Membership inference attack (MIA) has become one of the most widely used and effective methods for evaluating the privacy risks of machine learning models. This attack aims to determine whether a specific sample is part of the model's training set by analyzing the model's output. While traditional membership inference attacks focus on leveraging the model’s posterior output, such as confidence on the target sample, we propose IMIA, a novel attack strategy that utilizes the process of generating adversarial samples to infer membership. We propose to infer the member properties of the target sample using the number of iterations required to generate its adversarial sample. We conduct experiments across multiple models and datasets, and our results demonstrate that the number of iterations for generating an adversarial sample is a reliable feature for membership inference, achieving strong performance both in black-box and white-box attack scenarios. This work provides a new perspective for evaluating model privacy and highlights the potential of adversarial example-based features for privacy leakage assessment.

This article considers the decentralized composite optimization problem. We propose a novel decentralized variance-reduction proximal-gradient algorithmic framework, called PMGT-VR, which combines several techniques, including multi-consensus, gradient tracking, and variance reduction. The proposed framework imitates centralized algorithms and algorithms under this framework achieve convergence rates similar to that of their centralized counterparts. We also describe and analyze two representative algorithms, PMGT-SAGA and PMGT-LSVRG, and compare them to existing state-of-the-art proximal algorithms. To the best of our knowledge, PMGT-VR is the first linearly convergent decentralized stochastic algorithm that can solve decentralized composite optimization problems. Numerical experiments are provided to demonstrate the effectiveness of the proposed algorithms.

As large language models (LLMs) are increasingly being integrated into various real-world applications, the identification of their vulnerabilities to jailbreaking attacks becomes an essential component of ensuring the safety and reliability of LLMs. Previous studies have developed LLM assistants, known as the adversarial prompter, to automatically generate suffixes that manipulate target LLMs into generating harmful and undesirable outputs. However, these approaches often suffer from low performance or generate semantically meaningless prompts, which can be easily identified by perplexity-based defenses. In this paper, we introduce a novel two-stage method, $\texttt{ProAdvPrompter}$, that significantly improves the performance of adversarial prompters. In $\texttt{ProAdvPrompter}$, the first stage (Exploration) utilizes the loss information to guide the adversarial prompter in generating suffixes that are more likely to elicit harmful responses. Then the second stage (Exploitation) iteratively fine-tunes the prompter using high-quality generated adversarial suffixes to further boost performance. Additionally, we incorporate the prompt template to aid in the Exploration stage and propose a filtering mechanism to accelerate the training process in the Exploitation stage. We evaluate $\texttt{ProAdvPrompter}$ against the well-aligned LLMs (i.e., Llama2-Chat-7B and Llama3-chat-8B), achieving attack success rates of 99.68% and 97.12% respectively after 10 trials on the AdvBench dataset, thereby enhancing performance by $\sim 2$ times compared to previous works. Moreover, $\texttt{ProAdvPrompter}$ reduces training time by 20% on Llama3-Instruct-8B, generates more generalized adversarial suffixes, and demonstrates resilience against the perplexity defense. An ablation study further evaluates the effects of key components in $\texttt{ProAdvPrompter}$ (the prompt template and the filtering mechanism).

Fine-tuning large language models (LLMs) is necessary for specific downstream tasks, but classic first-order optimizer entails prohibitive GPU memory because of the back propagation. Recent works such as MeZO have turned to zeroth-order optimizers for fine-tuning, which reduce substantial memory by using two forward passes. However, heterogeneous curvatures across different parameter dimensions in LLMs often cause model convergence instability or even failure. In this work, we propose HiZOO, a diagonal Hessian informed Zeroth-Order Optimizer , which is the first work to leverage the diagonal Hessian to enhance ZOO for fine-tuning LLMs. We provide theoretical proof for HiZOO and visualize the optimization trajectories on test functions to illustrate how it improves convergence in handling heterogeneous curvatures. Extensive experiments on various models (RoBERTa, OPT, Phi-2 and LLama3, with 350M$\sim$66B parameters) indicate that HiZOO significantly reduces training steps and enhances model accuracy, while keeping the memory advantage of ZOO. For example, on SST2 task HiZOO achieves $8\times$ speedup and better accuracy over MeZO across different models. We also propose HiZOO-L, which reduces the Hessian memory cost to 10\% of the MeZO, while maintaining almost same performance. Compared with ZO-Adam, HiZOO-L achieves a 4.3\% improvement, just using 50\% of the GPU memory. Code is available at https://anonymous.4open.science/r/HiZOO-27F8.

