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Haishan Ye

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23 papers

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AAAI Conference 2026 Conference Paper

Privacy Leaks by Adversaries: Adversarial Iterations for Membership Inference Attack

Jing Xue
Zhishen Sun
Haishan Ye
Luo Luo
Xiangyu Chang
Guang Dai

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.

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NeurIPS Conference 2025 Conference Paper

A Near-Optimal Algorithm for Decentralized Convex-Concave Finite-Sum Minimax Optimization

Hongxu Chen
Ke Wei
Haishan Ye
Luo Luo

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.

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JMLR Journal 2025 Journal Article

Optimal Decentralized Composite Optimization for Strongly Convex Functions

Haishan Ye
Xiangyu Chang

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. [abs] [ pdf ][ bib ] &copy JMLR 2025. ( edit, beta )

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ICLR Conference 2025 Conference Paper

ProAdvPrompter: A Two-Stage Journey to Effective Adversarial Prompting for LLMs

Hao Di
Tong He
Haishan Ye
Yinghui Huang 0001
Xiangyu Chang
Guang Dai
Ivor W. Tsang

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).

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ICLR Conference 2025 Conference Paper

Second-Order Fine-Tuning without Pain for LLMs: A Hessian Informed Zeroth-Order Optimizer

Yanjun Zhao 0001
Sizhe Dang
Haishan Ye
Guang Dai
Yi Qian 0004
Ivor W. Tsang

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.

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ICML Conference 2024 Conference Paper

Can Gaussian Sketching Converge Faster on a Preconditioned Landscape?

Yilong Wang
Haishan Ye
Guang Dai
Ivor W. Tsang

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.

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ICLR Conference 2024 Conference Paper

Decentralized Riemannian Conjugate Gradient Method on the Stiefel Manifold

Jun Chen 0023
Haishan Ye
Mengmeng Wang 0005
Tianxin Huang
Guang Dai
Ivor W. Tsang
Yong Liu 0007

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.

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ICML Conference 2024 Conference Paper

Double Stochasticity Gazes Faster: Snap-Shot Decentralized Stochastic Gradient Tracking Methods

Hao Di
Haishan Ye
Xiangyu Chang
Guang Dai
Ivor W. Tsang

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 $\texttt{DSGT}$.

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ICML Conference 2024 Conference Paper

Double Variance Reduction: A Smoothing Trick for Composite Optimization Problems without First-Order Gradient

Hao Di
Haishan Ye
Yueling Zhang
Xiangyu Chang
Guang Dai
Ivor W. Tsang

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.

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NeurIPS Conference 2024 Conference Paper

Near-Optimal Distributed Minimax Optimization under the Second-Order Similarity

Qihao Zhou
Haishan Ye
Luo Luo

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.

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JMLR Journal 2023 Journal Article

Multi-Consensus Decentralized Accelerated Gradient Descent

Haishan Ye
Luo Luo
Ziang Zhou
Tong Zhang

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. [abs] [ pdf ][ bib ] &copy JMLR 2023. ( edit, beta )

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NeurIPS Conference 2023 Conference Paper

Stochastic Distributed Optimization under Average Second-order Similarity: Algorithms and Analysis

Dachao Lin
Yuze Han
Haishan Ye
Zhihua Zhang

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.

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ICLR Conference 2022 Conference Paper

Eigencurve: Optimal Learning Rate Schedule for SGD on Quadratic Objectives with Skewed Hessian Spectrums

Rui Pan 0002
Haishan Ye
Tong Zhang 0001

Learning rate schedulers have been widely adopted in training deep neural networks. Despite their practical importance, there is a discrepancy between its practice and its theoretical analysis. For instance, it is not known what schedules of SGD achieve best convergence, even for simple problems such as optimizing quadratic objectives. In this paper, we propose Eigencurve, the first family of learning rate schedules that can achieve minimax optimal convergence rates (up to a constant) for SGD on quadratic objectives when the eigenvalue distribution of the underlying Hessian matrix is skewed. The condition is quite common in practice. Experimental results show that Eigencurve can significantly outperform step decay in image classification tasks on CIFAR-10, especially when the number of epochs is small. Moreover, the theory inspires two simple learning rate schedulers for practical applications that can approximate eigencurve. For some problems, the optimal shape of the proposed schedulers resembles that of cosine decay, which sheds light to the success of cosine decay for such situations. For other situations, the proposed schedulers are superior to cosine decay.

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JMLR Journal 2022 Journal Article

Explicit Convergence Rates of Greedy and Random Quasi-Newton Methods

Dachao Lin
Haishan Ye
Zhihua Zhang

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. [abs] [ pdf ][ bib ] &copy JMLR 2022. ( edit, beta )

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JMLR Journal 2021 Journal Article

Approximate Newton Methods

Haishan Ye
Luo Luo
Zhihua Zhang

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. [abs] [ pdf ][ bib ] &copy JMLR 2021. ( edit, beta )

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JMLR Journal 2021 Journal Article

DeEPCA: Decentralized Exact PCA with Linear Convergence Rate

Haishan Ye
Tong Zhang

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. [abs] [ pdf ][ bib ] &copy JMLR 2021. ( edit, beta )

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NeurIPS Conference 2021 Conference Paper

Greedy and Random Quasi-Newton Methods with Faster Explicit Superlinear Convergence

Dachao Lin
Haishan Ye
Zhihua Zhang

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.

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AAAI Conference 2021 Conference Paper

Revisiting Co-Occurring Directions: Sharper Analysis and Efficient Algorithm for Sparse Matrices

Luo Luo
Cheng Chen
Guangzeng Xie
Haishan Ye

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 realworld sparse datasets show that the proposed algorithm is more efficient than baseline methods.

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NeurIPS Conference 2020 Conference Paper

Decentralized Accelerated Proximal Gradient Descent

Haishan Ye
Ziang Zhou
Luo Luo
Tong Zhang

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.

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JMLR Journal 2020 Journal Article

Nesterov's Acceleration for Approximate Newton

Haishan Ye
Luo Luo
Zhihua Zhang

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. [abs] [ pdf ][ bib ] &copy JMLR 2020. ( edit, beta )

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NeurIPS Conference 2020 Conference Paper

Stochastic Recursive Gradient Descent Ascent for Stochastic Nonconvex-Strongly-Concave Minimax Problems

Luo Luo
Haishan Ye
Zhichao Huang
Tong Zhang

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.

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ICML Conference 2017 Conference Paper

Approximate Newton Methods and Their Local Convergence

Haishan Ye
Luo Luo
Zhihua Zhang 0004

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.

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AAAI Conference 2016 Conference Paper

Accelerating Random Kaczmarz Algorithm Based on Clustering Information

Yujun Li
Kaichun Mo
Haishan Ye

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.

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