Machine learning problems such as neural network training, tensor decomposition, and matrix factorization, require local minimization of a nonconvex function. This local minimization is challenged by the presence of saddle points, of which there can be many and from which descent methods may take inordinately large number of iterations to escape. This paper presents a second-order method that modifies the update of Newton's method by replacing the negative eigenvalues of the Hessian by their absolute values and uses a truncated version of the resulting matrix to account for the objective's curvature. The method is shown to escape saddles in at most $1 + \log_{3/2} (δ/2\varepsilon)$ iterations where $\varepsilon$ is the target optimality and $δ$ characterizes a point sufficiently far away from the saddle. This base of this exponential escape is $3/2$ independently of problem constants. Adding classical properties of Newton's method, the paper proves convergence to a local minimum with probability $1-p$ in $O\left(\log(1/p)) + O(\log(1/\varepsilon)\right)$ iterations.
The note considers normalized gradient descent (NGD), a natural modification of classical gradient descent (GD) in optimization problems. A serious shortcoming of GD in non-convex problems is that GD may take arbitrarily long to escape from the neighborhood of a saddle point. This issue can make the convergence of GD arbitrarily slow, particularly in high-dimensional non-convex problems where the relative number of saddle points is often large. The paper focuses on continuous-time descent. It is shown that, contrary to standard GD, NGD escapes saddle points `quickly.' In particular, it is show…
The paper considers distributed gradient flow (DGF) for multi-agent nonconvex optimization. DGF is a continuous-time approximation of distributed gradient descent that is often easier to study than its discrete-time counterpart. The paper has two main contributions. First, the paper considers optimization of nonsmooth, nonconvex objective functions. It is shown that DGF converges to critical points in this setting. The paper then considers the problem of avoiding saddle points. It is shown that if agents' objective functions are assumed to be smooth and nonconvex, then DGF can only converge to…
A commonly used heuristic in non-convex optimization is Normalized Gradient Descent (NGD) - a variant of gradient descent in which only the direction of the gradient is taken into account and its magnitude ignored. We analyze this heuristic and show that with carefully chosen parameters and noise injection, this method can provably evade saddle points. We establish the convergence of NGD to a local minimum, and demonstrate rates which improve upon the fastest known first order algorithm due to Ge e al. (2015). The effectiveness of our method is demonstrated via an application to the problem of…
In centralized settings, it is well known that stochastic gradient descent (SGD) avoids saddle points and converges to local minima in nonconvex problems. However, similar guarantees are lacking for distributed first-order algorithms. The paper studies distributed stochastic gradient descent (D-SGD)--a simple network-based implementation of SGD. Conditions under which D-SGD avoids saddle points and converges to local minima are studied. First, we consider the problem of computing critical points. Assuming loss functions are nonconvex and possibly nonsmooth, it is shown that, for each fixed ini…
We examine the behavior of accelerated gradient methods in smooth nonconvex unconstrained optimization, focusing in particular on their behavior near strict saddle points. Accelerated methods are iterative methods that typically step along a direction that is a linear combination of the previous step and the gradient of the function evaluated at a point at or near the current iterate. (The previous step encodes gradient information from earlier stages in the iterative process.) We show by means of the stable manifold theorem that the heavy-ball method method is unlikely to converge to strict s…
In this paper, we give a sharp analysis for Stochastic Gradient Descent (SGD) and prove that SGD is able to efficiently escape from saddle points and find an $(ε, O(ε^{0.5}))$-approximate second-order stationary point in $\tilde{O}(ε^{-3.5})$ stochastic gradient computations for generic nonconvex optimization problems, when the objective function satisfies gradient-Lipschitz, Hessian-Lipschitz, and dispersive noise assumptions. This result subverts the classical belief that SGD requires at least $O(ε^{-4})$ stochastic gradient computations for obtaining an $(ε,O(ε^{0.5}))$-approximate second-o…