In this paper, we study the distributed convex-concave finite-sum minimax optimization over the network, and a decentralized variance-reduced optimistic gradient method with stochastic mini-batch sizes (DIVERSE) is proposed. For the strongly-convex-strongly-concave objective, it is shown that DIVERSE can achieve a linear convergence rate that depends on the global smoothness parameters, yielding sharper computation and communication complexity bounds than existing results. Furthermore, we also establish the lower complexity bounds, which show that our upper bounds are optimal up to a logarithmic factor in terms of the local incremental first-order oracle calls, the computation rounds, and the communication rounds. Numerical experiments demonstrate that our algorithm outperforms existing methods in practice.

This paper concentrates on decentralized composite optimization for strongly convex functions. Specifically, we first study the case where each local objective function $f_i(x)$ held by agent $i$ is $L$-smooth and convex, while the regularization term $g(x)$ is $\mu$-strongly convex. For this problem class, we propose the first decentralized algorithm that simultaneously achieves the optimal computation and communication complexities. Furthermore, we extend our algorithm to two broader scenarios. In the first extension, when each $f_i(x)$ is $L$-smooth and $\mu$-strongly convex while $g(x)$ is merely convex, our algorithm continues to attain the optimal complexities. In the second extension, we show that under time-varying communication networks, our algorithm matches the lower bounds on decentralized optimization established in Kovalev et al. (2021). Finally, extensive experiments validate both the computational and communication efficiency of the proposed algorithms.

Zeroth-order optimization algorithms recently emerge as a popular research theme in optimization and machine learning, playing important roles in many deep-learning related tasks such as black-box adversarial attack, deep reinforcement learning, as well as hyper-parameter tuning. Mainstream zeroth-order optimization algorithms, however, concentrate on exploiting zeroth-order-estimated first-order gradient information of the objective landscape. In this paper, we propose a novel meta-algorithm called Hessian-Aware Zeroth-Order (ZOHA) optimization algorithm, which utilizes several canonical variants of zeroth-order-estimated second-order Hessian information of the objective: power-method-based, and Gaussian-smoothing-based. We conclude theoretically that ZOHA enjoys an improved convergence rate compared with existing work without incorporating in zeroth-order optimization second-order Hessian information. Empirical studies on logistic regression as well as the black-box adversarial attack are provided to validate the effectiveness and improved success rates with reduced query complexity of the zeroth-order oracle.

This paper studies the stochastic nonconvex-strongly-concave minimax optimization over a multi-agent network. We propose an efficient algorithm, called Decentralized Recursive gradient descEnt Ascent Method (DREAM), which achieves the best-known theoretical guarantee for finding the $\epsilon$-stationary points. Concretely, it requires $\mathcal{O}(\min (\kappa^3\epsilon^{-3},\kappa^2 \sqrt{N} \epsilon^{-2} ))$ stochastic first-order oracle (SFO) calls and $\tilde \mathcal O(\kappa^2 \epsilon^{-2})$ communication rounds, where $\kappa$ is the condition number and $N$ is the total number of individual functions. Our numerical experiments also validate the superiority of DREAM over previous methods.

The conjugate gradient method is a crucial first-order optimization method that generally converges faster than the steepest descent method, and its computational cost is much lower than that of second-order methods. However, while various types of conjugate gradient methods have been studied in Euclidean spaces and on Riemannian manifolds, there is little study for those in distributed scenarios. This paper proposes a decentralized Riemannian conjugate gradient descent (DRCGD) method that aims at minimizing a global function over the Stiefel manifold. The optimization problem is distributed among a network of agents, where each agent is associated with a local function, and the communication between agents occurs over an undirected connected graph. Since the Stiefel manifold is a non-convex set, a global function is represented as a finite sum of possibly non-convex (but smooth) local functions. The proposed method is free from expensive Riemannian geometric operations such as retractions, exponential maps, and vector transports, thereby reducing the computational complexity required by each agent. To the best of our knowledge, DRCGD is the first decentralized Riemannian conjugate gradient algorithm to achieve global convergence over the Stiefel manifold.

