 ) 2   ≤ (1 -β)E ∥v t -∇f (x t )∥ 2 + 2β 2 σ 2 B 1 + 2L 2 ∥x t+1 -x t ∥ 2 B 1 ≤ (1 -β)E ∥v t -∇f (x t )∥ 2 + 2β 2 σ 2 B 1 + 2L 2 η 2 d B 1 ,where the first inequality is due to the factE β 1 B1 B1 k=1 ∇f (x t ; ξ k t+1 -∇f (x t ) + ∇f (x t ) -∇f (x t+1 ) + 1 B1 B1 k=1 ∇f (x t+1 ; ξ k t+1 ) -∇f (x t ; ξ k t+1 ) = 0, and (a + b) 2 ≤ 2a 2 + 2b 2 .Summing up and noticing that we use a batch size of B 0 in the first iteration, we can ensureE 1 T T t=1 ∥v t -∇f (x t )∥ 2 ≤ E ∥v 1 -∇f (x 1 )∥ 2 βT + 2σ 2 β B 1 + 2L 2 η 2 d βB 1 ≤ σ 2 B 0 βT + 2σ 2 β B 1 + 2L 2 η 2 d βB 1 (9)Incorporating the above into equation ( 8) and setting that β= O T -2/3 , η = O d -1/2 T -2/3 , B 0 = O T 1/3 , B 1 = O (1), we observe: E 1 T T t=1 ∥∇f (x t )∥ 1 ≤ ∆ f ηT + 2 √ d • E 1 T T t=1 ∥∇f (x t ) -v t ∥ 2 + ηLd 2 ≤ ∆ f ηT + 2 √ d • σ 2 B 0 βT + 2σ 2 β B 1 + 2L 2 η 2 d βB 1 + ηLd 2 = O (∆ f + σ + L) d 1/2 T 1/3 = O d 1/2T 1/3 , which finishes the proof of Theorem 1.To improve the convergence rate for finite-sum structures, we can reuse the results of equation ( 8), but bound the termE 1 T T t=1 ∥∇f (x t ) -v t ∥ 2 differently. Since v t = (1 -β)v t-1 + βh t + (1 - β) (∇f it (x t ) -∇f it (x t-1 )), where h t = ∇f it (x t ) -∇f it (x τ ) + ∇f (x τ ), we have: E ∥∇f (x t+1 ) -v t+1 ∥ 2 = E (1 -β)v t + βh t+1 + (1 -β)(∇f it+1 (x t+1 ) -∇f it+1 (x t )) -∇f (x t+1 ) 2 = E [∥(1 -β)(v t -∇f (x t )) + β (h t+1 -∇f (x t+1 )) +(1 -β) ∇f (x t ) -∇f (x t+1 ) + ∇f it+1 (x t+1 ) -∇f it+1 (x t ) ∥ 2 ≤ (1 -β) 2 E ∥v t -∇f (x t )∥ 2 + 2β 2 E ∥h t+1 -∇f (x t+1 )∥ 2 + 2(1 -β) 2 E ∥∇f (x t ) -∇f (x t+1 ) + ∇f it+1 (x t+1 ) -∇f it+1 (x t )∥ 2 ≤ (1 -β) 2 E ∥v t -∇f (x t )∥ 2 + 2β 2 E ∥h t+1 -∇f (x t+1 )∥ 2 + 2(1 -β) 2 E ∥∇f it+1 (x t+1 ) -∇f it+1 (x t )∥ 2 ≤ (1 -β)E ∥v t -∇f (x t )∥ 2 + 2β 2 E ∥h t+1 -∇f (x t+1 )∥ 2 + 2L 2 E ∥x t+1 -x t ∥ 2 ≤ (1 -β)E ∥v t -∇f (x t )∥ 2 + 2β 2 L 2 E ∥x t+1 -x τ ∥ 2 + 2L 2 E ∥x t+1 -x t ∥ 2 ,where the last inequality is due to the fact that:E ∥h t+1 -∇f (x t+1 )∥ 2 =E ∥∇f it+1 (x t+1 ) -∇f it+1 (x τ ) + ∇f (x τ ) -∇f (x t+1 )∥ 2 ≤E ∥∇f it+1 (x t+1 ) -∇f it+1 (x τ )∥ 2≤L 2 E ∥x t+1 -x τ ∥ 2 . By rearranging and summing up, we establish:E 1 T T t=1 ∥∇f (x t ) -v t ∥ 2 ≤ E ∥v 1 -∇f (x 1 )∥ 2 βT + 2βL 2 E 1 T T t=1 ∥x t+1 -x τ ∥ 2 + 2L 2 β E 1 T T t=1 ∥x t+1 -x t ∥ 2 ≤2βI 2 L 2 E 1 T T t=1 ∥x t+1 -x t ∥ 2 + 2L 2 β E 1 T T t=1 ∥x t+1 -x t ∥ 2 ≤2L 2 βI 2 + 1 β η 2 d,where we use full batch in the first iteration, and the second inequality is due to the fact that1 T T t=1 ∥x t+1 -x τ ∥ 2 = 1 T T t=1 t i=τ (x i+1 -x i ) 2 ≤ 1 T T t=1 I t i=τ ∥x i+1 -x i ∥ 2 ≤ I 2 T T t=1 ∥x t+1 -x t ∥ 2 .Incorporate the above into equation ( 8) and settingI = m, β = O 1 m , and η = O 1 m 1/4 d 1/2 T 1/2, we refine the bound as:E 1 T T t=1 ∥∇f (x t )∥ 1 ≤ ∆ f ηT + 2 √ d • E 1 T T t=1 ∥∇f (x t ) -v t ∥ 2 + ηLd 2 ≤ ∆ f ηT + 2 √ d • 2L 2 βI 2 + 1 β η 2 d + ηLd 2 = O (∆ f + L) m 1/4 d 1/2 T 1/2 = O m 1/4 d 1/2 T 1/2 .By setting β = 1 2 , we can ensure thatv j t ∞ ≤ R = 4G, since v j t ∞ = (1 -β)v j t-1 + ∇f (x t ; ξ j t ) -(1 -β)f (x t-1 ; ξ j t ) ∞ ≤(1 -β) v j t-1 ∞ + (2 -β)G ≤(1 -β) t-1 v j 1 ∞ + (2 -β)G t s=1 (1 -β) t-s ≤G + (2 -β)G β ≤ 4G = R.Since the overall objective function f (x) is L-smooth, we have the following:f (x t+1 ) ≤f (x t ) + ⟨∇f (x t ), x t+1 -x t ⟩ + L 2 ∥x t+1 -x t ∥ 2 ≤f (x t ) -η ∇f (x t ), Sign   1 n n j=1 S(v j t )   + η 2 Ld 2 =f (x t ) + η ∇f (x t ), Sign(∇f (x t )) -Sign   1 n n j=1 S(v j t )   -η ⟨∇f (x t ), Sign(∇f (x t ))⟩ + η 2 Ld 2 =f (x t ) + η ∇f (x t ), Sign(∇f (x t )) -Sign   1 n n j=1 S(v j t )   -η ∥∇f (x t )∥ 1 + η 2 Ld 2 ≤f (x t ) + 2ηR √ d ∇f (x t ) R - 1 n n j=1 S(v j t ) -η ∥∇f (x t )∥ 1 + η 2 Ld 2 ,(10) where the last inequality is because ofEQUATIONRearranging and taking the expectation over equation ( 10), we have:EQUATION)where the third inequality is due to the fact that (E [X])2 ≤ E X 2 , and the forth inequality is because of E S v j t = v j t R , as well as the S operation in each node is independent. Rearranging the terms and summing up, we have:1 T T i=1 E [∥∇f (x t )∥ 1 ] ≤ ∆ f ηT + 2 √ dE   1 T T i=1 ∇f (x t ) - 1 n n j=1 v j t   + 2dR √ n + ηLd 2 ≤ ∆ f ηT + 2 √ d E    1 T T i=1 ∇f (x t ) - 1 n n j=1 v j t 2    + 2dR √ n + ηLd 2 ,where the last inequality is due to Jensen's inequality.For each worker j, we have the following according to the definition of v j t :v j t+1 -∇f j (x t+1 ) = (1 -β) v j t -∇f j (x t ) + β ∇f j (x t+1 ; ξ j t+1 ) -∇f j (x t+1 ) + (1 -β) ∇f j (x t+1 ; ξ j t+1 ) -∇f j (x t ; ξ j t+1 ) + ∇f j (x t ) -∇f j (x t+1 ) .Summing over {n} and noting that ∇f (x) = 1 n n j=1 ∇f j (x), we can obtain:1 n n j=1 v j t+1 -∇f (x t+1 ) = 1 n n j=1 v j t+1 -∇f j (x t+1 ) =(1 -β) 1 n n j=1 v j t -∇f j (x t ) + β 1 n n j=1 ∇f j (x t+1 ; ξ j t+1 ) -∇f j (x t+1 ) + (1 -β) 1 n n j=1∇f j (x t+1 ; ξ j t+1 ) -∇f j (x t ; ξ j t+1 ) + ∇f j (x t ) -∇f j (x t+1 ) .Then we haveE    1 n n j=1 v j t+1 -∇f (x t+1 ) 2    ≤(1 -β) 2 E    1 n n j=1 v j t -∇f j (x t ) 2    + 2β 2 1 n 2 n j=1 E ∇f j (x t+1 ; ξ j t+1 ) -∇f j (x t+1 ) 2 + 2(1 -β) 2 1 n 2 n j=1 E ∇f j (x t+1 ; ξ j t+1 ) -∇f j (x t ; ξ j t+1 ) 2 ≤(1 -β)E    1 n n j=1 v j t -∇f j (x t ) 2    + 2β 2 σ 2 n + 2L 2 n ∥x t+1 -x t ∥ 2 ≤(1 -β)E    1 n n j=1 v j t -∇f (x t ) 2    + 2β 2 σ 2 n + 2L 2 η 2 d n .By summing up and rearranging, we observeEQUATIONFinally, by setting β = 1 2 and η = O T -1/2 d -1/2 , we ensure that1 T T i=1 ∥∇f (x t )∥ 1 ≤ ∆ f ηT + 2dR √ n + ηLd 2 + 2 √ d E    1 T T i=1 ∇f (x t ) - 1 n n j=1 v j t 2    ≤ ∆ f ηT + 2dR √ n + ηLd 2 + 2 √ d σ 2 nβT + 2σ 2 β n + 2L 2 η 2 d nβ = O d 1/2 T 1/2 + d n 1/2 .Due to the fact that the overall objective function f (x) is L-smooth, we have the following:f (x t+1 ) ≤f (x t ) + ⟨∇f (x t ), x t+1 -x t ⟩ + L 2 ∥x t+1 -x t ∥ 2 ≤f (x t ) -η ∇f (x t ), S 1   1 n n j=1 S G (v j t )   + η 2 Ld 2 =f (x t ) + η ∇f (x t ), ∇f (x t ) -S 1   1 n n j=1 S G (v j t )   -η ∥∇f (x t )∥ 2 + η 2 Ld 2 .Taking expectations leads to:EQUATIONRearranging the terms