Scalable Semidefinite Programming

Yurtsever, Alp; Tropp, Joel A.; Fercoq, Olivier; Udell, Madeleine; Cevher, Volkan

doi:10.48550/arxiv.1912.02949

Cited by 8 publications

(18 citation statements)

References 75 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This observation is a key motivation for our notion of sampled solutions for SDPs. Following [36], we define the working memory of an algorithm as follows.…”

Section: Motivating Example: Maximum Cut Problemmentioning

confidence: 99%

“…In their recent work, Yurtsever et al [36] extend the approach from [35] to deal with SDPs with d linear equality constraints. They provide a polynomial-time randomized sketching algorithm that, with high probability, computes a rank-r approximation of a near-feasible solution to SDP with d linear equality constraints.…”

Section: Related Work On Low Memory Algorithms For Sdpmentioning

confidence: 99%

“…is the nuclear norm. We will discuss the similarities and differences between our work and [36] in Section 7.…”

Section: Related Work On Low Memory Algorithms For Sdpmentioning

confidence: 99%

“…Working Memory Required Tropp et al [31] Θ(d + nr) Yurtsever et al [36] O(d + rn/ζ) Ding et al [11] Θ [2,29] states that the rank of the optimal solution could be as high as Ω( √ n). As such, the algorithms for solving (MaxCut-SDP) could require Ω(n 1.5 ) working memory.…”

Section: Referencementioning

confidence: 99%

See 3 more Smart Citations

Memory-efficient structured convex optimization via extreme point sampling

Shinde¹,

Narayanan²,

Saunderson³

2020

Preprint

View full text Add to dashboard Cite

Memory is a key computational bottleneck when solving large-scale convex optimization problems such as semidefinite programs (SDPs). In this paper, we focus on the regime in which storing an n × n matrix decision variable is prohibitive. To solve SDPs in this regime, we develop a randomized algorithm that returns a random vector whose covariance matrix is near-feasible and near-optimal for the SDP. We show how to develop such an algorithm by modifying the Frank-Wolfe algorithm to systematically replace the matrix iterates with random vectors. As an application of this approach, we show how to implement the Goemans-Williamson approximation algorithm for MaxCut using O(n) memory in addition to the memory required to store the problem instance. We then extend our approach to deal with a broader range of structured convex optimization problems, replacing decision variables with random extreme points of the feasible region.

show abstract

“…This observation is a key motivation for our notion of sampled solutions for SDPs. Following [36], we define the working memory of an algorithm as follows.…”

Section: Motivating Example: Maximum Cut Problemmentioning

confidence: 99%

Section: Related Work On Low Memory Algorithms For Sdpmentioning

confidence: 99%

“…is the nuclear norm. We will discuss the similarities and differences between our work and [36] in Section 7.…”

Section: Related Work On Low Memory Algorithms For Sdpmentioning

confidence: 99%

Section: Referencementioning

confidence: 99%

See 2 more Smart Citations

Memory-efficient structured convex optimization via extreme point sampling

Shinde¹,

Narayanan²,

Saunderson³

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…Recovery of low-rank matrices has a wide array of applications, including recommendation systems (Recht et al, 2010;Candès & Recht, 2009;Davenport & Romberg, 2016;Chandrasekaran et al, 2012;Chen & Chi, 2018;Candès & Plan, 2011), quantum state tomography (Recht et al, 2010;Kyrillidis et al, 2018;Flammia et al, 2012;Gross et al, 2010;Liu, 2011;Chen et al, 2015;Cai & Zhang, 2015), phase retrieval and blind deconvolution (Shechtman et al, 2015;Fienup, 1982;Candès et al, 2013;Chen et al, 2015;Cai & Zhang, 2015;Li et al, 2016;Segarra et al, 2017;Ling & Strohmer, 2015), neural word embeddings (Mikolov et al, 2013;Pennington et al, 2014), text classification (Joulin et al, 2017), convexified convolutional NNs (Zhang et al, 2017), and SDP instances (Burer & Monteiro, 2003;Bhojanapalli et al, 2018;Kyrillidis et al, 2018;Wang et al, 2017;Yurtsever et al, 2019;Goto et al, 2019). Because of its importance in practice, there has been a large push in the literature to develop efficient algorithms for this task (Davenport & Romberg, 2016).…”

Section: Introductionmentioning

confidence: 99%

Rank-One Measurements of Low-Rank PSD Matrices Have Small Feasible Sets

Roddenberry,

Segarra,

Kyrillidis

2020

Preprint

View full text Add to dashboard Cite

We study the role of the constraint set in determining the solution to low-rank, positive semidefinite (PSD) matrix sensing problems. The setting we consider involves rank-one sensing matrices: In particular, given a set of rank-one projections of an approximately low-rank PSD matrix, we characterize the radius of the set of PSD matrices that satisfy the measurements. This result yields a sampling rate to guarantee singleton solution sets when the true matrix is exactly low-rank, such that the choice of the objective function or the algorithm to be used is inconsequential in its recovery. We discuss applications of this contribution and compare it to recent literature regarding implicit regularization for similar problems. We demonstrate practical implications of this result by applying conic projection methods for PSD matrix recovery without incorporating low-rank regularization.

show abstract

Limitations of optimization algorithms on noisy quantum devices

2021

View full text Add to dashboard Cite

Recent technological developments have focused the interest of the quantum computing community on investigating how near-term devices could outperform classical computers for practical applications. A central question that remains open is whether their noise can be overcome or it fundamentally restricts any potential quantum advantage. We present a transparent way of comparing classical algorithms to quantum ones running on near-term quantum devices for a large family of problems that include optimization problems and approximations to the ground state energy of Hamiltonians. Our approach is based on the combination of entropic inequalities that determine how fast the quantum computation state converges to the fixed point of the noise model, together with established classical methods of Gibbs state sampling. The approach is extremely versatile and allows for its application to a large variety of problems, noise models and quantum computing architectures. We use our result to provide estimates for a variety of problems and architectures that have been the focus of recent experiments, such as quantum annealers, variational quantum eigensolvers, and quantum approximate optimization. The bounds we obtain indicate that substantial quantum advantages are unlikely for classical optimization unless the current noise rates are decreased by orders of magnitude or the topology of the problem matches that of the device. This is the case even if the number of qubits increases substantially. We reach similar but less stringent conclusions for quantum Hamiltonian problems.

show abstract

Scalable Semidefinite Programming

Cited by 8 publications

References 75 publications

Memory-efficient structured convex optimization via extreme point sampling

Memory-efficient structured convex optimization via extreme point sampling

Rank-One Measurements of Low-Rank PSD Matrices Have Small Feasible Sets

Limitations of optimization algorithms on noisy quantum devices

Contact Info

Product

Resources

About