It is well known that finding a global optimum is extremely challenging for nonconvex optimization. There are some recent efforts \cite{anandkumar2016efficient, cartis2018second, cartis2020sharp, chen2019high} regarding the optimization methods for computing higher-order critical points, which can exclude the so-called degenerate saddle points and reach a solution with better quality. Desipte theoretical development in \cite{anandkumar2016efficient, cartis2018second, cartis2020sharp, chen2019high}, the corresponding numerical experiments are missing. In this paper, we propose an implementable …
Avoiding strict saddle points of nonconvex regularized problems
School of Information Science and Technology, ShanghaiTech University（上海科技大学信息科学与技术学院）；School of Mathematical Sciences, Shenzhen University（深圳大学数学科学学院）；Department of Applied Mathematics, The Hong Kong Polytechnic University（香港理工大学应用数学系）
Gradient descent and its variants are widely used in machine learning. However, oracle access of gradient may not be available in many applications, limiting the direct use of gradient descent. This paper proposes a method of estimating gradient to perform gradient descent, that converges to a stationary point for general non-convex optimization problems. Beyond the first-order stationary properties, the second-order stationary properties are important in machine learning applications to achieve better performance. We show that the proposed model-free non-convex optimization algorithm returns …
Clemson University（克莱姆森大学）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Privacy protection and nonconvexity are two challenging problems in decentralized optimization and learning involving sensitive data. Despite some recent advances addressing each of the two problems separately, no results have been reported that have theoretical guarantees on both privacy protection and saddle/maximum avoidance in decentralized nonconvex optimization. We propose a new algorithm for decentralized nonconvex optimization that can enable both rigorous differential privacy and saddle/maximum avoiding performance. The new algorithm allows the incorporation of persistent additive noi…
Randomly initialized first-order optimization algorithms are the method of choice for solving many high-dimensional nonconvex problems in machine learning, yet general theoretical guarantees cannot rule out convergence to critical points of poor objective value. For some highly structured nonconvex problems however, the success of gradient descent can be understood by studying the geometry of the objective. We study one such problem -- complete orthogonal dictionary learning, and provide converge guarantees for randomly initialized gradient descent to the neighborhood of a global optimum. The …
University of California, Berkeley（加州大学伯克利分校）；Microsoft Research, India（微软研究院（印度））；Duke University（杜克大学）；University of Washington, Seattle（华盛顿大学西雅图分校）
Gradient descent (GD) and stochastic gradient descent (SGD) are the workhorses of large-scale machine learning. While classical theory focused on analyzing the performance of these methods in convex optimization problems, the most notable successes in machine learning have involved nonconvex optimization, and a gap has arisen between theory and practice. Indeed, traditional analyses of GD and SGD show that both algorithms converge to stationary points efficiently. But these analyses do not take into account the possibility of converging to saddle points. More recent theory has shown that GD an…
University of Minnesota（明尼苏达大学）；Johns Hopkins University（约翰斯·霍普金斯大学）；Princeton University（普林斯顿大学）；Georgia Institute of Technology（佐治亚理工学院）
We propose a general theory for studying the \xl{landscape} of nonconvex \xl{optimization} with underlying symmetric structures \tz{for a class of machine learning problems (e.g., low-rank matrix factorization, phase retrieval, and deep linear neural networks)}. In specific, we characterize the locations of stationary points and the null space of Hessian matrices \xl{of the objective function} via the lens of invariant groups\removed{for associated optimization problems, including low-rank matrix factorization, phase retrieval, and deep linear neural networks}. As a major motivating example, w…
The article discusses distributed gradient-descent algorithms for computing local and global minima in nonconvex optimization. For local optimization, we focus on distributed stochastic gradient descent (D-SGD)--a simple network-based variant of classical SGD. We discuss local minima convergence guarantees and explore the simple but critical role of the stable-manifold theorem in analyzing saddle-point avoidance. For global optimization, we discuss annealing-based methods in which slowly decaying noise is added to D-SGD. Conditions are discussed under which convergence to global minima is guar…
