Haishan Ye

Jun Chen, Haishan Ye, Mengmeng Wang et al.

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.

Dachao Lin, Yuze Han, Haishan Ye et al.

We study finite-sum distributed optimization problems involving a master node and $n-1$ local nodes under the popular $δ$-similarity and $μ$-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 $\tilde{\mathcal{O}}(n {+} \sqrt{n}δ/μ)$ compared to existing non-accelerated algorithms. Applying the framework proposed in Katyusha X, we also develop a directly accelerated version named AccSVRS with the $\tilde{\mathcal{O}}(n {+} n^{3/4}\sqrt{δ/μ})$ 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.

Haishan Ye, Luo Luo, Zhihua Zhang

Many machine learning models involve solving optimization problems. Thus, it is important to deal with 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 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 the 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. Based on this framework, we present our theoretical results which match the performance in real applications well.

Haishan Ye

The zeroth-order optimization has been widely used in machine learning applications. However, the theoretical study of the zeroth-order optimization focus on the algorithms which approximate (first-order) gradients using (zeroth-order) function value difference at a random direction. The theory of algorithms which approximate the gradient and Hessian information by zeroth-order queries is much less studied. In this paper, we focus on the theory of zeroth-order optimization which utilizes both the first-order and second-order information approximated by the zeroth-order queries. We first propose a novel reparameterized objective function with parameters $(μ, Σ)$. This reparameterized objective function achieves its optimum at the minimizer and the Hessian inverse of the original objective function respectively, but with small perturbations. Accordingly, we propose a new algorithm to minimize our proposed reparameterized objective, which we call \texttt{MiNES} (mirror descent natural evolution strategy). We show that the estimated covariance matrix of \texttt{MiNES} converges to the inverse of Hessian matrix of the objective function with a convergence rate $\widetilde{\mathcal{O}}(1/k)$, where $k$ is the iteration number and $\widetilde{\mathcal{O}}(\cdot)$ hides the constant and $\log$ terms. We also provide the explicit convergence rate of \texttt{MiNES} and how the covariance matrix promotes the convergence rate.

Lesi Chen, Haishan Ye, Luo Luo

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 $ε$-stationary points. Concretely, it requires $\mathcal{O}(\min (κ^3ε^{-3},κ^2 \sqrt{N} ε^{-2} ))$ stochastic first-order oracle (SFO) calls and $\tilde{\mathcal{O}}(κ^2 ε^{-2})$ communication rounds, where $κ$ 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.

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.

Haishan Ye, Yujun Li, Zhihua Zhang

Prior optimal CUR decomposition and near optimal column reconstruction methods have been established by combining BSS sampling and adaptive sampling. In this paper, we propose a new approach to the optimal CUR decomposition and near optimal column reconstruction by just using leverage score sampling. In our approach, both the BSS sampling and adaptive sampling are not needed. Moreover, our approach is the first $O(\mathrm{nnz}(\A))$ optimal CUR algorithm where $\A$ is a data matrix in question. We also extend our approach to the Nystr{ö}m method, obtaining a fast algorithm which runs $\tilde{O}(n^{2})$ or $O(\mathrm{\nnz}(\A))$

Zhishen Sun, Guang Dai, Ivor Tsang et al.

LLMs have made significant progress in the field of mathematical reasoning, but whether they have true the mathematical understanding ability is still controversial. To explore this issue, we propose a new perturbation framework to evaluate LLMs' reasoning ability in complex environments by injecting additional semantically irrelevant perturbation sentences and gradually increasing the perturbation intensity. At the same time, we use an additional perturbation method: core questioning instruction missing, to further analyze the LLMs' problem-solving mechanism. The experimental results show that LLMs perform stably when facing perturbation sentences without numbers, but there is also a robustness boundary. As the perturbation intensity increases, the performance exhibits varying degrees of decline; when facing perturbation sentences with numbers, the performance decreases more significantly, most open source models with smaller parameters decrease by nearly or even more than 10%, and further increasing with the enhancement of perturbation intensity, with the maximum decrease reaching 51.55%. Even the most advanced commercial LLMs have seen a 3%-10% performance drop. By analyzing the reasoning process of LLMs in detail, We find that models are more sensitive to perturbations with numerical information and are more likely to give incorrect answers when disturbed by irrelevant numerical information. The higher the perturbation intensity, the more obvious these defects are. At the same time, in the absence of core questioning instruction, models can still maintain an accuracy of 20%-40%, indicating that LLMs may rely on memory templates or pattern matching to complete the task, rather than logical reasoning. In general, our work reveals the shortcomings and limitations of current LLMs in their reasoning capabilities, which is of great significance for the further development of LLMs.