and summing up:1 T T i=1 E ∇f (x t )∥ 2 ≤ 2∆ f ηT + E    1 T T i=1 ∇f (x t ) - 1 n n j=1 vj t 2    + ηLd ≤ 2∆ f ηT + E   1 n n j=1 1 T T i=1 ∇f j (x t ) -vj t 2   + ηLd ≤ 2∆ f ηT + E   1 n n j=1 1 T T i=1 ∇f j (x t ) -Π G v j t 2   + ηLd ≤ 2∆ f ηT + E   1 n n j=1 1 T T i=1 ∇f j (x t ) -v j t 2   + ηLd.where the last inequality is due to the non-expansive property of the projection operation.For each worker j, according to the definition of v j t , we have:v j t+1 -∇f j (x t+1 ) = (1 -β) v j t -∇f j (x t ) + β ∇f j (x t+1 ; ξ j t+1 ) -∇f j (x t+1 )+ (1 -β) ∇f j (x t+1 ; ξ j t+1 ) -∇f j (x t ; ξ j t+1 ) + ∇f j (x t ) -∇f j (x t+1 ) .Then we haveE v j t+1 -∇f j (x t+1 ) 2 ≤(1 -β) 2 E v j t -∇f j (x t ) 2 + 2β 2 E ∇f j (x t+1 ; ξ j t+1 ) -∇f j (x t+1 ) 2 + 2(1 -β) 2 E ∇f j (x t+1 ; ξ j t+1 ) -∇f j (x t ; ξ j t+1 ) 2 ≤(1 -β)E v j t -∇f j (x t ) 2 + 2β 2 σ 2 + 2L 2 ∥x t+1 -x t ∥ 2 ≤(1 -β)E v j t -∇f j (x t ) 2 + 2β 2 σ 2 + 2L 2 η 2 d.As a result, we know thatE   1 n n j=1 1 T T t=1 v j t -∇f j (x t ) 2   ≤ σ 2 βT + 2σ 2 β + 2L 2 η 2 d β .Finally, combining the above and setting thatβ = O 1 T 1/2 , η = O 1 d 1/2 T 1/2, we obtain the final bound:E 1 T T i=1 ∥∇f (x t )∥ ≤ E 1 T T i=1 ∥∇f (x t )∥ 2 ≤ 2∆ f ηT + ηLd + σ 2 βT + 2σ 2 β + 2L 2 η 2 d β ≤ O d 1/2 (∆ f + L) + σ 2 + L 2 T 1/2 = O d 1/4 T 1/4 .In this section, we demonstrate that our proposed methods can maintain similar convergence rates under less stringent assumptions -expected α-symmetric generalized-smoothness [Chen et al., 2023] and affine variance [Faw et al., 2022] . We first detail these relaxed assumptions below.Assumption 1 ′ (Expected α-symmetric generalized-smoothness)E ξ ∥∇f (x; ξ) -∇f (y; ξ)∥ 2 ≤ ∥x -y∥ 2 E ξ L 0 + L 1 max θ∈[0,1] ∥∇f (x θ ; ξ)∥ α 2 ,where x θ := θx + (1 -θ)y, and 0 ≤ α ≤ 1.Assumption 2 ′ (Affine variance)E ξ ∥∇f (x; ξ) -∇f (x)∥ 2 ≤ Γ 2 ∥∇f (x)∥ 2 + Λ 2 .Remark: Assumption 1 ′ can be reduced to standard average smoothness (Assumption 1) when L 1 = 0. Note that α-symmetric generalized-smooth functions not only include asymmetric and Hessian-based generalized-smooth functions, but also contain high-order polynomials and exponential functions [Chen et al., 2023] . Moreover, affine variance is also weaker than Assumption 2 and can be reduced to it when Γ = 0.We then demonstrate that these relaxed conditions are sufficient for our algorithms to achieve the same convergence rate.Theorem 5 Under Assumptions 1 ′ and 2 ′ , by setting thatβ = O( d 1/3 T 2/3 ), η = O( 1 d 1/6 T 2/3 ), B 0 = O(1), B 1 = O(d), and denoting N as the samples used, our SSVR method guarantees:E [∥∇f (x τ )∥ 1 ] ≤ O d 1/2 N 1/3 .Furthermore, we introduce the following relaxed assumption for the finite-sum problem.Assumption 3 ′ (Generalized smoothness) For each i ∈ {1, 2, • • • , m}, we have∥∇f i (x) -∇f i (y)∥ ≤ ∥x -y∥ L 0 + L 1 max θ∈[0,1] ∥∇f (x θ )∥ α ,where x θ := θx + (1 -θ)y, and 0 ≤ α ≤ 1.This assumption is weaker than the standard Assumption 3. We validate that our SSVR-FS algorithm can still achieve similar convergence under this relaxed condition.Theorem 6 Under Assumption 3 ′ , by setting η = O(min{1 m 1/4 d 1/2 T 1/2 , 1 md }), β = O( 1 m ), and I = m, our SSVR-FS algorithm ensures:E [∥∇F (x φ )∥ 1 ] ≤ O m 1/4 d 1/2 T 1/2 + md T .Remark: When the iteration number T is large, the dominant term becomes O(m 1/4 d 1/2 T -1/2 ), which aligns with the results in Theorem 2.We first present some useful tools for analysis. According to Proposition 4 in Chen et al. [2023] , Assumption 1 ′ leads to the following lemmas.Lemma 1 For α ∈ (0, 1), we have:f (x t+1 ) ≤f (x t ) + ⟨∇f (x t ), x t+1 -x t ⟩ + 1 2 ∥x t+1 -x t ∥ 2 K 0 + K 1 ∥∇f (x t )∥ α + 2K 2 ∥x t+1 -x t ∥ α 1-α , where K 0 := L 0 2 α 2 1-α + 1 , K 1 := L 1 • 2 α 2 1-α • 3 α , K 2 := L 1 1-α 1 • 2 α 2 1-α • 3 α (1 -α) α 1-α .For α = 1, we also have:f (x t+1 ) ≤f (x t ) + ⟨∇f (x t ), x t+1 -x t ⟩ + 1 2 ∥x t+1 -x t ∥ 2 L 0 + L 1 ∥∇f (x t )∥ exp L 1 ∥x t+1 -x t ∥ ,Similarly, according to the Proposition 4 in Chen et al. [2023] , we have the following guarantees.Lemma 2 For α ∈ (0, 1), we have:E ξ ∥∇f (x; ξ) -∇f (y; ξ)∥ 2 ≤ ∥x -y∥ 2 K 0 + K 1 E ξ ∥∇f (y; ξ)∥ α + K 2 ∥x -y∥ α 1-α 2 .whereK 0 = 2 2-α 1-α L 0 , K 1 = 2 2-α 1-α L 1 , K 2 = (5L 1 ) 1 1-α .For α = 1, we also have:E ξ ∥∇f (x; ξ) -∇f (y; ξ)∥ 2 ≤ 2∥x -y∥ 2 (L 2 0 + 2L 2 1 E ξ ∥∇f (y; ξ)∥ 2 ) exp(12L 2 1 ∥x -y∥ 2 ).Then, we can begin our proof. For α ∈ (0, 1), according to Lemma 1, by setting η ≤ d -1 2 , we have:f (x t+1 ) ≤ f (x t ) + ⟨∇f (x t ), x t+1 -x t ⟩ + 1 2 ∥x t+1 -x t ∥ 2 K 0 + K 1 ∥∇f (x t )∥ α + 2K 2 ∥x t+1 -x t ∥ α 1-α ≤ f (x t ) + ⟨∇f (x t ), -η Sign(v t )⟩ + 1 2 ∥x t+1 -x t ∥ 2 (K 0 + K 1 (1 + ∥∇f (x t )∥) + 2K 2 ) ≤ f (x t ) + η ⟨∇f (x t ), Sign(∇f (x t )) -Sign(v t )⟩ -η ⟨∇f (x t ), Sign(∇f (x t ))⟩ + 1 2 ∥x t+1 -x t ∥ 2 (K 0 + K 1 (1 + ∥∇f (x t )∥) + 2K 2 ) = f (x t ) + η ⟨∇f (x t ), Sign(∇f (x t )) -Sign(v t )⟩ -η∥∇f (x t )∥ 1 + 1 2 ∥x t+1 -x t ∥ 2 ((K 0 + K 1 + 2K 2 ) + K 1 ∥∇f (x t )∥) ≤ f (x t ) + 2η √ d∥∇f (x t ) -v t ∥ -η∥∇f (x t )∥ 1 + η 2 d 2 ((K 0 + K 1 + 2K 2 ) + K 1 ∥∇f (x t )∥) ,where the second inequality is due to the fact that α < 1 and ∥x t+1 -x t ∥ 2 ≤ η 2 d ≤ 1.Rearranging and summing up, we then haveEQUATIONFor α = 1, according to Lemma 1 and setting η ≤ 1L1 √ d , we have f (x t+1 ) ≤f (x t ) + ⟨∇f (x t ), x t+1 -x t ⟩ + 1 2 ∥x t+1 -x t ∥ 2 L 0 + L 1 ∥∇f (x t )∥ exp L 1 ∥x t+1 -x t ∥ ≤f (x t ) + ⟨∇f (x t ), x t+1 -x t ⟩ + exp (1) 2 η 2 d (L 0 + L 1 ∥∇f (x t )∥) ≤f (x t ) + 2η √ d∥∇f (x t ) -v t ∥ -η∥∇f (x t )∥ 1 + 3η 2 d 2 (L 0 + L 1 ∥∇f (x t )∥) ,where the second inequality is due to L 1 ∥x t+1 -x t ∥ ≤ 1, and others follow the previous proof.Rearranging and summing up, we then haveE 1 T T t=1 ∥∇f (x t )∥ 1 ≤ ∆ f ηT + 2 √ d • E 1 T T t=1 ∥∇f (x t ) -v t ∥ + 3ηdL 0 2 + 3ηdL 1 2T E T t=1 ∥∇f (x t )∥ .By setting η ≤ 1 3dL1 , we can getEQUATION)Then we begin to bound the term E 1 T T t=1 ∥∇f (x t ) -v t ∥ . According to the definition of v t , we have:v