Linear convergence of a modified Frank–Wolfe algorithm for computing minimum-volume enclosing ellipsoids

Ahipaşaoğlu, Selin Damla; Sun, Peng; Todd, Michael J.

doi:10.1080/10556780701589669

Cited by 76 publications

(179 citation statements)

References 16 publications

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“…All mentioned algorithms in the above paragraph have in common that they converge, were successfully applied in practice, but no convergence speed or bound on the running time has ever been proved so far. Here we prove the convergence speed for Gilbert's algorithm, on one hand by observing for the first time that it is nothing else than the Frank-Wolfe algorithm [11] applied to the standard quadratic programming formulation of polytope distance 3 , and on the other hand by giving 3 The quadratic programming formulation is minx f (x) = (Ax) 2 , xi ≥ 0, P i xi = 1 when A is the d × n-matrix containing all points as columns. Then the gradient ∇f (x) T = a new slightly easier geometric variant of recent proofs by [3,9] on the convergence speed of sparse greedy approximation:…”

Section: Gilbert's Algorithmmentioning

confidence: 89%

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Coresets for polytope distance

Gärtner

Jäggi

2009

Proceedings of the Twenty-Fifth Annual Symposium on Computational Geometry

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Following recent work of Clarkson, we translate the coreset framework to the problems of finding the point closest to the origin inside a polytope, finding the shortest distance between two polytopes, Perceptrons, and soft-as well as hard-margin Support Vector Machines (SVM). We prove asymptotically matching upper and lower bounds on the size of coresets, stating that -coresets of size (1 + o(1))E * / do always exist as → 0, and that this is best possible. The crucial quantity E * is what we call the excentricity of a polytope, or a pair of polytopes.Additionally, we prove linear convergence speed of Gilbert's algorithm, one of the earliest known approximation algorithms for polytope distance, and generalize both the algorithm and the proof to the two polytope case.Interestingly, our coreset bounds also imply that we can for the first time prove matching upper and lower bounds for the sparsity of Perceptron and SVM solutions.

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Section: Gilbert's Algorithmmentioning

confidence: 89%

“…The key fact enabling linear convergence is the following bound, originally due to [3]: In each step of Gilbert's algorithm, the improvement in the primal error hi is at least…”

Section: Gilbert's Algorithmmentioning

confidence: 99%

Coresets for polytope distance

Gärtner

Jäggi

2009

Proceedings of the Twenty-Fifth Annual Symposium on Computational Geometry

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“…Further discussions on optimization of this algorithm can be found in Ahipasaoglu et al (2008). In addition to methods to select beam particles and graphics for each time step, it is often useful to track the bins occupied by the beam candidates by using lifetime diagrams.…”

Section: Deriving Maximum Peaks (B3)mentioning

confidence: 99%

Automated Detection and Analysis of Particle Beams in Laser-Plasman Accelerator Simulations

Ushizima¹,

Geddes²,

Cormier‐Michel³

et al. 2010

Machine Learning

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“…For the general problem of maximizing a concave function over a polytope, Wolfe [41] sketched and Guélat and Marcotte [20] detailed the proof of linear convergence under the assumptions of Lipschitz continuity of the gradient of the objective function, strong concavity of the objective function, and strict complementarity. More recently, Ahipaşaoglu, Sun, and Todd [1] established the linear convergence of such an algorithm for the problem of maximizing a concave function over the unit simplex under a slightly different set of assumptions. Unfortunately, none of these previous results is directly applicable to our case, since either set of these assumptions implies the uniqueness of the optimal solution, which is not, in general, satisfied by the dual formulation of the minimum enclosing ball problem.…”

Section: Linear Convergence Of the Second Algorithmmentioning

confidence: 99%

“…We therefore use an argument similar to that of [1] to establish the linear convergence of Algorithm 4.1. We work with a perturbation of the primal formulation (P 2 ) and show that the distance from an optimal primal-dual solution of the perturbed problem to the set of optimal primal-dual solutions of (P 2 ) and (D) satisfies Downloaded 09/28/17 to 139.179.72.198.…”

Section: Linear Convergence Of the Second Algorithmmentioning

confidence: 99%

Two Algorithms for the Minimum Enclosing Ball Problem

Yıldırım¹

2008

SIAM J. Optim.

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Abstract. Given A := {a 1 , . . . , a m } ⊂ R n and > 0, we propose and analyze two algorithms for the problem of computing a (1 + )-approximation to the radius of the minimum enclosing ball of A. The first algorithm is closely related to the Frank-Wolfe algorithm with a proper initialization applied to the dual formulation of the minimum enclosing ball problem. We establish that this algorithm converges in O(1/ ) iterations with an overall complexity bound of O(mn/ ) arithmetic operations. In addition, the algorithm returns a "core set" of size O(1/ ), which is independent of both m and n. The latter algorithm is obtained by incorporating "away" steps into the former one at each iteration and achieves the same asymptotic complexity bound as the first one. While the asymptotic bound on the size of the core set returned by the second algorithm also remains the same as the first one, the latter algorithm has the potential to compute even smaller core sets in practice, since, in contrast to the former one, it allows "dropping" points from the working core set at each iteration. Our analysis reveals that the leading terms in the asymptotic complexity analysis are reasonably small. In contrast to the first algorithm, we also establish that the second algorithm asymptotically exhibits linear convergence, which provides further insight into our computational results, indicating that the latter algorithm indeed terminates faster with smaller core sets in comparison with the first one. We also discuss how our algorithms can be extended to compute an approximation to the minimum enclosing ball of more general input sets without sacrificing the iteration complexity and the bound on the core set size. In particular, we establish the existence of a core set of size O(1/ ) for a much wider class of input sets. We adopt the real number model of computation in our analysis.

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Linear convergence of a modified Frank–Wolfe algorithm for computing minimum-volume enclosing ellipsoids

Abstract: We show the linear convergence of a simple first-order algorithm for the minimum-volume enclosing ellipsoid problem and its dual, the D-optimal design problem of statistics. Computational tests confirm the attractive features of this method.

Cited by 76 publications

References 16 publications

Coresets for polytope distance

Coresets for polytope distance

Automated Detection and Analysis of Particle Beams in Laser-Plasman Accelerator Simulations

Two Algorithms for the Minimum Enclosing Ball Problem

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