In decentralized optimization, $m$ agents form a network and only communicate with their neighbors, which gives advantages in data ownership, privacy, and scalability. At the same time, decentralized stochastic gradient descent ($\texttt{SGD}$) methods, as popular decentralized algorithms for training large-scale machine learning models, have shown their superiority over centralized counterparts. Distributed stochastic gradient tracking $\texttt{DSGT}$ has been recognized as the popular and state-of-the-art decentralized $\texttt{SGD}$ method due to its proper theoretical guarantees. However, the theoretical analysis of $\texttt{DSGT}$ shows that its iteration complexity is $\tilde{\mathcal{O}} \left(\frac{\bar{\sigma}^2}{m\mu \varepsilon} + \frac{\sqrt{L}\bar{\sigma}}{\mu(1 - \lambda_2(W))^{1/2} C_W \sqrt{\varepsilon} }\right)$, where the doubly stochastic matrix $W$ represents the network topology and $ C_W $ is a parameter that depends on $W$. Thus, it indicates that the convergence property of $\texttt{DSGT}$ is heavily affected by the topology of the communication network. To overcome the weakness of $\texttt{DSGT}$, we resort to the snap-shot gradient tracking skill and propose two novel algorithms, snap-shot $\texttt{DSGT}$ ($\texttt{SS-DSGT}$) and accelerated snap-shot $\texttt{DSGT}$ ($\texttt{ASS-DSGT}$). We further justify that $\texttt{SS-DSGT}$ exhibits a lower iteration complexity compared to $\texttt{DSGT}$ in the general communication network topology. Additionally, $\texttt{ASS-DSGT}$ matches $\texttt{DSGT}$'s iteration complexity $\mathcal{O}\left( \frac{\bar{\sigma}^2}{m\mu \varepsilon} + \frac{\sqrt{L}\bar{\sigma}}{\mu (1 - \lambda_2(W))^{1/2}\sqrt{\varepsilon}} \right)$ under the same conditions as $\texttt{DSGT}$. Numerical experiments validate $\texttt{SS-DSGT}$'s superior performance performance in the general communication network topology and exhibit better practical performance of $\texttt{ASS-DSGT}$ on the specified $W$ compared to $\texttt{DSGT}$.

Variance reduction techniques are designed to decrease the sampling variance, thereby accelerating convergence rates of first-order (FO) and zeroth-order (ZO) optimization methods. However, in composite optimization problems, ZO methods encounter an additional variance called the coordinate-wise variance, which stems from the random gradient estimation. To reduce this variance, prior works require estimating all partial derivatives, essentially approximating FO information. This approach demands $\mathcal{O}(d)$ function evaluations ($d$ is the dimension size), which incurs substantial computational costs and is prohibitive in high-dimensional scenarios. This paper proposes the Zeroth-order Proximal Double Variance Reduction ($\texttt{ZPDVR}$) method, which utilizes the averaging trick to reduce both sampling and coordinate-wise variances. Compared to prior methods, $\texttt{ZPDVR}$ relies solely on random gradient estimates, calls the stochastic zeroth-order oracle (SZO) in expectation $\mathcal{O}(1)$ times per iteration, and achieves the optimal $\mathcal{O}(d(n + \kappa)\log (\frac{1}{\epsilon}))$ SZO query complexity in the strongly convex and smooth setting, where $\kappa$ represents the condition number and $\epsilon$ is the desired accuracy. Empirical results validate $\texttt{ZPDVR}$’s linear convergence and demonstrate its superior performance over other related methods.

This paper focuses on the large-scale optimization which is very popular in the big data era. The gradient sketching is an important technique in the large-scale optimization. Specifically, the random coordinate descent algorithm is a kind of gradient sketching method with the random sampling matrix as the sketching matrix. In this paper, we propose a novel gradient sketching called GSGD (Gaussian Sketched Gradient Descent). Compared with the classical gradient sketching methods such as the random coordinate descent and SEGA (Hanzely et al., 2018), our GSGD does not require the importance sampling but can achieve a fast convergence rate matching the ones of these methods with importance sampling. Furthermore, if the objective function has a non-smooth regularization term, our GSGD can also exploit the implicit structure information of the regularization term to achieve a fast convergence rate. Finally, our experimental results substantiate the effectiveness and efficiency of our algorithm.