t -∇f (x t ) =(1 -β) (v t-1 -∇f (x t-1 )) + β 1 B 1 B1 k=1 ∇f (x t ; ξ k t ) -∇f (x t ) + (1 -β) 1 B 1 B1 k=1 ∇f (x t ; ξ k t ) -∇f (x t-1 ; ξ k t ) + ∇f (x t-1 ) -∇f (x t ) . Denote that G t = 1 B1 B1 k=1 ∇f (x t ; ξ k t ) -∇f (x t-1 ; ξ k t ) + ∇f (x t-1 ) -∇f (x t ) and ∆ t = 1 B1 B1 k=1 ∇f (x t ; ξ k t ) -∇f (x t ). By summing up, we havev t -∇f (x t ) =(1 -β)(v t-1 -∇f (x t-1 )) + β∆ t + (1 -β)G t = • • • =(1 -β) t-1 (v 1 -∇f (x 1 )) + β t s=1 (1 -β) t-s ∆ s + (1 -β) t s=1 (1 -β) t-s G s .Thus, we can know thatE [∥v t -∇f (x t )∥] ≤(1 -β) t-1 E [∥v 1 -∇f (x 1 )∥] + βE t s=1 (1 -β) t-s ∆ s + (1 -β)E t s=1 (1 -β) t-s G s .Then we give the following two important lemmas, and their proofs can be found in Appendix E.1.1 and Appendix E.1.2, respectively.Lemma 3E t s=1 (1 -β) t-s ∆ s ≤ E    t-r s=1 (1 -β) t-s ∆ s 2 + 1 B 1 r s=1 (1 -β) 2s-2 Λ 2    + t s=t+1-r Γ √ B 1 (1 -β) t-s E [∥∇f (x s )∥]Lemma 4E t s=1 (1 -β) t-s G s ≤ E    t-r s=1 (1 -β) t-s G s 2 + 1 B 1 r s=1 2(1 -β) 2s-2 η 2 L 2 3 d    + √ 2dηL 4 √ B 1 t s=t+1-r (1 -β) t-s E [∥∇f (x s )∥]Using these lemmas and setting r = t, we then haveE t s=1 (1 -β) t-s ∆ s ≤ Λ 2 B 1 t s=1 (1 -β) 2s-2 + t s=1 Γ √ B 1 (1 -β) t-s E [∥∇f (x s )∥] ≤ Λ √ B 1 β + t s=1 Γ √ B 1 (1 -β) t-s E [∥∇f (x s )∥] E t s=1 (1 -β) t-s G s ≤ 2η 2 L 2 3 d B 1 t s=1 (1 -β) 2s-2 + √ 2dηL 4 √ B 1 t s=1 (1 -β) t-s E [∥∇f (x s )∥] ≤ √ 2dηL 3 √ B 1 β + √ 2dηL 4 √ B 1 t s=1 (1 -β) t-s E [∥∇f (x s )∥] .Combining above inequalities and setting β = η √ d, we deriveE 1 T T t=1 ∥∇f (x t ) -v t ∥ ≤ 1 T T t=1 (1 -β) t-1 E [∥v 1 -∇f (x 1 )∥] + β 1 T T t=1 E t s=1 (1 -β) t-s ∆ s + 1 T T t=1 E t s=1 (1 -β) t-s G s ≤ σ √ B 0 1 T T t=1 (1 -β) t-1 + Λ √ β √ B 1 + √ 2dηL 3 √ B 1 β + βΓ √ B 1 + √ 2dηL 4 √ B 1 1 T T t=1 t s=1 (1 -β) t-s E [∥∇f (x s )∥] ≤ σ βT √ B 0 + Λ √ β √ B 1 + √ 2dηL 3 √ B 1 β + βΓ √ B 1 + √ 2dηL 4 √ B 1 T i=1 (1 -β) i 1 T T t=1 E [∥∇f (x t )∥] ≤ σ βT √ B 0 + Λ √ β √ B 1 + √ 2dηL 3 √ B 1 β + Γ √ B 1 + √ 2dηL 4 √ B 1 β 1 T T t=1 E [∥∇f (x t )∥] ≤ σ ηT √ B 0 d + (Λ + √ 2L 3 )d 1/4 η 1/2 √ B 1 + Γ √ B 1 + √ 2L 4 √ B 1 1 T T t=1 E [∥∇f (x t )∥] .For α ∈ (0, 1), by setting thatB 0 = 1, B 1 ≥ max{256Γ 2 , 512L 2 4 }d, β = d 1/3 T 2/3 , η = 1 d 1/6 T 2/3 ,and suppose that iteration numberT ≥ O(d 2 ), then we can guarantee E 1 T T t=1 ∥∇f (x t )∥ 1 ≤ 2∆ f ηT + ηd(K 0 + K 1 + 2K 2 ) + 4σ ηT √ B 0 + 4(Λ + √ 2L 3 )d 3/4 η 1/2 √ B 1 + 4 √ dΓ √ B 1 + 4 √ 2dL 4 √ B 1 1 T T t=1 E [∥∇f (x t )∥] ≤O d 1/6 T 1/3 + 1 2 E 1 T T t=1 ∥∇f (x t )∥ ≤O d 1/6 T 1/3 + 1 2 E 1 T T t=1 ∥∇f (x t )∥ 1 , which indicates that E 1 T T t=1 ∥∇f (x t )∥ 1 ≤ O d 1/6 T 1/3 .Note that the batch size for each iteration isB 1 = O(d), by assuming that N = B 1 * T , we know that the convergence concerning N is O d 1/6 T 1/3 = O d 1/2 (B 1 T ) 1/3 = O d 1/2 N 1/3 .Similar results can be easily obtained for α = 1, i.e., we can also guarantee the following for α = 1:E 1 T T t=1 ∥∇f (x t )∥ 1 ≤ O d 1/6 N 1/3 .We prove this lemma by mathematical induction.1) When r = 0, we the following:E t s=1 (1 -β) t-s ∆ s = E    t s=1 (1 -β) t-s ∆ s 2    ,which satisfies the above lemma.2) Then, suppose the lemma holds for r = k. For r = k + 1, we haveE t s=1 (1 -β) t-s ∆ s ≤ E    t-k s=1 (1 -β) t-s ∆ s 2 + 1 B 1 k s=1 (1 -β) 2s-2 Λ 2    + t s=t+1-k Γ √ B 1 (1 -β) t-s E [∥∇f (x s )∥] = E   Eξ t-k    t-k-1 s=1 (1 -β) t-s ∆ s + (1 -β) k ∆ t-k 2 + 1 B 1 k s=1 (1 -β) 2s-2 Λ 2       + t s=t+1-k Γ √ B 1 (1 -β) t-s E [∥∇f (x s )∥] ≤ E   E ξ t-k t-k-1 s=1 (1 -β) t-s ∆ s + (1 -β) k ∆ t-k 2 + 1 B 1 k s=1 (1 -β) 2s-2 Λ 2    + t s=t+1-k Γ √ B 1 (1 -β) t-s E [∥∇f (x s )∥] ≤ E    t-k-1 s=1 (1 -β) t-s ∆ s 2 + E ξ t-k (1 -β) 2k ∥∆ t-k ∥ 2 + 1 B 1 k s=1 (1 -β) 2s-2 Λ 2    + t s=t+1-k Γ √ B 1 (1 -β) t-s E [∥∇f (x s )∥] ≤ E    t-k-1 s=1 (1 -β) t-s ∆ s 2 + (1 -β) 2k B 1 Λ 2 + Γ 2 ∥∇f (x t-k )∥ 2 + 1 B 1 k s=1 (1 -β) 2s-2 Λ 2    + t s=t+1-k Γ √ B 1 (1 -β) t-s E [∥∇f (x s )∥] ≤ E    t-k-1 s=1 (1 -β) t-s ∆ s 2 + 1 B 1 k+1 s=1 (1 -β) 2s-2 Λ 2    + (1 -β) k Γ √ B 1 E [∥∇f (x t-k )∥] + t s=t+1-k Γ √ B 1 (1 -β) t-s E [∥∇f (x s )∥] = E    t-k-1 s=1 (1 -β) t-s ∆ s 2 + 1 B 1 k+1 s=1 (1 -β) 2s-2 Λ 2    + t s=t-k Γ √ B 1 (1 -β) t-s E [∥∇f (x s )∥] ,where the second inequality is due to the Jensen Inequality.We prove this lemma by mathematical induction.1) When r = 0, we can easily proveEQUATION2) Suppose the inequality holds for r = k. Then, for r = k + 1, we deriveE t s=1 (1 -β) t-s G s ≤E    t-k s=1 (1 -β) t-s G s 2 + 1 B 1 k s=1 2(1 -β) 2s-2 η 2 L 2 3 d    + √ 2dηL 4 √ B 1 t s=t+1-k (1 -β) t-s E [∥∇f (x s )∥] =E   Eξ t-k    t-k s=1 (1 -β) t-s G s 2 + 1 B 1 k s=1 2(1 -β) 2s-2 η 2 L 2 3 d       + √ 2dηL 4 √ B 1 t s=t+1-k (1 -β) t-s E [∥∇f (x s )∥] =E   Eξ t-k    t-k-1 s=1 (1 -β) t-s G s + (1 -β) k G t-k 2 + 1 B 1 k s=1 2(1 -β) 2s-2 η 2 L 2 3 d       + √ 2dηL 4 √ B 1 t s=t+1-k (1 -β) t-s E [∥∇f (x s )∥] ≤E    E ξ t-k t-k-1 s=1 (1 -β) t-s G s + (1 -β) k G t-k 2 + 1 B 1 k s=1 2(1 -β) 2s-2 η 2 L 2 3 d    + √ 2dηL 4 √ B 1 t s=t+1-k (1 -β) t-s E [∥∇f (x s )∥] =E    t-k-1 s=1 (1 -β) t-s G s 2 + (1 -β) 2k E ξ t-k ∥G t-k ∥ 2 + 1 B 1 k s=1 2(1 -β) 2s-2 η 2 L 2 3 d    + √ 2dηL 4 √ B 1 t s=t+1-k (1 -β) t-s E [∥∇f (x s )∥] ≤E    t-k-1 s=1 (1 -β) t-s G s 2 + 2 B 1 (1 -β) 2k η 2 L 2 3 d + 1 B 1 k s=1 2(1 -β) 2s-2 η 2 L 2 3 d    + √ 2dηL 4 (1 -β) k √ B 1 E [∥∇f (x t-k )∥] + √ 2dηL 4 √ B 1 t s=t+1-k (1 -β) t-s E [∥∇f (x s )∥] =E    t-k-1 s=1 (1 -β) t-s G s 2 + 1 B 1 k+1 s=1 2(1 -β) 2s-2 η 2 L 2 3 d    + √ 2dηL 4 √ B 1 t s=t-k (1 -β) t-s E [∥∇f (x s )∥] ,where the third inequality is due to the following, for simplify we denote ξ t-k = ξ 1 t-k , ..., ξ B1 t-k :E ξ t-k ∥G t-k ∥ 2 = E ξ t-k   1 B 2 1 B1 j=1 ∇f (x t-k ; ξ j t-k ) -∇f (x t-k-1 ; ξ j t-k ) + ∇f (x t-k-1 ) -∇f (x t-k ) 2   = E ξ t-k   1 B 2 1 B1 j=1 ∇f (x t-k ; ξ j t-k ) -∇f (x t-k-1 ; ξ j t-k ) 2   + 1 B 1 ∥∇f (x t-k-1 ) -∇f (x t-k )∥ 2 -2E ξ t-k   1 B 2 1 B1 j=1 ∇f (x t-k ; ξ j t-k ) -∇f (x t-k-1 ; ξ j t-k ), ∇f (x t-k ) -∇f (x t-k-1 )   = E ξ t-k   1 B 2 1 B1 j=1 ∇f (x t-k ; ξ j t-k ) -∇f (x t-k-1 ; ξ j t-k ) 2   - 1 B 1 ∥∇f (x t-k-1 ) -∇f (x t-k )∥ 2 ≤ E ξ t-k   1 B 2 1 B1 j=1 ∇f (x t-k ; ξ j t-k ) -∇f (x t-k-1 ; ξ j t-k ) 2   For α ∈ (0, 1), denoting L 2 3 = K 0 + K 1 + K 2 2 + K 2 1 Λ 2 , and L 2 4 = K 2 1 (1 + Γ 2), we have:E ξ t-k   1 B 2 1 B1 j=1 ∇f (x t-k ; ξ j t-k ) -∇f (x t-k-1 ; ξ j t-k ) 2   ≤ 1 B 2 1 B1 j=1 ∥x t-k -x t-k-1 ∥ 2 K 0 + K 1 E ξ t-k ∇f (x t-k ; ξ j t-k ) α + K 2 ∥x t-k -x t-k-1 ∥ α 1-α 2 ≤ η 2 d B 2 1 B1 j=1 K 0 + K 1 + K 2 + K 1 E ξ t-k ∇f (x t-k ; ξ j t-k ) 2 ≤ 2η 2 d B 1 K 0 + K 1 + K 2 2 + 2η 2 d B 2 1 B1 j=1 K 2 1 E ξ t-k ∇f (x t-k ; ξ j t-k ) 2 ≤ 2η 2 d B 1 K 0 + K 1 + K 2 2 + 2η 2 d B 1 K 2 1 (1 + Γ 2 ) ∥∇f (x t-k )∥ + Λ 2 ≤ 2η 2 dL 2 3 B 1 + 2η 2 dL 2 4 B 1 ∥∇f (x t-k )∥ 2 ,where the second inequality holds by settingη ≤ d -1/2 such that ∥x t-k -x t-k-1 ∥ ≤ η √ d ≤ 1. For α = 1, denoting L 2 3 = 3 L 2 0 + 2L 2 1 Λ 2 , and L 2 4 = 6L 2 1 (1 + Γ 2), we have:E ξ t-k   1 B 2 1 B1 j=1 ∇f (x t-k ; ξ j t-k ) -∇f (x t-k-1 ; ξ j t-k ) 2   ≤ 2 B 2 1 B1 j=1 ∥x t-k -x t-k-1 ∥ 2 L 2 0 + 2L 2 1 E ξ t-k ∇f (x t-k ; ξ j t-k ) 2 exp 12L 2 1 ∥x t-k -x t-k-1 ∥ 2 ≤ 6η 2 d B 2 1 B1 j=1 L 2 0 + 2L 2 1 E ξ t-k ∇f (x t-k ; ξ j t-k ) 2 ≤ 6η 2 d B 1 L 2 0 + 2L 2 1 (1 + Γ 2 ) ∥∇f (x t-k )∥ 2 + Λ 2 ≤ 2η 2 dL 2 3 B 1 + 2η 2 dL 2 4 B 1 ∥∇f (x t-k )∥ 2 ,where the second inequality holds by setting η ≤1 √ 12L 2 1 d , such that ∥x t-k -x t-k-1 ∥ 2 ≤ 1 12L 2 1 .Similar to Lemma 1 and 2, we can have the following lemma for generalized individual smoothness.Lemma 5 For α ∈ (0, 1), generalized individual smoothness (Assumption 3 ′ ) leads tof i (x t+1 ) ≤f i (x t ) + ⟨∇f i (x t ), x t+1 -x t ⟩ + 1 2 ∥x t+1 -x t ∥ 2 K 0 + K 1 ∥∇f (x t )∥ α + 2K 2 ∥x t+1 -x t ∥ α 1-α ,as well as∥∇f i (x t+1 ) -∇f i (x t )∥ ≤ ∥x t+1 -x t ∥ K 0 + K 1 ∥∇f (x t )∥ α + K 2 ∥x t+1 -x t ∥ α 1-α .whereK 0 := L 0 2 α 2 1-α + 1 , K 1 := L 1 • 2 α 2 1-α • 3 α , K 2 := L 1 1-α 1 • 2 α 2 1-α • 3 α (1 -α) α 1-α . Lemma 6 For α = 1, generalized individual smoothness (Assumption 3 ′ ) leads to f i (x t+1 ) ≤f i (x t ) + ⟨∇f i (x t ), x t+1 -x t ⟩ + 1 2 ∥x t+1 -x t ∥ 2 L 0 + L 1 ∥∇f (x t )∥ exp L 1 ∥x t+1 -x t ∥ ,as well as∥∇f i (x t+1 ) -∇f i (x t )∥ ≤ ∥x t+1 -x t ∥ L 0 + L 1 ∥∇f (x t )∥ exp L 1 ∥x t+1 -x t ∥ .Then, we can begin our proof. For α ∈ (0, 1), according to Lemma 5, by setting η ≤ d -1 2 , we have:f (x t+1 ) -f (x t ) ≤ 1 m m i=1 f i (x t+1 ) - 1 m m i=1 f i (x t ) ≤ 1 m m i=1 ⟨∇f i (x t ), x t+1 -x t ⟩ + 1 2 ∥x t+1 -x t ∥ 2 K 0 + K 1 ∥∇f (x t )∥ α + 2K 2 ∥x t+1 -x t ∥ α 1-α ≤ ⟨∇f (x t ), -η Sign(v t )⟩ + 1 2 ∥x t+1 -x t ∥ 2 (K 0 + K 1 (1 + ∥∇f (x t )∥) + 2K 2 ) ≤ η ⟨∇f (x t ), Sign(∇f (x t )) -Sign(v t )⟩ -η ⟨∇f (x t ), Sign(∇f (x t ))⟩ + 1 2 ∥x t+1 -x t ∥ 2 (K 0 + K 1 (1 + ∥∇f (x t )∥) + 2K 2 ) = η ⟨∇f (x t ), Sign(∇f (x t )) -Sign(v t )⟩ -η∥∇f (x t )∥ 1 + 1 2 ∥x t+1 -x t ∥ 2 ((K 0 + K 1 + 2K 2 ) + K 1 ∥∇f (x t )∥) ≤ 2η √ d∥∇f (x t ) -v t ∥ -η∥∇f (x t )∥ 1 + η 2 d 2 ((K 0 + K 1 + 2K 2 ) + K 1 ∥∇f (x t )∥) ,where the second inequality is due to the fact that α < 1 and ∥x t+1 -x t ∥ 2 ≤ η 2 d ≤ 1.Rearranging and summing up, we then haveEQUATIONFor α = 1, by setting η ≤ 1 3dL1 , we can apply the very similar analysis and obtainEQUATIONThen we bound the termE 1 T T t=1 ∥∇f (x t ) -v t ∥ .According to the definition of v t , we have:v t -∇f (x t ) =(1 -β) (v t-1 -∇f (x t-1 )) + β (h t -∇f (x t )) + (1 -β) ∇f (x t ; ξ k t ) -∇f (x t-1 ; ξ k t ) + ∇f (x t-1 ) -∇f (x t ) Denote that G t = ∇f (x t ; ξ k t ) -∇f (x t-1 ; ξ k t ) + ∇f (x t-1 ) -∇f (x t ) and ∆ t = h t -∇f (x t ). By summing up, we havev t -∇f (x t ) =(1 -β)(v t-1 -∇f (x t-1 )) + β∆ t + (1 -β)G t =(1 -β) t-1 (v 1 -∇f (x 1 )) + β t s=1 (1 -β) t-s ∆ s + (1 -β) t s=1 (1 -β) t-s G s .Thus, we can know thatE [∥v t -∇f (x t )∥] ≤(1 -β) t-1 E [∥v 1 -∇f (x 1 )∥] + βE t s=1 (1 -β) t-s ∆ s + (1 -β)E t s=1 (1 -β) t-s G s ≤βE t s=1 (1 -β) t-s ∆ s + (1 -β)E t s=1 (1 -β) t-s G s ,where the first term vanishes since we use full batch in the first iteration.Then we give the following two important lemmas, and their proofs can be found in Appendix E.2.1 and Appendix E.2.2, respectively.Lemma 7E t s=1 (1 -β) t-s ∆ s ≤E    t-r s=1 (1 -β) t-s ∆ s 2 + r s=1 2(1 -β) 2s-2 η 2 I 2 L 2 5 d    + √ 2dηIL 6 t s=t+1-r (1 -β) t-s E [∥∇f (x s )∥]Lemma 8E t s=1 (1 -β) t-s G s ≤ E    t-r s=1 (1 -β) t-s G s 2 + r s=1 2(1 -β) 2s-2 η 2 L 2 7 d    + √ 2dηL 8 t s=t+1-r (1 -β) t-s E [∥∇f (x s )∥]Using these lemmas and setting r = t, we then haveE t s=1 (1 -β) t-s ∆ s ≤ 2η 2 I 2 L 2 5 d t s=1 (1 -β) 2s-2 + √ 2dηIL 6 t s=1 (1 -β) t-s E [∥∇f (x s )∥] ≤ √ 2dηIL 5 √ β + √ 2dηIL 6 t s=1 (1 -β) t-s E [∥∇f (x s )∥] ,as well asE t s=1 (1 -β) t-s G s ≤ 2η 2 L 2 7 d t s=1 (1 -β) 2s-2 + √ 2dηL 8 t s=1 (1 -β) t-s E [∥∇f (x s )∥] ≤ √ 2dηL 7 √ β + √ 2dηL 8 t s=1 (1 -β) t-s E [∥∇f (x s )∥] .Combining above inequalities and setting β = 1/m and I = m, we deriveE 1 T T t=1 ∥∇f (x t ) -v t ∥ ≤β 1 T T t=1 E t s=1 (1 -β) t-s ∆ s + 1 T T t=1 E t s=1 (1 -β) t-s G s ≤ 2dβηIL 5 + √ 2dηL 7 √ β + √ 2dηβIL 6 + √ 2dηL 8 1 T T t=1 t s=1 (1 -β) t-s E [∥∇f (x s )∥] ≤ √ 2dmη(L 5 + L 7 ) + √ 2dη(L 6 + L 8 ) T i=1 (1 -β) i 1 T T t=1 E [∥∇f (x t )∥] ≤ √ 2dmη(L 5 + L 7 ) + √ 2dηm(L 6 + L 8 ) 1 T T t=1 E [∥∇f (x t )∥] .For α ∈ (0, 1), by setting η = min1 m 1/4 d 1/2 T 1/2 , 1 8 √ 2md(L6+L8+1) , we can guarantee E 1 T T t=1 ∥∇f (x t )∥ 1 ≤ 2∆ f ηT + ηd(K 0 + K 1 + 2K 2 ) + 4d √ 2mη(L 5 + L 7 ) + 4d √ 2ηm(L 6 + L 8 ) 1 T T t=1 E [∥∇f (x t )∥] ≤O m 1/4 d 1/2 T 1/2 + md T + 1 2 E 1 T T t=1 ∥∇f (x t )∥ 1 , which indicates that E 1 T T t=1 ∥∇f (x t )∥ 1 ≤ O m 1/4 d 1/2 T 1/2 + md T. Similar results can be easily obtained for α = 1.We prove this lemma by mathematical induction. 