Luo Luo

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.

Haikuo Yang, Luo Luo, Chris Junchi Li et al.

We present a method for solving general nonconvex-strongly-convex bilevel optimization problems. Our method -- the \emph{Restarted Accelerated HyperGradient Descent} (\texttt{RAHGD}) method -- finds an $ε$-first-order stationary point of the objective with $\tilde{\mathcal{O}}(κ^{3.25}ε^{-1.75})$ oracle complexity, where $κ$ is the condition number of the lower-level objective and $ε$ is the desired accuracy. We also propose a perturbed variant of \texttt{RAHGD} for finding an $\big(ε,\mathcal{O}(κ^{2.5}\sqrtε\,)\big)$-second-order stationary point within the same order of oracle complexity. Our results achieve the best-known theoretical guarantees for finding stationary points in bilevel optimization and also improve upon the existing upper complexity bound for finding second-order stationary points in nonconvex-strongly-concave minimax optimization problems, setting a new state-of-the-art benchmark. Empirical studies are conducted to validate the theoretical results in this paper.

Lesi Chen, Boyuan Yao, Luo Luo

This paper considers stochastic first-order algorithms for minimax optimization under Polyak--Łojasiewicz (PL) conditions. We propose SPIDER-GDA for solving the finite-sum problem of the form $\min_x \max_y f(x,y)\triangleq \frac{1}{n} \sum_{i=1}^n f_i(x,y)$, where the objective function $f(x,y)$ is $μ_x$-PL in $x$ and $μ_y$-PL in $y$; and each $f_i(x,y)$ is $L$-smooth. We prove SPIDER-GDA could find an $ε$-optimal solution within ${\mathcal O}\left((n + \sqrt{n}\,κ_xκ_y^2)\log (1/ε)\right)$ stochastic first-order oracle (SFO) complexity, which is better than the state-of-the-art method whose SFO upper bound is ${\mathcal O}\big((n + n^{2/3}κ_xκ_y^2)\log (1/ε)\big)$, where $κ_x\triangleq L/μ_x$ and $κ_y\triangleq L/μ_y$. For the ill-conditioned case, we provide an accelerated algorithm to reduce the computational cost further. It achieves $\tilde{\mathcal O}\big((n+\sqrt{n}\,κ_xκ_y)\log^2 (1/ε)\big)$ SFO upper bound when $κ_y \gtrsim \sqrt{n}$. Our ideas also can be applied to the more general setting that the objective function only satisfies PL condition for one variable. Numerical experiments validate the superiority of proposed methods.

Luo Luo, Yunyan Bai, Lesi Chen et al.

We study the decentralized optimization problem $\min_{{\bf x}\in{\mathbb R}^d} f({\bf x})\triangleq \frac{1}{m}\sum_{i=1}^m f_i({\bf x})$, where the local function on the $i$-th agent has the form of $f_i({\bf x})\triangleq \frac{1}{n}\sum_{j=1}^n f_{i,j}({\bf x})$ and every individual $f_{i,j}$ is smooth but possibly nonconvex. We propose a stochastic algorithm called DEcentralized probAbilistic Recursive gradiEnt deScenT (DEAREST) method, which achieves an $ε$-stationary point at each agent with the communication rounds of $\tilde{\mathcal O}(Lε^{-2}/\sqrtγ\,)$, the computation rounds of $\tilde{\mathcal O}(n+(L+\min\{nL, \sqrt{n/m}\bar L\})ε^{-2})$, and the local incremental first-oracle calls of ${\mathcal O}(mn + {\min\{mnL, \sqrt{mn}\bar L\}}{ε^{-2}})$, where $L$ is the smoothness parameter of the objective function, $\bar L$ is the mean-squared smoothness parameter of all individual functions, and $γ$ is the spectral gap of the mixing matrix associated with the network. We then establish the lower bounds to show that the proposed method is near-optimal. Notice that the smoothness parameters $L$ and $\bar L$ used in our algorithm design and analysis are global, leading to sharper complexity bounds than existing results that depend on the local smoothness. We further extend DEAREST to solve the decentralized finite-sum optimization problem under the Polyak-Łojasiewicz condition, also achieving the near-optimal complexity bounds.

Zhuanghua Liu, Luo Luo, Bryan Kian Hsiang Low

We consider the finite-sum optimization problem, where each component function is strongly convex and has Lipschitz continuous gradient and Hessian. The recently proposed incremental quasi-Newton method is based on BFGS update and achieves a local superlinear convergence rate that is dependent on the condition number of the problem. This paper proposes a more efficient quasi-Newton method by incorporating the symmetric rank-1 update into the incremental framework, which results in the condition-number-free local superlinear convergence rate. Furthermore, we can boost our method by applying the block update on the Hessian approximation, which leads to an even faster local convergence rate. The numerical experiments show the proposed methods significantly outperform the baseline methods.

Luo Luo, Xue Cui, Tingkai Jia et al.