This paper considers the distributed convex-concave minimax optimization under the second-order similarity. We propose stochastic variance-reduced optimistic gradient sliding (SVOGS) method, which takes the advantage of the finite-sum structure in the objective by involving the mini-batch client sampling and variance reduction. We prove SVOGS can achieve the $\varepsilon$-duality gap within communication rounds of ${\mathcal O}(\delta D^2/\varepsilon)$, communication complexity of ${\mathcal O}(n+\sqrt{n}\delta D^2/\varepsilon)$, and local gradient calls of $\tilde{\mathcal O}(n+(\sqrt{n}\delta+L)D^2/\varepsilon\log(1/\varepsilon))$, where $n$ is the number of nodes, $\delta$ is the degree of the second-order similarity, $L$ is the smoothness parameter and $D$ is the diameter of the constraint set. We can verify that all of above complexity (nearly) matches the corresponding lower bounds. For the specific $\mu$-strongly-convex-$\mu$-strongly-convex case, our algorithm has the upper bounds on communication rounds, communication complexity, and local gradient calls of $\mathcal O(\delta/\mu\log(1/\varepsilon))$, ${\mathcal O}((n+\sqrt{n}\delta/\mu)\log(1/\varepsilon))$, and $\tilde{\mathcal O}(n+(\sqrt{n}\delta+L)/\mu)\log(1/\varepsilon))$ respectively, which are also nearly tight. Furthermore, we conduct the numerical experiments to show the empirical advantages of proposed method.

We study finite-sum distributed optimization problems involving a master node and $n-1$ local nodes under the popular $\delta$-similarity and $\mu$-strong convexity conditions. We propose two new algorithms, SVRS and AccSVRS, motivated by previous works. The non-accelerated SVRS method combines the techniques of gradient sliding and variance reduction and achieves a better communication complexity of $\tilde{\mathcal{O}}(n {+} \sqrt{n}\delta/\mu)$ compared to existing non-accelerated algorithms. Applying the framework proposed in Katyusha X, we also develop a directly accelerated version named AccSVRS with the $\tilde{\mathcal{O}}(n {+} n^{3/4}\sqrt{\delta/\mu})$ communication complexity. In contrast to existing results, our complexity bounds are entirely smoothness-free and exhibit superiority in ill-conditioned cases. Furthermore, we establish a nearly matched lower bound to verify the tightness of our AccSVRS method.

his paper considers the decentralized convex optimization problem, which has a wide range of applications in large-scale machine learning, sensor networks, and control theory. We propose novel algorithms that achieve optimal computation complexity and near optimal communication complexity. Our theoretical results give affirmative answers to the open problem on whether there exists an algorithm that can achieve a communication complexity (nearly) matching the lower bound depending on the global condition number instead of the local one. Furthermore, the linear convergence of our algorithms only depends on the strong convexity of global objective and it does not require the local functions to be convex. The design of our methods relies on a novel integration of well-known techniques including Nesterov's acceleration, multi-consensus and gradient-tracking. Empirical studies show the outperformance of our methods for machine learning applications.

Distributed second-order optimization, as an effective strategy for training large-scale machine learning systems, has been widely investigated due to its low communication complexity. However, the existing distributed second-order optimization algorithms, including distributed approximate Newton (DANE), accelerated inexact DANE (AIDE), and statistically preconditioned accelerated gradient (SPAG), are all required to precisely solve an expensive subproblem up to the target precision. Therefore, this causes these algorithms to suffer from high computation costs and this hinders their development. In this article, we design a novel distributed second-order algorithm called the accelerated distributed approximate Newton (ADAN) method to overcome the high computation costs of the existing ones. Compared with DANE, AIDE, and SPAG, which are constructed based on the relative smooth theory, ADAN's theoretical foundation is built upon the inexact Newton theory. The different theoretical foundations lead to handle the expensive subproblem efficiently, and steps required to solve the subproblem are independent of the target precision. At the same time, ADAN resorts to the acceleration and can effectively exploit the objective function's curvature information, making ADAN to achieve a low communication complexity. Thus, ADAN can achieve both the communication and computation efficiencies, while DANE, AIDE, and SPAG can achieve only the communication efficiency. Our empirical study also validates the advantages of ADAN over extant distributed second-order algorithms.