1) When r = 0, we have the following:E t s=1 (1 -β) t-s ∆ s = E    t s=1 (1 -β) t-s ∆ s 2    ,which satisfies the above lemma.2) Then, suppose the lemma holds for r = k. For r = k + 1, we haveE t s=1 (1 -β) t-s ∆ s ≤E    t-k s=1 (1 -β) t-s ∆ s 2 + k s=1 2(1 -β) 2s-2 η 2 I 2 L 2 5 d    + √ 2dηIL 6 t s=t+1-k (1 -β) t-s E [∥∇f (x s )∥] =E   Ei t-k    t-k-1 s=1 (1 -β) t-s ∆ s + (1 -β) k ∆ t-k 2 + k s=1 2(1 -β) 2s-2 η 2 I 2 L 2 5 d       + √ 2dηIL 6 t s=t+1-k (1 -β) t-s E [∥∇f (x s )∥] ≤E    E i t-k t-k-1 s=1 (1 -β) t-s ∆ s + (1 -β) k ∆ t-k 2 + k s=1 2(1 -β) 2s-2 η 2 I 2 L 2 5 d    + √ 2dηIL 6 t s=t+1-k (1 -β) t-s E [∥∇f (x s )∥] =E    t-k-1 s=1 (1 -β) t-s ∆ s 2 + (1 -β) 2k E i t-k ∥∆ t-k ∥ 2 + k s=1 2(1 -β) 2s-2 η 2 I 2 L 2 5 d    + √ 2dηIL 6 t s=t+1-k (1 -β) t-s E [∥∇f (x s )∥] ≤E    t-k-1 s=1 (1 -β) t-s ∆ s 2 + 2(1 -β) 2k η 2 I 2 L 2 5 d + k s=1 2(1 -β) 2s-2 η 2 I 2 L 2 5 d    + √ 2dηIL 6 (1 -β) k E [∥∇f (x t-k )∥] + √ 2dηIL 6 t s=t+1-k (1 -β) t-s E [∥∇f (x s )∥] =E    t-k-1 s=1 (1 -β) t-s ∆ s 2 + k+1 s=1 2(1 -β) 2s-2 η 2 I 2 L 2 5 d    + √ 2dηIL 6 t s=t-k (1 -β) t-s E [∥∇f (x s )∥] ,where the third inequality is due to the following:E i t-k ∥∆ t-k ∥ 2 = E i t-k ∇f i t-k (x t-k ) -∇f i t-k (x τ ) + ∇f (x τ ) -∇f (x t-k ) 2 ≤ E i t-k ∇f i t-k (x t-k ) -∇f i t-k (x τ ) 2For α ∈ (0, 1), denoting thatL 2 5 = K 0 + K 1 + K 2 2 ,L 2 6 = K 2 1 , we have:E i t-k ∇f i t-k (x t-k ) -∇f i t-k (x τ ) 2 ≤ ∥x t-k -x τ ∥ 2 K 0 + K 1 ∥∇f (x t-k )∥ α + K 2 ∥x t-k -x τ ∥ α 1-α 2 ≤ η 2 I 2 d (K 0 + K 1 + K 2 + K 1 ∥∇f (x t-k )∥) 2 ≤ 2η 2 I 2 d (K 0 + K 1 + K 2 ) 2 + 2η 2 I 2 dK 2 1 ∥∇f (x t-k )∥ 2 ≤ 2η 2 dI 2 L 2 5 + 2η 2 I 2 dL 2 6 ∥∇f (x t-k )∥ 2 ,where the second inequality holds by settingη ≤ I -1 d -1/2 such that ∥x t-k -x τ ∥ ≤ ηI √ d ≤ 1.For α = 1, denoting thatL 2 5 = 9L 2 0 L 2 6 = 9L 21 , we have:E i t-k ∇f i t-k (x t-k ) -∇f i t-k (x τ ) 2 ≤ 2 ∥x t-k -x τ ∥ 2 L 2 0 + L 2 1 ∥∇f (x t-k )∥ 2 exp L 2 1 ∥x t-k -x τ ∥ 2 2 ≤ 18η 2 dI 2 L 2 0 + L 2 1 ∥∇f (x t-k )∥ 2 ≤ 2η 2 I 2 dL 2 5 + 2η 2 I 2 dL 2 6 ∥∇f (x t-k )∥ 2 ,where the second inequality holds by settingη ≤ 1 L 2 1 I 2 d, such that we have∥x t-k -x t-k-1 ∥ 2 ≤ η 2 I 2 d ≤ 1 L 2 1 .We prove this lemma by mathematical induction.1) When r = 0, we can easily prove thatEQUATION2) Suppose the inequality holds for r = k. Then, for r = k + 1, we deriveE t s=1 (1 -β) t-s G s ≤E    t-k s=1 (1 -β) t-s G s 2 + k s=1 2(1 -β) 2s-2 η 2 L 2 7 d    + √ 2dηL 8 t s=t+1-k (1 -β) t-s E [∥∇f (x s )∥] ≤E   Ei t-k    t-k s=1 (1 -β) t-s G s 2 + k s=1 2(1 -β) 2s-2 η 2 L 2 7 d       + √ 2dηL 8 t s=t+1-k (1 -β) t-s E [∥∇f (x s )∥] =E   Ei t-k    t-k-1 s=1 (1 -β) t-s G s + (1 -β) k G t-k 2 + k s=1 2(1 -β) 2s-2 η 2 L 2 7 d       + √ 2dηL 8 t s=t+1-k (1 -β) t-s E [∥∇f (x s )∥] ≤E    E i t-k t-k-1 s=1 (1 -β) t-s G s + (1 -β) k G t-k 2 + k s=1 2(1 -β) 2s-2 η 2 L 2 7 d    + √ 2dηL 8 t s=t+1-k (1 -β) t-s E [∥∇f (x s )∥] =E    t-k-1 s=1 (1 -β) t-s G s 2 + (1 -β) 2k E i t-k ∥G t-k ∥ 2 + k s=1 2(1 -β) 2s-2 η 2 L 2 7 d    + √ 2dηL 8 t s=t+1-k (1 -β) t-s E [∥∇f (x s )∥] ≤E    t-k-1 s=1 (1 -β) t-s G s 2 + 2(1 -β) 2k η 2 L 2 7 d + k s=1 2(1 -β) 2s-2 η 2 L 2 7 d    + √ 2dηL 8 (1 -β) k E [∥∇f (x t-k )∥] + √ 2dηL 8 t s=t+1-k (1 -β) t-s E [∥∇f (x s )∥] =E    t-k-1 s=1 (1 -β) t-s G s 2 + k+1 s=1 2(1 -β) 2s-2 η 2 L 2 7 d    + √ 2dηL 8 t s=t-k (1 -β) t-s E [∥∇f (x s )∥] ,where the third inequality is due to the following:E i t-k ∥G t-k ∥ 2 = E i t-k ∇f i t-k (x t-k ) -∇f i t-k (x t-k-1 ) + ∇f (x t-k-1 ) -∇f (x t-k ) 2 = E i t-k ∇f i t-k (x t-k ) -∇f i t-k (x t-k-1 ) 2 + ∥∇f (x t-k-1 ) -∇f (x t-k )∥ 2 -2E i t-k ∇f i t-k (x t-k ) -∇f i t-k (x t-k-1 ), ∇f (x t-k ) -∇f (x t-k-1 ) = E i t-k ∇f i t-k (x t-k ) -∇f i t-k (x t-k-1 ) 2 -∥∇f (x t-k-1 ) -∇f (x t-k )∥ 2 ≤ E i t-k ∇f i t-k (x t-k ) -∇f i t-k (x t-k-1 ) 2 For α ∈ (0, 1), denoting L 2 7 = K 0 + K 1 + K 2 2, and L 2 8 = K 2 1 , we have:E i t-k ∇f i t-k (x t-k ) -∇f i t-k (x t-k-1 ) 2 ≤ ∥x t-k -x t-k-1 ∥ 2 K 0 + K 1 ∥∇f (x t-k )∥ α + K 2 ∥x t-k -x t-k-1 ∥ α 1-α 2 ≤ η 2 d (K 0 + K 1 + K 2 + K 1 ∥∇f (x t-k )∥) 2 ≤ 2η 2 d (K 0 + K 1 + K 2 ) 2 + 2η 2 dK 2 1 ∥∇f (x t-k )∥ 2 ≤ 2η 2 dL 2 7 + 2η 2 dL 2 8 ∥∇f (x t-k )∥ 2 ,where the second inequality holds by setting ηEQUATION1 , we have:E i t-k ∇f i t-k (x t-k ) -∇f i t-k (x t-k-1 ) 2 ≤ 2 ∥x t-k -x t-k-1 ∥ 2 L 2 0 + L 2 1 ∥∇f (x t-k )∥ 2 exp L 2 1 ∥x t-k -x t-k-1 ∥ 2 2 ≤ 18η 2 d L 2 0 + 2L 2 1 ∥∇f (x t-k )∥ 2 ≤ 2η 2 dL 2 7 + 2η 2 dL 2 8 ∥∇f (x t-k )∥ 2 ,where the second inequality holds by setting η ≤1 √ L 2 1 d, such that we have ∥xt-k -x t-k-1 ∥ 2 ≤ η 2 d ≤ 1 L 2 1 .In this section, we present additional experiments on the CIFAR-10 dataset to validate whether the proposed SSVR method is sensitive to hyper-parameters such as learning rate η, momentum parameter β, and batch size. Specifically, we fix the value of β as 0.5 and try different learning rates from the set {5e -3, 1e -3, 5e -4, 1e -4, 5e -5}. Then, we fix the learning rate as 1e -3 and enumerate β from the set {0.3, 0.5, 0.7, 0.9, 0.99}. The results are reported in Figure 3 and Figure 4 , respectively, which indicate that our method is insensitive to the choice of hyper-parameters within a certain range. Finally, we also try different batch sizes from the set {64, 128, 256, 512}, and the results are shown in Figure 5 . It can be seen that our algorithm does not necessitate large batches for convergence and is not sensitive to variations in batch sizes. 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). 10. Broader Impacts Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed? Answer: [NA] Justification: This is primarily a theoretical paper with no potential negative social impact. Guidelines:• 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).Note that most lower bounds are measured in the l2-norm, and there are no existing lower bounds for the l1-norm to the best of our knowledge. However, since ∥x∥ 1 ≈ √ d ∥x∥ 2 for dense vector x, the additional O( √ d) term is acceptable when considering the l1-norm. This is also the case for our methods (measured in the l1-norm) and the lower bounds for variance reduction methods (for l2-norm).The original convergence rate is measured under the squared l2-norm, and we convert it to the rate under the l2-norm criterion for a fair comparison. 