This paper studies decentralized optimization problem $f(\mathbf{x})=\frac{1}{m}\sum_{i=1}^m f_i(\mathbf{x})$, where each local function has the form of $f_i(\mathbf{x}) = {\mathbb E}\left[F(\mathbf{x};{\boldsymbol ξ}_i)\right]$ which is $(L_0,L_1)$-smooth but possibly nonconvex and the random variable ${\boldsymbol ξ}_i$ follows distribution ${\mathcal D}_i$. We propose a novel algorithm called decentralized normalized stochastic gradient descent (DNSGD), which can achieve an $ε$-stationary point at each local agent. We present a new framework for analyzing decentralized first-order methods in the relaxed smooth setting, based on the Lyapunov function related to the product of the gradient norm and the consensus error. We show the upper bounds on the sample complexity of ${\mathcal O}(m^{-1}(L_fσ^2Δ_fε^{-4} + σ^2ε^{-2} + L_f^{-2}L_1^3σ^2Δ_fε^{-1} + L_f^{-2}L_1^2σ^2))$ per agent and the communication complexity of $\tilde{\mathcal O}((L_fε^{-2} + L_1ε^{-1})γ^{-1/2}Δ_f)$, where $L_f=L_0 +L_1ζ$, $σ^2$ is the variance of the stochastic gradient, $Δ_f$ is the initial optimal function value gap, $γ$ is the spectral gap of the network, and $ζ$ is the degree of the gradient dissimilarity. In the special case of $L_1=0$, the above results (nearly) match the lower bounds of decentralized stochastic nonconvex optimization under the standard smoothness. We also conduct numerical experiments to show the empirical superiority of our method.

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.

Luo Luo, Yujun Li, Cheng Chen

We study the smooth minimax optimization problem $\min_{\bf x}\max_{\bf y} f({\bf x},{\bf y})$, where $f$ is $\ell$-smooth, strongly-concave in ${\bf y}$ but possibly nonconvex in ${\bf x}$. Most of existing works focus on finding the first-order stationary points of the function $f({\bf x},{\bf y})$ or its primal function $P({\bf x})\triangleq \max_{\bf y} f({\bf x},{\bf y})$, but few of them focus on achieving second-order stationary points. In this paper, we propose a novel approach for minimax optimization, called Minimax Cubic Newton (MCN), which could find an $\big(\varepsilon,κ^{1.5}\sqrt{ρ\varepsilon}\,\big)$-second-order stationary point of $P({\bf x})$ with calling ${\mathcal O}\big(κ^{1.5}\sqrtρ\varepsilon^{-1.5}\big)$ times of second-order oracles and $\tilde{\mathcal O}\big(κ^{2}\sqrtρ\varepsilon^{-1.5}\big)$ times of first-order oracles, where $κ$ is the condition number and $ρ$ is the Lipschitz continuous constant for the Hessian of $f({\bf x},{\bf y})$. In addition, we propose an inexact variant of MCN for high-dimensional problems to avoid calling expensive second-order oracles. Instead, our method solves the cubic sub-problem inexactly via gradient descent and matrix Chebyshev expansion. This strategy still obtains the desired approximate second-order stationary point with high probability but only requires $\tilde{\mathcal O}\big(κ^{1.5}\ell\varepsilon^{-2}\big)$ Hessian-vector oracle calls and $\tilde{\mathcal O}\big(κ^{2}\sqrtρ\varepsilon^{-1.5}\big)$ first-order oracle calls. To the best of our knowledge, this is the first work that considers the non-asymptotic convergence behavior of finding second-order stationary points for minimax problems without the convex-concave assumptions.

Luo Luo, Guangzeng Xie, Tong Zhang et al.

This paper considers stochastic first-order algorithms for convex-concave minimax problems of the form $\min_{\bf x}\max_{\bf y}f(\bf x, \bf y)$, where $f$ can be presented by the average of $n$ individual components which are $L$-average smooth. For $μ_x$-strongly-convex-$μ_y$-strongly-concave setting, we propose a new method which could find a $\varepsilon$-saddle point of the problem in $\tilde{\mathcal O} \big(\sqrt{n(\sqrt{n}+κ_x)(\sqrt{n}+κ_y)}\log(1/\varepsilon)\big)$ stochastic first-order complexity, where $κ_x\triangleq L/μ_x$ and $κ_y\triangleq L/μ_y$. This upper bound is near optimal with respect to $\varepsilon$, $n$, $κ_x$ and $κ_y$ simultaneously. In addition, the algorithm is easily implemented and works well in practical. Our methods can be extended to solve more general unbalanced convex-concave minimax problems and the corresponding upper complexity bounds are also near optimal.

Cheng Chen, Luo Luo, Weinan Zhang et al.

