Tensor optimization is crucial to massive machine learning and signal processing tasks. In this paper, we consider tensor optimization with a convex and well-conditioned objective function and reformulate it into a nonconvex optimization using the Burer-Monteiro type parameterization. We analyze the local convergence of applying vanilla gradient descent to the factored formulation and establish a local regularity condition under mild assumptions. We also provide a linear convergence analysis of the gradient descent algorithm started in a neighborhood of the true tensor factors. Complementary to the local analysis, this work also characterizes the global geometry of the best rank-one tensor approximation problem and demonstrates that for orthogonally decomposable tensors the problem has no spurious local minima and all saddle points are strict except for the one at zero which is a third-order saddle point.
The most widely used technique for solving large-scale semidefinite programs (SDPs) in practice is the non-convex Burer-Monteiro method, which explicitly maintains a low-rank SDP solution for memory efficiency. There has been much recent interest in obtaining a better theoretical understanding of the Burer-Monteiro method. When the maximum allowed rank $p$ of the SDP solution is above the Barvinok-Pataki bound (where a globally optimal solution of rank at most $p$ is guaranteed to exist), a recent line of work established convergence to a global optimum for generic or smoothed instances of the…
Consider a semidefinite program (SDP) involving an $n\times n$ positive semidefinite matrix $X$. The Burer-Monteiro method uses the substitution $X=Y Y^T$ to obtain a nonconvex optimization problem in terms of an $n\times p$ matrix $Y$. Boumal et al. showed that this nonconvex method provably solves equality-constrained SDPs with a generic cost matrix when $p \gtrsim \sqrt{2m}$, where $m$ is the number of constraints. In this note we extend their result to arbitrary SDPs, possibly involving inequalities or multiple semidefinite constraints. We derive similar guarantees for a fixed cost matrix …
When solving large scale semidefinite programs that admit a low-rank solution, an efficient heuristic is the Burer-Monteiro factorization: instead of optimizing over the full matrix, one optimizes over its low-rank factors. This reduces the number of variables to optimize, but destroys the convexity of the problem, thus possibly introducing spurious second-order critical points. The article [Boumal, Voroninski, and Bandeira, 2018] shows that when the size of the factors is of the order of the square root of the number of linear constraints, this does not happen: for almost any cost matrix, sec…
We propose a bilinear decomposition for the Burer-Monteiro method and combine it with the standard Alternating Direction Method of Multipliers algorithm for semidefinite programming. Bilinear decomposition reduces the degree of the augmented Lagrangian from four to two, which makes each of the subproblems a quadratic programming and hence computationally efficient. Our approach is able to solve a class of large-scale SDPs with diagonal constraints. We prove that our ADMM algorithm converges globally to a first-order stationary point, and show by exploiting the negative curvature that the algor…
The Burer-Monteiro method is one of the most widely used techniques for solving large-scale semidefinite programs (SDP). The basic idea is to solve a nonconvex program in $Y$, where $Y$ is an $n \times p$ matrix such that $X = Y Y^T$. In this paper, we show that this method can solve SDPs in polynomial time in a smoothed analysis setting. More precisely, we consider an SDP whose domain satisfies some compactness and smoothness assumptions, and slightly perturb the cost matrix and the constraints. We show that if $p \gtrsim \sqrt{2(1+η)m}$, where $m$ is the number of constraints and $η>0$ is an…
University of Toronto（多伦多大学）；Massachusetts Institute of Technology（麻省理工学院）
Semidefinite programming (SDP) with diagonal constraints arise in many optimization problems, such as Max-Cut, community detection and group synchronization. Although SDPs can be solved to arbitrary precision in polynomial time, generic convex solvers do not scale well with the dimension of the problem. In order to address this issue, Burer and Monteiro proposed to reduce the dimension of the problem by appealing to a low-rank factorization and solve the subsequent non-convex problem instead. In this paper, we present coordinate ascent based methods to solve this non-convex problem with provab…