Optimization is important in machine learning problems, and quasi-Newton methods have a reputation as the most efficient numerical methods for smooth unconstrained optimization. In this paper, we study the explicit superlinear convergence rates of quasi-Newton methods and address two open problems mentioned by Rodomanov and Nesterov (2021b). First, we extend Rodomanov and Nesterov (2021b)’s results to random quasi-Newton methods, which include common DFP, BFGS, SR1 methods. Such random methods employ a random direction for updating the approximate Hessian matrix in each iteration. Second, we focus on the specific quasi-Newton methods: SR1 and BFGS methods. We provide improved versions of greedy and random methods with provable better explicit (local) superlinear convergence rates. Our analysis is closely related to the approximation of a given Hessian matrix, unconstrained quadratic objective, as well as the general strongly convex, smooth, and strongly self-concordant functions.

We study the streaming model for approximate matrix multiplication (AMM). We are interested in the scenario that the algorithm can only take one pass over the data with limited memory. The state-of-the-art deterministic sketching algorithm for streaming AMM is the co-occurring directions (COD), which has much smaller approximation errors than randomized algorithms and outperforms other deterministic sketching methods empirically. In this paper, we provide a tighter error bound for COD whose leading term considers the potential approximate low-rank structure and the correlation of input matrices. We prove COD is space optimal with respect to our improved error bound. We also propose a variant of COD for sparse matrices with theoretical guarantees. The experiments on real-world sparse datasets show that the proposed algorithm is more efficient than baseline methods.

In this paper, we follow Rodomanov and Nesterov’s work to study quasi-Newton methods. We focus on the common SR1 and BFGS quasi-Newton methods to establish better explicit (local) superlinear convergence rates. First, based on the greedy quasi-Newton update which greedily selects the direction to maximize a certain measure of progress, we improve the convergence rate to a condition-number-free superlinear convergence rate. Second, based on the random quasi-Newton update that selects the direction randomly from a spherically symmetric distribution, we show the same superlinear convergence rate established as above. Our analysis is closely related to the approximation of a given Hessian matrix, unconstrained quadratic objective, as well as the general strongly convex, smooth, and strongly self-concordant functions.

Many machine learning models involve solving optimization problems. Thus, it is important to address a large-scale optimization problem in big data applications. Recently, subsampled Newton methods have emerged to attract much attention due to their efficiency at each iteration, rectified a weakness in the ordinary Newton method of suffering a high cost in each iteration while commanding a high convergence rate. Other efficient stochastic second order methods have been also proposed. However, the convergence properties of these methods are still not well understood. There are also several important gaps between the current convergence theory and the empirical performance in real applications. In this paper, we aim to fill these gaps. We propose a unifying framework to analyze both local and global convergence properties of second order methods. Accordingly, we present our theoretical results which match the empirical performance in real applications well.

Due to the rapid growth of smart agents such as weakly connected computational nodes and sensors, developing decentralized algorithms that can perform computations on local agents becomes a major research direction. This paper considers the problem of decentralized principal components analysis (PCA), which is a statistical method widely used for data analysis. We introduce a technique called subspace tracking to reduce the communication cost, and apply it to power iterations. This leads to a decentralized PCA algorithm called DeEPCA, which has a convergence rate similar to that of the centralized PCA, while achieving the best communication complexity among existing decentralized PCA algorithms. DeEPCA is the first decentralized PCA algorithm with the number of communication rounds for each power iteration independent of target precision. Compared to existing algorithms, the proposed method is easier to tune in practice, with an improved overall communication cost. Our experiments validate the advantages of DeEPCA empirically.

Composite function optimization problem often arises in machine learning known as regularized empirical minimization. We introduce the acceleration technique to the Newton-type proximal method and propose a novel algorithm called accelerated proximal subsampled Newton method (APSSN). APSSN only subsamples a small subset of samples to construct an approximate Hessian that achieves computational efficiency. At the same time, APSSN still keeps a fast convergence rate. Furthermore, we obtain the scaled proximal mapping by solving its dual problem using the semismooth Newton method instead of resorting to the first-order methods. Due to our sampling strategy and the fast convergence rate of the semismooth Newton method, we can get the scaled proximal mapping efficiently. Both our theoretical analysis and empirical study show that APSSN is an effective and computationally efficient algorithm for composite function optimization problems.