Haishan Ye, Zhihua Zhang

First, we extend the results of approximate matrix multiplication from the Frobenius norm to the spectral norm. Second, We develop a class of fast approximate generalized linear regression algorithms with respect to the spectral norm. Finally, We give a fast approximate SVD.

Hao Di, Yi Yang, Haishan Ye et al.

Personalization aims to characterize individual preferences and is widely applied across many fields. However, conventional personalized methods operate in a centralized manner, potentially exposing raw data when pooling individual information. In this paper, with privacy considerations, we develop a flexible and interpretable personalized framework within the paradigm of federated learning, called \texttt{PPFL} (Population Personalized Federated Learning). By leveraging ``canonical models" to capture fundamental characteristics of a heterogeneous population and employing ``membership vectors" to reveal clients' preferences, \texttt{PPFL} models heterogeneity as clients' varying preferences for these characteristics. This approach provides substantial insights into client characteristics, which are lacking in existing Personalized Federated Learning (PFL) methods. Furthermore, we explore the relationship between \texttt{PPFL} and three main branches of PFL methods: clustered FL, multi-task PFL, and decoupling PFL, and demonstrate the advantages of \texttt{PPFL}. To solve \texttt{PPFL} (a non-convex optimization problem with linear constraints), we propose a novel random block coordinate descent algorithm and establish its convergence properties. We conduct experiments on both pathological and practical data sets, and the results validate the effectiveness of \texttt{PPFL}.

Yanjun Zhao, Sizhe Dang, Haishan Ye et al.

Fine-tuning large language models (LLMs) with classic first-order optimizers entails prohibitive GPU memory due to the backpropagation process. Recent works have turned to zeroth-order optimizers for fine-tuning, which save substantial memory by using two forward passes. However, these optimizers are plagued by the heterogeneity of parameter curvatures across different dimensions. 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 zeroth-order optimizer for fine-tuning LLMs. What's more, HiZOO avoids the expensive memory cost and only increases one forward pass per step. Extensive experiments on various models (350M~66B parameters) indicate that HiZOO improves model convergence, significantly reducing training steps and effectively enhancing model accuracy. Moreover, we visualize the optimization trajectories of HiZOO on test functions, illustrating its effectiveness in handling heterogeneous curvatures. Lastly, we provide theoretical proofs of convergence for HiZOO. Code is publicly available at https://anonymous.4open.science/r/HiZOO27F8.

Haishan Ye

Zeroth-order (ZO) optimization with ordinal feedback has emerged as a fundamental problem in modern machine learning systems, particularly in human-in-the-loop settings such as reinforcement learning from human feedback, preference learning, and evolutionary strategies. While rank-based ZO algorithms enjoy strong empirical success and robustness properties, their theoretical understanding, especially under stochastic objectives and standard smoothness assumptions, remains limited. In this paper, we study rank-based zeroth-order optimization for stochastic functions where only ordinal feedback of the stochastic function is available. We propose a simple and computationally efficient rank-based ZO algorithm. Under standard assumptions including smoothness, strong convexity, and bounded second moments of stochastic gradients, we establish explicit non-asymptotic query complexity bounds for both convex and nonconvex objectives. Notably, our results match the best-known query complexities of value-based ZO algorithms, demonstrating that ordinal information alone is sufficient for optimal query efficiency in stochastic settings. Our analysis departs from existing drift-based and information-geometric techniques, offering new tools for the study of rank-based optimization under noise. These findings narrow the gap between theory and practice and provide a principled foundation for optimization driven by human preferences.

Sizhe Dang, Yangyang Guo, Yanjun Zhao et al.

Fine-tuning large language models (LLMs) often faces GPU memory bottlenecks: the backward pass of first-order optimizers like Adam increases memory usage to more than 10 times the inference level (e.g., 633 GB for OPT-30B). Zeroth-order (ZO) optimizers avoid this cost by estimating gradients only from forward passes, yet existing methods like MeZO usually require many more steps to converge. Can this trade-off between speed and memory in ZO be fundamentally improved? Normalized-SGD demonstrates strong empirical performance with greater memory efficiency than Adam. In light of this, we introduce FZOO, a Fast Zeroth-Order Optimizer toward Adam-Scale Speed. FZOO reduces the total forward passes needed for convergence by employing batched one-sided estimates that adapt step sizes based on the standard deviation of batch losses. It also accelerates per-batch computation through the use of Rademacher random vector perturbations coupled with CUDA's parallel processing. Extensive experiments on diverse models, including RoBERTa-large, OPT (350M-66B), Phi-2, and Llama3, across 11 tasks validate FZOO's effectiveness. On average, FZOO outperforms MeZO by 3 percent in accuracy while requiring 3 times fewer forward passes. For RoBERTa-large, FZOO achieves average improvements of 5.6 percent in accuracy and an 18 times reduction in forward passes compared to MeZO, achieving convergence speeds comparable to Adam. We also provide theoretical analysis proving FZOO's formal equivalence to a normalized-SGD update rule and its convergence guarantees. FZOO integrates smoothly into PEFT techniques, enabling even larger memory savings. Overall, our results make single-GPU, high-speed, full-parameter fine-tuning practical and point toward future work on memory-efficient pre-training.