Many fundamental low-rank optimization problems, such as matrix completion, phase synchronization/retrieval, power system state estimation, and robust PCA, can be formulated as the matrix sensing problem. Two main approaches for solving matrix sensing are based on semidefinite programming (SDP) and Burer-Monteiro (B-M) factorization. The SDP method suffers from high computational and space complexities, whereas the B-M method may return a spurious solution due to the non-convexity of the problem. The existing theoretical guarantees for the success of these methods have led to similar conservat…
Accelerating nuclear-norm regularized low-rank matrix optimization through Burer-Monteiro decomposition
Institute of Statistical Mathematics（统计数学研究所）；Department of Mathematics University of Maryland at College Park（大学数学系）；Institute of Operations Research and Analytics National University of Singapore（运营研究与分析国家大学研究所）；Department of Mathematics and Institute of Operations Research and Analytics National University of Singapore（大学数学系和运营研究与分析国家大学研究所）
Preconditioned Gradient Descent for Overparameterized Nonconvex Burer--Monteiro Factorization with Global Optimality Certification
Electrical and Computer Engineering University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校电气与计算机工程学院）；Industrial and Operations Engineering University of Michigan（密歇根大学工业与运营工程学院）
The so-called Burer-Monteiro method is a well-studied technique for solving large-scale semidefinite programs (SDPs) via low-rank factorization. The main idea is to solve rank-restricted, albeit non-convex, surrogates instead of the SDP. Recent works have shown that, in an important class of SDPs with elegant geometric structure, one can find globally optimal solutions to the SDP by finding rank-deficient second-order critical points of an unconstrained Riemannian optimization problem. Hence, in such problems, the Burer-Monteiro approach can provide a scalable and reliable alternative to inter…
On the convergence of orthogonalization-free conjugate gradient method for extreme eigenvalues of Hermitian matrices: a Riemannian optimization interpretation
We study a family of (potentially non-convex) constrained optimization problems with convex composite structure. Through a novel analysis of non-smooth geometry, we show that proximal-type algorithms applied to exact penalty formulations of such problems exhibit local linear convergence under a quadratic growth condition, which the compositional structure we consider ensures. The main application of our results is to low-rank semidefinite optimization with Burer-Monteiro factorizations. We precisely identify the conditions for quadratic growth in the factorized problem via structures in the se…
This work considers two popular minimization problems: (i) the minimization of a general convex function $f(\mathbf{X})$ with the domain being positive semi-definite matrices; (ii) the minimization of a general convex function $f(\mathbf{X})$ regularized by the matrix nuclear norm $\|\mathbf{X}\|_*$ with the domain being general matrices. Despite their optimal statistical performance in the literature, these two optimization problems have a high computational complexity even when solved using tailored fast convex solvers. To develop faster and more scalable algorithms, we follow the proposal o…
In this paper, we propose a new nonlinear optimization model to solve semidefinite optimization problems (SDPs), providing some properties related to local optimal solutions. The proposed model is based on another nonlinear optimization model given by Burer and Monteiro (2003), but it has several nice properties not seen in the existing one. Firstly, the decision variable of the proposed model is a triangular low-rank matrix, and hence the dimension of its decision variable space is smaller. Secondly, the existence of a strict local optimum of the proposed model is guaranteed under some condit…
We consider the Max-Cut problem. Let $G = (V,E)$ be a graph with adjacency matrix $(a_{ij})_{i,j=1}^{n}$. Burer, Monteiro & Zhang proposed to find, for $n$ angles $\left\{θ_1, θ_2, \dots, θ_n\right\} \subset [0, 2π]$, minima of the energy $$ f(θ_1, \dots, θ_n) = \sum_{i,j=1}^{n} a_{ij} \cos{(θ_i - θ_j)}$$ because configurations achieving a global minimum leads to a partition of size 0.878 Max-Cut(G). This approach is known to be computationally viable and leads to very good results in practice. We prove that by replacing $\cos{(θ_i - θ_j)}$ with an explicit function $g_{\varepsilon}(θ_i - θ_j)…
The problem of sensor network localization (SNL) can be formulated as a semidefinite programming problem with a rank constraint. We propose a new method for solving such SNL problems. We factorize a semidefinite matrix with the rank constraint into a product of two matrices via the Burer--Monteiro factorization. Then, we add the difference of the two matrices, with a penalty parameter, to the objective function, thereby reformulating SNL as an unconstrained multiconvex optimization problem, to which we apply the block coordinate descent method. In this paper, we also provide theoretical analys…
