Sublinear time algorithms for approximate semidefinite programming

Garber, Dan; Hazan, Elad

doi:10.1007/s10107-015-0932-z

Cited by 15 publications

(12 citation statements)

References 21 publications

(2 reference statements)

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“…Here, the key is that each iteration increases the rank of the solution only by one, so that if only a few iterations are required to reach satisfactory accuracy, then only low dimensional objects need to be manipulated. This line of work was later improved by Laue [2012], Garber [2016] and Garber and Hazan [2016] through hybrid methods. Still, if high accuracy solutions are desired, a large number of iterations will be required, eventually leading to large-rank iterates.…”

Section: Related Workmentioning

confidence: 99%

Smoothed analysis of the low-rank approach for smooth semidefinite programs

Pumir¹,

Jelassi²,

Boumal³

2018

Preprint

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We consider semidefinite programs (SDPs) of size n with equality constraints. In order to overcome scalability issues, Burer and Monteiro proposed a factorized approach based on optimizing over a matrix Y of size n × k such that X = Y Y * is the SDP variable. The advantages of such formulation are twofold: the dimension of the optimization variable is reduced, and positive semidefiniteness is naturally enforced. However, optimization in Y is non-convex. In prior work, it has been shown that, when the constraints on the factorized variable regularly define a smooth manifold, provided k is large enough, for almost all cost matrices, all second-order stationary points (SOSPs) are optimal. Importantly, in practice, one can only compute points which approximately satisfy necessary optimality conditions, leading to the question: are such points also approximately optimal? To answer it, under similar assumptions, we use smoothed analysis to show that approximate SOSPs for a randomly perturbed objective function are approximate global optima, with k scaling like the square root of the number of constraints (up to log factors). Moreover, we bound the optimality gap at the approximate solution of the perturbed problem with respect to the original problem. We particularize our results to an SDP relaxation of phase retrieval.

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Section: Related Workmentioning

confidence: 99%

Smoothed analysis of the low-rank approach for smooth semidefinite programs

Pumir¹,

Jelassi²,

Boumal³

2018

Preprint

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“…The resulting MMW algorithm can be interpreted as performing gradient descent in a dual space and using the matrix exponential map to transfer information back to the primal space. To scale this approach to larger problems, researchers have proposed linearization, random projection, sparsification techniques, and stochastic Lanczos quadrature to approximate the matrix exponential [11,7,80,40,41,8,13,61,29]. Even so, the reduction to a sequence of feasibility problems makes this technique impractical for general SDPs.…”

Section: Datasets and Evaluationmentioning

confidence: 99%

Scalable Semidefinite Programming

Yurtsever¹,

Tropp²,

Fercoq³

et al. 2021

SIAM Journal on Mathematics of Data Science

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Semidefinite programming (SDP) is a powerful framework from convex optimization that has striking potential for data science applications. This paper develops a provably correct randomized algorithm for solving large, weakly constrained SDP problems by economizing on the storage and arithmetic costs. Numerical evidence shows that the method is effective for a range of applications, including relaxations of MaxCut, abstract phase retrieval, and quadratic assignment. Running on a laptop equivalent, the algorithm can handle SDP instances where the matrix variable has over 10 14 entries.

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“…We tabulate the arithmetic costs of an algorithm by counting the number of times we apply each primitive, plus the total amount of extra arithmetic. For lower bounds on the arithmetic operations needed to solve an SDP to moderate accuracy, see [44,Thm. 2].…”

Section: Resource Usagementioning

confidence: 99%

Scalable Semidefinite Programming

Yurtsever,

Tropp,

Fercoq

et al. 2019

Preprint

View full text Add to dashboard Cite

Semidefinite programming (SDP) is a powerful framework from convex optimization that has striking potential for data science applications. This paper develops a provably correct algorithm for solving large SDP problems by economizing on both the storage and the arithmetic costs. Numerical evidence shows that the method is effective for a range of applications, including relaxations of MaxCut, abstract phase retrieval, and quadratic assignment. Running on a laptop, the algorithm can handle SDP instances where the matrix variable has over 10 13 entries.

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Sublinear time algorithms for approximate semidefinite programming

Abstract: We present an algorithm for approximating semidefinite programs with running time that is sublinear in the number of entries in the semidefinite instance. We also present lower bounds that show our algorithm to have a nearly optimal running time 1 .

Cited by 15 publications

References 21 publications

Smoothed analysis of the low-rank approach for smooth semidefinite programs

Smoothed analysis of the low-rank approach for smooth semidefinite programs

Scalable Semidefinite Programming

Scalable Semidefinite Programming

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