Zhishen Sun, Sizhe Dang, Guang Dai et al.

Reinforcement learning (RL) has become a key training step for improving mathematical reasoning in large language models (LLMs), but it often has high GPU memory usage, which makes it hard to use in settings with limited resources. To reduce these issues, we propose Evolution Strategies with Sharpness-Aware Maximization (ESSAM), a full parameter fine-tuning framework that tightly combines the zero-order search in parameter space from Evolution Strategies (ES) with the Sharpness-Aware Maximization (SAM) to improve generalization. We conduct fine-tuning experiments on the mainstream mathematica reasoning task GSM8K. The results show that ESSAM achieves an average accuracy of 78.27\% across all models and its overall performance is comparable to RL methods. It surpasses classic RL algorithm PPO with an accuracy of 77.72\% and is comparable to GRPO with an accuracy of 78.34\%, and even surpassing them on some models. In terms of GPU memory usage, ESSAM reduces the average GPU memory usage by $18\times$ compared to PPO and by $10\times$ compared to GRPO, achieving an extremely low GPU memory usage.

Luo Luo, Haishan Ye

This paper studies decentralized convex-concave minimax optimization problems of the form $\min_x\max_y f(x,y) \triangleq\frac{1}{m}\sum_{i=1}^m f_i(x,y)$, where $m$ is the number of agents and each local function can be written as $f_i(x,y)=\frac{1}{n}\sum_{j=1}^n f_{i,j}(x,y)$. We propose a novel decentralized optimization algorithm, called multi-consensus stochastic variance reduced extragradient, which achieves the best known stochastic first-order oracle (SFO) complexity for this problem. Specifically, each agent requires $\mathcal O((n+κ\sqrt{n})\log(1/\varepsilon))$ SFO calls for strongly-convex-strongly-concave problem and $\mathcal O((n+\sqrt{n}L/\varepsilon)\log(1/\varepsilon))$ SFO call for general convex-concave problem to achieve $\varepsilon$-accurate solution in expectation, where $κ$ is the condition number and $L$ is the smoothness parameter. The numerical experiments show the proposed method performs better than baselines.

Rui Pan, Haishan Ye, Tong Zhang

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.

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 \texttt{DeEPCA}, which has a convergence rate similar to that of the centralized PCA, while achieving the best communication complexity among existing decentralized PCA algorithms. \texttt{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 \texttt{DeEPCA} empirically.

Haishan Ye, Wei Xiong, Tong Zhang

This paper considers the decentralized composite optimization problem. We propose a novel decentralized variance-reduction proximal-gradient algorithmic framework, called PMGT-VR, which is based on a combination of several techniques including multi-consensus, gradient tracking, and variance reduction. The proposed framework relies on an imitation of centralized algorithms and we demonstrate that 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.

Haishan Ye, Luo Luo, Ziang Zhou et al.

This 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 \emph{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.

Chaoyang He, Haishan Ye, Li Shen et al.

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 \mldas, 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, \mldas\ 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, \mldas\ 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.

Luo Luo, Haishan Ye, Zhichao Huang et al.

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 $Φ(\cdot)=\max_{\bf y\in{\mathcal Y}} f(\cdot, {\bf y})$. The most popular algorithm to solve this problem is stochastic gradient decent ascent, which requires $\mathcal O(κ^3\varepsilon^{-4})$ stochastic gradient evaluations, where $κ$ 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 ${\mathcal O}(κ^3\varepsilon^{-3})$, and its dependency on $\varepsilon$ is optimal for this problem.

Haishan Ye, Shusen Wang, Zhihua Zhang et al.

Fast matrix algorithms have become the fundamental tools of machine learning in big data era. The generalized matrix regression problem is widely used in the matrix approximation such as CUR decomposition, kernel matrix approximation, and stream singular value decomposition (SVD), etc. In this paper, we propose a fast generalized matrix regression algorithm (Fast GMR) which utilizes sketching technique to solve the GMR problem efficiently. Given error parameter $0<ε<1$, the Fast GMR algorithm can achieve a $(1+ε)$ relative error with the sketching sizes being of order $\cO(ε^{-1/2})$ for a large group of GMR problems. We apply the Fast GMR algorithm to the symmetric positive definite matrix approximation and single pass singular value decomposition and they achieve a better performance than conventional algorithms. Our empirical study also validates the effectiveness and efficiency of our proposed algorithms.