Many recently proposed methods for Neural Architecture Search (NAS) can be formulated as bilevel optimization. For efficient implementation, its solution requires approximations of second-order methods. In this paper, we demonstrate that gradient errors caused by such approximations lead to suboptimality, in the sense that the optimization procedure fails to converge to a (locally) optimal solution. To remedy this, this paper proposes MiLeNAS, a mixed-level reformulation for NAS that can be optimized efficiently and reliably. It is shown that even when using a simple first-order method on the mixed-level formulation, MiLeNAS can achieve a lower validation error for NAS problems. Consequently, architectures obtained by our method achieve consistently higher accuracies than those obtained from bilevel optimization. Moreover, MiLeNAS proposes a framework beyond DARTS. It is upgraded via model size-based search and early stopping strategies to complete the search process in around 5 hours. Extensive experiments within the convolutional architecture search space validate the effectiveness of our approach.

We consider nonconvex-concave minimax optimization problems of the form $\min_{\bf x}\max_{\bf y\in{\mathcal Y}} f({\bf x},{\bf y})$, where $f$ is strongly-concave in $\bf y$ but possibly nonconvex in $\bf x$ and ${\mathcal Y}$ is a convex and compact set. We focus on the stochastic setting, where we can only access an unbiased stochastic gradient estimate of $f$ at each iteration. This formulation includes many machine learning applications as special cases such as robust optimization and adversary training. We are interested in finding an ${\mathcal O}(\varepsilon)$-stationary point of the function $\Phi(\cdot)=\max_{\bf y\in{\mathcal Y}} f(\cdot, {\bf y})$. The most popular algorithm to solve this problem is stochastic gradient decent ascent, which requires $\mathcal O(\kappa^3\varepsilon^{-4})$ stochastic gradient evaluations, where $\kappa$ is the condition number. In this paper, we propose a novel method called Stochastic Recursive gradiEnt Descent Ascent (SREDA), which estimates gradients more efficiently using variance reduction. This method achieves the best known stochastic gradient complexity of ${\mathcal O}(\kappa^3\varepsilon^{-3})$, and its dependency on $\varepsilon$ is optimal for this problem.

Decentralized optimization has wide applications in machine learning, signal processing, and control. In this paper, we study the decentralized composite optimization problem with a non-smooth regularization term. Many proximal gradient based decentralized algorithms have been proposed in the past. However, these algorithms do not achieve near optimal computational complexity and communication complexity. In this paper, we propose a new method which establishes the optimal computational complexity and a near optimal communication complexity. Our empirical study shows that the proposed algorithm outperforms existing state-of-the-art algorithms.

Optimization plays a key role in machine learning. Recently, stochastic second-order methods have attracted considerable attention because of their low computational cost in each iteration. However, these methods might suffer from poor performance when the Hessian is hard to be approximate well in a computation-efficient way. To overcome this dilemma, we resort to Nesterov's acceleration to improve the convergence performance of these second-order methods and propose accelerated approximate Newton. We give the theoretical convergence analysis of accelerated approximate Newton and show that Nesterov's acceleration can improve the convergence rate. Accordingly, we propose an accelerated regularized sub-sampled Newton (ARSSN) which performs much better than the conventional regularized sub-sampled Newton empirically and theoretically. Moreover, we show that ARSSN has better performance than classical first-order methods empirically.

Many machine learning models are reformulated as optimization problems. Thus, it is important to solve a large-scale optimization problem in big data applications. Recently, subsampled Newton methods have emerged to attract much attention for optimization due to their efficiency at each iteration, rectified a weakness in the ordinary Newton method of suffering a high cost in each iteration while commanding a high convergence rate. Other efficient stochastic second order methods are also proposed. However, the convergence properties of these methods are still not well understood. There are also several important gaps between the current convergence theory and performance in real applications. In this paper, we aim to fill these gaps. We propose a unifying framework to analyze local convergence properties of second order methods. Based on this framework, our theoretical analysis matches the performance in real applications.

Kaczmarz algorithm is an efficient iterative algorithm to solve overdetermined consistent system of linear equations. During each updating step, Kaczmarz chooses a hyperplane based on an individual equation and projects the current estimate for the exact solution onto that space to get a new estimate.Many vairants of Kaczmarz algorithms are proposed on how to choose better hyperplanes.Using the property of randomly sampled data in high-dimensional space,we propose an accelerated algorithm based on clustering information to improve block Kaczmarz and Kaczmarz via Johnson-Lindenstrauss lemma. Additionally, we theoretically demonstrate convergence improvement on block Kaczmarz algorithm.