The Frank-Wolfe algorithm is a classic method for constrained optimization problems. It has recently been popular in many machine learning applications because its projection-free property leads to more efficient iterations. In this paper, we study projection-free algorithms for convex-strongly-concave saddle point problems with complicated constraints. Our method combines Conditional Gradient Sliding with Mirror-Prox and shows that it only requires $\tilde{O}(1/\sqrtε)$ gradient evaluations and $\tilde{O}(1/ε^2)$ linear optimizations in the batch setting. We also extend our method to the stochastic setting and propose first stochastic projection-free algorithms for saddle point problems. Experimental results demonstrate the effectiveness of our algorithms and verify our theoretical guarantees.

Luo Luo, Cheng Chen, Guangzeng Xie et al.

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.

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.

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.

Luo Luo, Cheng Chen, Yujun Li et al.

We consider saddle point problems which objective functions are the average of $n$ strongly convex-concave individual components. Recently, researchers exploit variance reduction methods to solve such problems and achieve linear-convergence guarantees. However, these methods have a slow convergence when the condition number of the problem is very large. In this paper, we propose a stochastic proximal point algorithm, which accelerates the variance reduction method SAGA for saddle point problems. Compared with the catalyst framework, our algorithm reduces a logarithmic term of condition number for the iteration complexity. We adopt our algorithm to policy evaluation and the empirical results show that our method is much more efficient than state-of-the-art methods.

Guangzeng Xie, Luo Luo, Zhihua Zhang

This paper studies the lower bound complexity for the optimization problem whose objective function is the average of $n$ individual smooth convex functions. We consider the algorithm which gets access to gradient and proximal oracle for each individual component. For the strongly-convex case, we prove such an algorithm can not reach an $\varepsilon$-suboptimal point in fewer than $Ω((n+\sqrt{κn})\log(1/\varepsilon))$ iterations, where $κ$ is the condition number of the objective function. This lower bound is tighter than previous results and perfectly matches the upper bound of the existing proximal incremental first-order oracle algorithm Point-SAGA. We develop a novel construction to show the above result, which partitions the tridiagonal matrix of classical examples into $n$ groups. This construction is friendly to the analysis of proximal oracle and also could be used to general convex and average smooth cases naturally.

Luo Luo, Cheng Chen, Zhihua Zhang et al.

The frequent directions (FD) technique is a deterministic approach for online sketching that has many applications in machine learning. The conventional FD is a heuristic procedure that often outputs rank deficient matrices. To overcome the rank deficiency problem, we propose a new sketching strategy called robust frequent directions (RFD) by introducing a regularization term. RFD can be derived from an optimization problem. It updates the sketch matrix and the regularization term adaptively and jointly. RFD reduces the approximation error of FD without increasing the computational cost. We also apply RFD to online learning and propose an effective hyperparameter-free online Newton algorithm. We derive a regret bound for our online Newton algorithm based on RFD, which guarantees the robustness of the algorithm. The experimental studies demonstrate that the proposed method outperforms state-of-the-art second order online learning algorithms.

Zihao Chen, Luo Luo, Zhihua Zhang

Recently, there has been an increasing interest in designing distributed convex optimization algorithms under the setting where the data matrix is partitioned on features. Algorithms under this setting sometimes have many advantages over those under the setting where data is partitioned on samples, especially when the number of features is huge. Therefore, it is important to understand the inherent limitations of these optimization problems. In this paper, with certain restrictions on the communication allowed in the procedures, we develop tight lower bounds on communication rounds for a broad class of non-incremental algorithms under this setting. We also provide a lower bound on communication rounds for a class of (randomized) incremental algorithms.

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.

Luo Luo, Zihao Chen, Zhihua Zhang et al.

In this paper, we discuss the problem of minimizing the sum of two convex functions: a smooth function plus a non-smooth function. Further, the smooth part can be expressed by the average of a large number of smooth component functions, and the non-smooth part is equipped with a simple proximal mapping. We propose a proximal stochastic second-order method, which is efficient and scalable. It incorporates the Hessian in the smooth part of the function and exploits multistage scheme to reduce the variance of the stochastic gradient. We prove that our method can achieve linear rate of convergence.

Shusen Wang, Luo Luo, Zhihua Zhang

Symmetric positive semidefinite (SPSD) matrix approximation is an important problem with applications in kernel methods. However, existing SPSD matrix approximation methods such as the Nyström method only have weak error bounds. In this paper we conduct in-depth studies of an SPSD matrix approximation model and establish strong relative-error bounds. We call it the prototype model for it has more efficient and effective extensions, and some of its extensions have high scalability. Though the prototype model itself is not suitable for large-scale data, it is still useful to study its properties, on which the analysis of its extensions relies. This paper offers novel theoretical analysis, efficient algorithms, and a highly accurate extension. First, we establish a lower error bound for the prototype model and improve the error bound of an existing column selection algorithm to match the lower bound. In this way, we obtain the first optimal column selection algorithm for the prototype model. We also prove that the prototype model is exact under certain conditions. Second, we develop a simple column selection algorithm with a provable error bound. Third, we propose a so-called spectral shifting model to make the approximation more accurate when the eigenvalues of the matrix decay slowly, and the improvement is theoretically quantified. The spectral shifting method can also be applied to improve other SPSD matrix approximation models.