This work shows that applying Gradient Descent (GD) with a fixed step size to minimize a (possibly nonconvex) quadratic function is equivalent to running the Power Method (PM) on the gradients. The connection between GD with a fixed step size and the PM, both with and without fixed momentum, is thus established. Consequently, valuable eigen-information is available via GD. Recent examples show that GD with a fixed step size, applied to locally quadratic nonconvex functions, can take exponential time to escape saddle points (Simon S. Du, Chi Jin, Jason D. Lee, Michael I. Jordan, Aarti Singh, an…
Department of Statistics, University of California, Davis（加利福尼亚大学戴维斯分校统计系）；Department of Operations Research and Financial Engineering, Princeton University（普林斯顿大学运筹学与金融工程系）
In this paper, we propose and analyze zeroth-order stochastic approximation algorithms for nonconvex and convex optimization, with a focus on addressing constrained optimization, high-dimensional setting and saddle-point avoiding. To handle constrained optimization, we first propose generalizations of the conditional gradient algorithm achieving rates similar to the standard stochastic gradient algorithm using only zeroth-order information. To facilitate zeroth-order optimization in high-dimensions, we explore the advantages of structural sparsity assumptions. Specifically, (i) we highlight an…
Minimax problems, such as generative adversarial network, adversarial training, and fair training, are widely solved by a multi-step gradient descent ascent (MGDA) method in practice. However, its convergence guarantee is limited. In this paper, inspired by the primal-dual hybrid gradient method, we propose a new semi-anchoring (SA) technique for the MGDA method. This makes the MGDA method find a stationary point of a structured nonconvex-nonconcave composite minimax problem; its saddle-subdifferential operator satisfies the weak Minty variational inequality condition. The resulting method, na…
Riemannian ascent--descent for nonconvex nonconcave minimax landscapes: convergence to basin saddle points and applications to distributionally robust optimization
The trust region subproblem (TRS) is to minimize a possibly nonconvex quadratic function over a Euclidean ball. There are typically two cases for (TRS), the so-called ``easy case'' and ``hard case''. Even in the ``easy case'', the sequence generated by the classical projected gradient method (PG) may converge to a saddle point at a sublinear local rate, when the initial point is arbitrarily selected from a nonzero measure feasible set. To our surprise, when applying (PG) to solve a cheap and possibly nonconvex reformulation of (TRS), the generated sequence initialized with {\it any} feasible p…
School of Industrial and Systems Engineering at Georgia Tech（佐治亚理工学院工业与系统工程学院）
Momentum Stochastic Gradient Descent (MSGD) algorithm has been widely applied to many nonconvex optimization problems in machine learning, e.g., training deep neural networks, variational Bayesian inference, and etc. Despite its empirical success, there is still a lack of theoretical understanding of convergence properties of MSGD. To fill this gap, we propose to analyze the algorithmic behavior of MSGD by diffusion approximations for nonconvex optimization problems with strict saddle points and isolated local optima. Our study shows that the momentum helps escape from saddle points, but hurts…
We analyze stochastic gradient algorithms for optimizing nonconvex problems. In particular, our goal is to find local minima (second-order stationary points) instead of just finding first-order stationary points which may be some bad unstable saddle points. We show that a simple perturbed version of stochastic recursive gradient descent algorithm (called SSRGD) can find an $(ε,δ)$-second-order stationary point with $\widetilde{O}(\sqrt{n}/ε^2 + \sqrt{n}/δ^4 + n/δ^3)$ stochastic gradient complexity for nonconvex finite-sum problems. As a by-product, SSRGD finds an $ε$-first-order stationary poi…
University of Washington（华盛顿大学）；University of California, Berkeley（加州大学伯克利分校）
We consider minimizing a nonconvex, smooth function $f$ on a Riemannian manifold $\mathcal{M}$. We show that a perturbed version of Riemannian gradient descent algorithm converges to a second-order stationary point (and hence is able to escape saddle points on the manifold). The rate of convergence depends as $1/ε^2$ on the accuracy $ε$, which matches a rate known only for unconstrained smooth minimization. The convergence rate depends polylogarithmically on the manifold dimension $d$, hence is almost dimension-free. The rate also has a polynomial dependence on the parameters describing the cu…