Decision: This paper investigates the convergence of sign-SGD algorithms, with a focus on enhancing their performance through variance reduction techniques. The authors successfully improve the convergence rate of sign-SGD for stochastic optimization from \(O(d^{1/2}T^{-1/4})\) to \(O(d^{1/2}T^{-1/3})\). For finite-sum problems, the proposed approach further accelerates convergence by a factor of \(m^{1/4}\), where \(m\) represents the number of individual functions. Additionally, the paper demonstrates that this technique can be effectively applied to distributed settings, improving upon existing results in the literature.

The paper is generally well-written, presenting a solid contribution in the form of improved convergence rates over established methods. Following an in-depth discussion between the authors and reviewers, consensus has been reached that the paper is suitable for publication. I am also recommending acceptance of this paper. However, I have some additional comments that I believe could enhance the paper’s clarity and presentation, which I encourage the authors to consider in their final version:

1. **Clarification of Convergence Rate Definitions:**  
   The paper would benefit from a more detailed description of the convergence rate definitions used. Specifically, it would be helpful if the authors could clearly delineate the differences between this work and others in terms of convergence rate definitions—whether in terms of the norm of the gradient or the squared norm, and whether considering \(l_1\) or \(l_2\) norms. This clarification will aid readers in making direct comparisons between the results in this paper and those in related works.