Haishan Ye, Tong Zhang

Evolution Strategies such as CMA-ES (covariance matrix adaptation evolution strategy) and NES (natural evolution strategy) have been widely used in machine learning applications, where an objective function is optimized without using its derivatives. However, the convergence behaviors of these algorithms have not been carefully studied. In particular, there is no rigorous analysis for the convergence of the estimated covariance matrix, and it is unclear how does the estimated covariance matrix help the converge of the algorithm. The relationship between Evolution Strategies and derivative free optimization algorithms is also not clear. In this paper, we propose a new algorithm closely related toNES, which we call MiNES (mirror descent natural evolution strategy), for which we can establish rigorous convergence results. We show that the estimated covariance matrix of MiNES converges to the inverse of Hessian matrix of the objective function with a sublinear convergence rate. Moreover, we show that some derivative free optimization algorithms are special cases of MiNES. Our empirical studies demonstrate that MiNES is a query-efficient optimization algorithm competitive to classical algorithms including NES and CMA-ES.

Haishan Ye, Zhichao Huang, Cong Fang et al.

Zeroth-order optimization is an important research topic in machine learning. In recent years, it has become a key tool in black-box adversarial attack to neural network based image classifiers. However, existing zeroth-order optimization algorithms rarely extract second-order information of the model function. In this paper, we utilize the second-order information of the objective function and propose a novel \textit{Hessian-aware zeroth-order algorithm} called \texttt{ZO-HessAware}. Our theoretical result shows that \texttt{ZO-HessAware} has an improved zeroth-order convergence rate and query complexity under structured Hessian approximation, where we propose a few approximation methods for estimating Hessian. Our empirical studies on the black-box adversarial attack problem validate that our algorithm can achieve improved success rates with a lower query complexity.

Haishan Ye, Zhihua Zhang

Optimization plays a key role in machine learning. Recently, stochastic second-order methods have attracted much attention due to their low computational cost in each iteration. However, these algorithms might perform poorly especially if it is hard to approximate the Hessian well and efficiently. As far as we know, there is no effective way to handle this problem. In this paper, we resort to Nesterov's acceleration technique to improve the convergence performance of a class of second-order methods called approximate Newton. We give a theoretical analysis that Nesterov's acceleration technique can improve the convergence performance for approximate Newton just like for first-order methods. We accordingly propose an accelerated regularized sub-sampled Newton. Our accelerated algorithm performs much better than the original regularized sub-sampled Newton in experiments, which validates our theory empirically. Besides, the accelerated regularized sub-sampled Newton has good performance comparable to or even better than classical algorithms.

Haishan Ye, Zhihua Zhang

Optimization plays a key role in machine learning. Recently, stochastic second-order methods have attracted much attention due to their low computational cost in each iteration. However, these algorithms might perform poorly especially if it is hard to approximate the Hessian well and efficiently. As far as we know, there is no effective way to handle this problem. In this paper, we resort to Nestrov's acceleration technique to improve the convergence performance of a class of second-order methods called approximate Newton. We give a theoretical analysis that Nestrov's acceleration technique can improve the convergence performance for approximate Newton just like for first-order methods. We accordingly propose an accelerated regularized sub-sampled Newton. Our accelerated algorithm performs much better than the original regularized sub-sampled Newton in experiments, which validates our theory empirically. Besides, the accelerated regularized sub-sampled Newton has good performance comparable to or even better than state-of-art algorithms.

Haishan Ye, Qiaoming Ye, Zhihua Zhang

Generalized matrix approximation plays a fundamental role in many machine learning problems, such as CUR decomposition, kernel approximation, and matrix low rank approximation. Especially with today's applications involved in larger and larger dataset, more and more efficient generalized matrix approximation algorithems become a crucially important research issue. In this paper, we find new sketching techniques to reduce the size of the original data matrix to develop new matrix approximation algorithms. Our results derive a much tighter bound for the approximation than previous works: we obtain a $(1+ε)$ approximation ratio with small sketched dimensions which implies a more efficient generalized matrix approximation.

Haishan Ye, Luo Luo, Zhihua Zhang

Many machine learning models depend on solving a large scale optimization problem. Recently, sub-sampled 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 at each iteration while commanding a high convergence rate. In this work we propose two new efficient Newton-type methods, Refined Sub-sampled Newton and Refined Sketch Newton. Our methods exhibit a great advantage over existing sub-sampled Newton methods, especially when Hessian-vector multiplication can be calculated efficiently. Specifically, the proposed methods are shown to converge superlinearly in general case and quadratically under a little stronger assumption. The proposed methods can be generalized to a unifying framework for the convergence proof of several existing sub-sampled Newton methods, revealing new convergence properties. Finally, we empirically evaluate the performance of our methods on several standard datasets and the results show consistent improvement in computational efficiency.