National University of Defense Technology（国防科技大学）；Hunan University（湖南大学）
Nonconvex optimization algorithms with random initialization have attracted increasing attention recently. It has been showed that many first-order methods always avoid saddle points with random starting points. In this paper, we answer a question: can the nonconvex heavy-ball algorithms with random initialization avoid saddle points? The answer is yes! Direct using the existing proof technique for the heavy-ball algorithms is hard due to that each iteration of the heavy-ball algorithm consists of current and last points. It is impossible to formulate the algorithms as iteration like xk+1= g(x…
University of Michigan（密歇根大学）；Tata Consultancy Services Research（塔塔咨询服务公司研究院）
This study develops a fixed-time convergent saddle point dynamical system for solving min-max problems under a relaxation of standard convexity-concavity assumption. In particular, it is shown that by leveraging the dynamical systems viewpoint of an optimization algorithm, accelerated convergence to a saddle point can be obtained. Instead of requiring the objective function to be strongly-convex--strongly-concave (as necessitated for accelerated convergence of several saddle-point algorithms), uniform fixed-time convergence is guaranteed for functions satisfying only the two-sided Polyak-Łojas…
On the Regularization of Primal-Dual Dynamics in Time-Dependent Nonconvex Optimization
School of Science and Technology, IE University（IE大学科学与技术学院）；Department of Mechanical and Aerospace Engineering, University of California San Diego（加州大学圣地亚哥分校机械与航空航天工程系）
University of California, Berkeley（加州大学伯克利分校）；Microsoft Research, India（微软印度研究院）
Nesterov's accelerated gradient descent (AGD), an instance of the general family of "momentum methods", provably achieves faster convergence rate than gradient descent (GD) in the convex setting. However, whether these methods are superior to GD in the nonconvex setting remains open. This paper studies a simple variant of AGD, and shows that it escapes saddle points and finds a second-order stationary point in $\tilde{O}(1/ε^{7/4})$ iterations, faster than the $\tilde{O}(1/ε^{2})$ iterations required by GD. To the best of our knowledge, this is the first Hessian-free algorithm to find a second…
Modern minimax problems, such as generative adversarial network and adversarial training, are often under a nonconvex-nonconcave setting, and developing an efficient method for such setting is of interest. Recently, two variants of the extragradient (EG) method are studied in that direction. First, a two-time-scale variant of the EG, named EG+, was proposed under a smooth structured nonconvex-nonconcave setting, with a slow $\mathcal{O}(1/k)$ rate on the squared gradient norm, where $k$ denotes the number of iterations. Second, another variant of EG with an anchoring technique, named extra anc…
Oden Institute for Computational Engineering & Sciences, The University of Texas at Austin（德克萨斯大学奥斯汀分校奥登计算工程与科学研究所）；Department of Geological Sciences, The University of Texas at Austin（德克萨斯大学奥斯汀分校地质科学系）
In modern deep learning, highly subsampled stochastic approximation (SA) methods are preferred to sample average approximation (SAA) methods because of large data sets as well as generalization properties. Additionally, due to perceived costs of forming and factorizing Hessians, second order methods are not used for these problems. In this work we motivate the extension of Newton methods to the SA regime, and argue for the use of the scalable low rank saddle free Newton (LRSFN) method, which avoids forming the Hessian in favor of making a low rank approximation. Additionally, LRSFN can facilit…
The paper studies a distributed gradient descent (DGD) process and considers the problem of showing that in nonconvex optimization problems, DGD typically converges to local minima rather than saddle points. The paper considers unconstrained minimization of a smooth objective function. In centralized settings, the problem of demonstrating nonconvergence to saddle points of gradient descent (and variants) is typically handled by way of the stable-manifold theorem from classical dynamical systems theory. However, the classical stable-manifold theorem is not applicable in distributed settings. Th…