2. **Discussion on Upper vs. Lower Bounds:**  
   In the introduction, the authors discuss lower bounds for stochastic gradient methods and finite-sum optimization. However, the upper bounds provided in the theoretical section are primarily for the \(l_1\) norm, while the literature often discusses upper bounds in terms of the \(l_2\) norm. The authors should add a discussion that addresses this discrepancy, clarifying that it is acceptable that there is no exact lower bound for the sign-based optimization setting.

3. **Comprehensive Discussion of Variance Reduction Techniques:**  
   Since SVRG plays a pivotal role in achieving the improved convergence results in this paper, the related work section would benefit from a more thorough discussion of key variance reduction algorithms such as SCSG [1], SDCA [2, 4], nonconvex SVRG [3], SNVRG [5], etc. 

[1] Lei, Lihua, and Michael Jordan. "Less than a single pass: Stochastically controlled stochastic gradient." Artificial Intelligence and Statistics. PMLR, 2017.

[2] Shalev-Shwartz, Shai, and Tong Zhang. "Stochastic dual coordinate ascent methods for regularized loss minimization." Journal of Machine Learning Research 14.1 (2013).

[3] Reddi, Sashank J., et al. "Stochastic variance reduction for nonconvex optimization." International conference on machine learning. PMLR, 2016.

[4] Richtárik, Peter, and Martin Takáč. "Iteration complexity of randomized block-coordinate descent methods for minimizing a composite function." Mathematical Programming 144.1 (2014): 1-38.