This paper develops distributed synchronous and asynchronous algorithms for the large-scale semi-definite programming with diagonal constraints, which has wide applications in combination optimization, image processing and community detection. The information of the semi-definite programming is allocated to multiple interconnected agents such that each agent aims to find a solution by communicating to its neighbors. Based on low-rank property of solutions and the Burer-Monteiro factorization, we transform the original problem into a distributed optimization problem over unit spheres to reduce …
We propose a practical inexact augmented Lagrangian method (iALM) for nonconvex problems with nonlinear constraints. We characterize the total computational complexity of our method subject to a verifiable geometric condition, which is closely related to the Polyak-Lojasiewicz and Mangasarian-Fromowitz conditions. In particular, when a first-order solver is used for the inner iterates, we prove that iALM finds a first-order stationary point with $\tilde{\mathcal{O}}(1/ε^4)$ calls to the first-order oracle. If, in addition, the problem is smooth and a second-order solver is used for the inner i…
It is well-known that the Burer-Monteiro (B-M) factorization approach can efficiently solve low-rank matrix optimization problems under the RIP condition. It is natural to ask whether B-M factorization-based methods can succeed on any low-rank matrix optimization problems with a low information-theoretic complexity, i.e., polynomial-time solvable problems that have a unique solution. In this work, we provide a negative answer to the above question. We investigate the landscape of B-M factorized polynomial-time solvable matrix completion (MC) problems, which are the most popular subclass of low…
University of Washington（华盛顿大学）；Rutgers University（罗格斯大学）；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）
Existing results for low-rank matrix recovery largely focus on quadratic loss, which enjoys favorable properties such as restricted strong convexity/smoothness (RSC/RSM) and well conditioning over all low rank matrices. However, many interesting problems involve more general, non-quadratic losses, which do not satisfy such properties. For these problems, standard nonconvex approaches such as rank-constrained projected gradient descent (a.k.a. iterative hard thresholding) and Burer-Monteiro factorization could have poor empirical performance, and there is no satisfactory theory guaranteeing glo…
A number of statistical estimation problems can be addressed by semidefinite programs (SDP). While SDPs are solvable in polynomial time using interior point methods, in practice generic SDP solvers do not scale well to high-dimensional problems. In order to cope with this problem, Burer and Monteiro proposed a non-convex rank-constrained formulation, which has good performance in practice but is still poorly understood theoretically. In this paper we study the rank-constrained version of SDPs arising in MaxCut and in synchronization problems. We establish a Grothendieck-type inequality that pr…
For solving strongly convex optimization problems, we propose and study the global convergence of variants of the A-HPE and large-step A-HPE algorithms of Monteiro and Svaiter. We prove linear and the superlinear $\mathcal{O}\left(k^{\,-k\left(\frac{p-1}{p+1}\right)}\right)$ global rates for the proposed variants of the A-HPE and large-step A-HPE methods, respectively. The parameter $p\geq 2$ appears in the (high-order) large-step condition of the new large-step A-HPE algorithm. We apply our results to high-order tensor methods, obtaning a new inexact (relative-error) tensor method for (smooth…
This paper is concerned with finding an optimal algorithm for minimizing a composite convex objective function. The basic setting is that the objective is the sum of two convex functions: the first function is smooth with up to the d-th order derivative information available, and the second function is possibly non-smooth, but its proximal tensor mappings can be computed approximately in an efficient manner. The problem is to find -- in that setting -- the best possible (optimal) iteration complexity for convex optimization. Along that line, for the smooth case (without the second non-smooth p…
We present a family of algorithms, called descent algorithms, for optimizing convex and non-convex functions. We also introduce a new first-order algorithm, called rescaled gradient descent (RGD), and show that RGD achieves a faster convergence rate than gradient descent provided the function is strongly smooth -- a natural generalization of the standard smoothness assumption on the objective function. When the objective function is convex, we present two novel frameworks for "accelerating" descent methods, one in the style of Nesterov and the other in the style of Monteiro and Svaiter, using …