We provide a theoretical algorithm for checking local optimality and escaping saddles at nondifferentiable points of empirical risks of two-layer ReLU networks. Our algorithm receives any parameter value and returns: local minimum, second-order stationary point, or a strict descent direction. The presence of $M$ data points on the nondifferentiability of the ReLU divides the parameter space into at most $2^M$ regions, which makes analysis difficult. By exploiting polyhedral geometry, we reduce the total computation down to one convex quadratic program (QP) for each hidden node, $O(M)$ (in)equa…
This paper proposes a stochastic variant of a classic algorithm---the cubic-regularized Newton method [Nesterov and Polyak 2006]. The proposed algorithm efficiently escapes saddle points and finds approximate local minima for general smooth, nonconvex functions in only $\mathcal{\tilde{O}}(ε^{-3.5})$ stochastic gradient and stochastic Hessian-vector product evaluations. The latter can be computed as efficiently as stochastic gradients. This improves upon the $\mathcal{\tilde{O}}(ε^{-4})$ rate of stochastic gradient descent. Our rate matches the best-known result for finding local minima withou…
Second-Order KKT Guarantees for Bregman ADMM in Nonconvex and Non-Lipschitz Optimization
Department of Electrical and Computer Engineering, Iowa State University（电气与计算机工程系，爱荷华州立大学）；Department of Computer Science and Engineering, Ohio State University（计算机科学与工程系，俄亥俄州立大学）；DAMO Academy, Alibaba Group US（阿里巴巴集团美国DAMO研究院）
Gradient-related first-order methods have become the workhorse of large-scale numerical optimization problems. Many of these problems involve nonconvex objective functions with multiple saddle points, which necessitates an understanding of the behavior of discrete trajectories of first-order methods within the geometrical landscape of these functions. This paper concerns convergence of first-order discrete methods to a local minimum of nonconvex optimization problems that comprise strict-saddle points within the geometrical landscape. To this end, it focuses on analysis of discrete gradient tr…
Nanjing University of Information Science & Technology（南京信息工程大学）；Harbin Institute of Technology（哈尔滨工业大学）；Mohamed bin Zayed University of Artificial Intelligence（穆罕默德·本·扎耶德人工智能大学）
We consider escaping saddle points of nonconvex problems where only the function evaluations can be accessed. Although a variety of works have been proposed, the majority of them require either second or first-order information, and only a few of them have exploited zeroth-order methods, particularly the technique of negative curvature finding with zeroth-order methods which has been proven to be the most efficient method for escaping saddle points. To fill this gap, in this paper, we propose two zeroth-order negative curvature finding frameworks that can replace Hessian-vector product computa…
Minimax optimization problems arises from both modern machine learning including generative adversarial networks, adversarial training and multi-agent reinforcement learning, as well as from tradition research areas such as saddle point problems, numerical partial differential equations and optimality conditions of equality constrained optimization. For the unconstrained continuous nonconvex-nonconcave situation, Jin, Netrapalli and Jordan (2019) carefully considered the very basic question: what is a proper definition of local optima of a minimax optimization problem, and proposed a proper de…
In this note, we focus on smooth nonconvex optimization problems that obey: (1) all local minimizers are also global; and (2) around any saddle point or local maximizer, the objective has a negative directional curvature. Concrete applications such as dictionary learning, generalized phase retrieval, and orthogonal tensor decomposition are known to induce such structures. We describe a second-order trust-region algorithm that provably converges to a global minimizer efficiently, without special initializations. Finally we highlight alternatives, and open problems in this direction.
Duke University（杜克大学）；University of California, Berkeley（加州大学伯克利分校）
In this paper we develop a new framework that captures the common landscape underlying the common non-convex low-rank matrix problems including matrix sensing, matrix completion and robust PCA. In particular, we show for all above problems (including asymmetric cases): 1) all local minima are also globally optimal; 2) no high-order saddle points exists. These results explain why simple algorithms such as stochastic gradient descent have global converge, and efficiently optimize these non-convex objective functions in practice. Our framework connects and simplifies the existing analyses on opti…