[5] Zhou, Dongruo, et al. "Stochastic nested variance reduction for nonconvex optimization." Journal of machine learning research 21.103 (2020): 1-63.

Reviews: review: paper_summary: paper_summary: This paper investigates the stochastic optimization problem and a special case of finite sum optimization problem. They propose the
Sign-based Stochastic Variance Reduction (SSVR) method, leveraging variance reduction techiniques, which improves the convergence rate of Sign stochastic gradient descent (signSGD) algorithm.
Further, they also study the heterogeneous majority vote in distributed settings and introduce two novel algorithms, whose convergence rates improve the state-of-the-art.
Numerical experiments prove the efficiency of their algorithms and support their theoretical guarantees.
strength_weakness: strength_weakness: Some places of the paper need further clarification.
questions: questions: 1. How did you get the convergence rate of $O(m^{1/2}d^{1/2}T^{−1/2})$ for the SignSVRG of y? It seems that in their paper they claim $O(T^{-1/2}) convergence rate without $m$ and $n$ in the numerator. Did you calculate this result by yourself? Please provide discussions.

2. There is another paper studying sign-based optimization (Qin et al. (2023)), the setting studied in this paper is the same as your problem (2).  Can you also discuss this paper and compare the results?


References:
1. E. Chzhen and S. Schechtman. SignSVRG: fixing SignSGD via variance reduction. ArXiv e-prints, 317 arXiv:2305.13187, 2023.
2. Qin, Z., Liu, Z., & Xu, P. (2023). Convergence of Sign-based Random Reshuffling Algorithms for Nonconvex Optimization. arXiv preprint arXiv:2310.15976.
limitations: limitations: Yes.
paper_summary: paper_summary: This paper considers application of variance reduction to sign-based optimization, i.e., a setting where the algorithm can only access to $\mathrm{sign}(\nabla f(x^t; \xi^t))$ at step $t$, in the smooth and nonconvex setting.
Because $\\{\pm 1\\}$-valued vectors can be transmitted more efficiently than real-valued vectors, the sign-based optimization is considered to be useful especially in distributed optimization.
While a naive SGD-type algorithm previously achieved the convergence rate of $O(d^\frac12 T^{-\frac14})$, the proposed variance-reduced algorithm yields the rate of $O(d^\frac12 T^{-\frac13})$ (measured by the $L_1$-norm of the gradient.)
Next, they applied the algorithm to finite-sum optimization (with $m$ components) and obtained the convergence rate of $O(m^\frac14 d^\frac12 T^{-\frac14})$. 
Finally, they applied these results to distributed optimization (the server received $\\{\pm 1\\}$-valued gradient estimators from clients), with an additional technique called \textit{majority vote} to obtain unbiased gradient estimators, and obtained improved communication complexities.
strength_weakness: strength_weakness: ### **Technical novelty**

Variance reduction itself is quite common in the optimization field so I am afraid that this paper might be a bit incremental. Section 3.3 (Results under weaker assumptions) looks redundant given Theorem 2, as there are no technical difficulty about weakening the assumption described. 

### **Hyperparameter choice**

Although this is a common criticism, most variance reduction algorithm require a very careful hyperparameter tuning. It looks like the theoretical guarantees of the proposed algorithms also require a specific set of hyperparameters, and I am not confident in the robustness of the proposed algorithms to the hyperparameters.
questions: questions: ### **On additional terms**

- Why is the term of $md/T$ required in Theorem 4?

- Why is the term of $d n^{-\frac12}$ required in Theorem 5?

### **On the difficulties to further improve the convergence rates**

- In theorem 6, why cannot you improve the convergence rate into $\mathrm{poly}(d) T^{-\frac13}$?

### **Assumptions**

- Is it possible to weaken Assumption 3 to averaged smoothness, i.e., $\frac1m \sum_{i=1}^m \\|\nabla f_i(x)-\nabla f_i(y)\\|^2 \leq L^2 \\|x-y\\|^2$
limitations: limitations: N/A
paper_summary: paper_summary: The paper introduces an enhanced sign-based method called SSVR, designed to improve the convergence rate of signSGD with variance reduction techniques. SSVR achieves an improved convergence rate of $O(d^{1/2}T^{-1/3})$ for stochastic non-convex functions. For non-convex finite-sum optimization problems, the SSVR-FS method demonstrates an improved convergence rate of $ O(m^{1/4}d^{1/2}T^{-1/2}) $. To address heterogeneous data in distributed settings, the authors present novel algorithms that outperform existing methods in terms of convergence rates. The effectiveness of these proposed methods is validated through numerical experiments.
strength_weakness: strength_weakness: 1. The authors introduce the bounded gradient assumption (Assumption 4) for distributed settings. Why is this assumption necessary? Is it commonly used in previous literature for this setting?
2. The description of Algorithm 2 is not very rigorous. The variable $\tau$ in step 4 and $\tau$ in step 10 seems different but used the same notation.
3. Some typos:
    - Line 98: "signed-based" --> "sign-based."
    - Line 247: "Assumption 5′" --> "Assumption 5" or "Assumption 4′."
    - Line 487: "$S(v_t^j)$" --> "$S_{R}(v_t^j)$."
questions: questions: 1. Does previous work also require similar bounded gradient assumptions for the majority vote in heterogeneous settings? 
2. Is the performance of the proposed method sensitive to variations in batch size?
limitations: limitations: This paper does not present negative societal impacts.
paper_summary: paper_summary: This paper introduces the Sign-based Stochastic Variance Reduction (SSVR) algorithm, which enhances the convergence rate of the traditional signSGD method. By incorporating variance reduction techniques with sign-based updates, the authors achieve a convergence rate of $O(d^{1/2}T^{-1/3})$ for general stochastic optimization and $O(m^{1/4}d^{1/2}T^{-1/2})$ for finite-sum problems. Additionally, the paper proposes novel algorithms for distributed environments by introducing the unbiased sign operation, resulting in superior convergence rates for heterogeneous data. Numerical experiments further validate the effectiveness of the proposed methods.
strength_weakness: strength_weakness: 1. Given that signSGD methods typically require large batch sizes to ensure convergence, the authors should include more experimental results to demonstrate the dependency on batch size for the proposed methods. This would clarify whether the proposed method also necessitates large batches for convergence in practice.
2. The authors introduce the stochastic unbiased sign operation in Definition 1, which differs from the traditional sign operation. It would be beneficial to provide a more detailed explanation of their differences and specify when the sign operation is preferred.
3. There are some typos in the paper, as listed below:
   - Page 2, Lines 44 and 45: $T^{1/2}$ should be $T^{-1/2}$.
   - Page 3, Line 98: "signed-based" should be "sign-based".
   - Page 5, Algorithm 2, Step 4: "set $t=\tau$" should be "set $\tau=t$".
   - Page 9, Figure 2: "SSVR" should be "SSVR-MV".
   - Page 28, Line 482: $f(x_t;\xi_t^j)$ should be $\nabla f(x_t;\xi_t^j)$.
questions: questions: 1. Can you provide a more detailed discussion about the design of equation (3), especially the error correction term? The authors claim that "This gap can be effectively mitigated by the error correction term we introduced in equation (3)," but it remains unclear how it works based solely on the content in the main body.
2. Could you provide additional experimental results to demonstrate the performance of the proposed SSVR method with different batch sizes?
limitations: limitations: The paper is theoretical and does not present potential negative societal impacts.

Comments: title: final rating
Please read the authors’ rebuttal and reply by August 13, 2024 (11:59 PM, AOE time)
comment: Thank you for the detailed responses. They have satisfactorily addressed all my concerns, and I am inclined to recommend acceptance.
Thank you very much for your detailed response. I would like to keep my score. 

Best,
reviewer.
Thank you very much for your kind reply! We will revise our paper according to the constructive reviews.

Best regards,

Authors
Thanks for the rebuttal. It addressed all my concerns. I will increase my score.
Thank the authors for the rely. It addresses all my questions and concerns. I'm willing to keep my score.
Dear Reviewers,

Thank you for your hard work during the review process. The authors have responded to your initial reviews. **If you haven’t already done so, please take the time to carefully read their responses.** It is crucial for the authors to receive acknowledgment of your review by the deadline for the author-reviewer discussion period, which is August 13, 2024 (11:59 PM, Anywhere on Earth). Please address any points of disagreement with the authors as early as possible.

Best,

Your AC