A deterministic rescaled perceptron algorithm

Peña, Javier; Soheili, Negar

doi:10.1007/s10107-015-0860-y

Cited by 14 publications

(49 citation statements)

References 21 publications

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“…If ρA > 0, then, similarly to the von Neumann algorithm, the perceptron algorithm terminates with a solution to the system A T y > 0 after at most 1/ρ 2 A iterations (see Novikoff [13]). Peña and Soheili gave a smoothed variant of the perceptron update which guarantees termination in time O( √ log n/ρA) [14], and showed how this gives rise to a polynomial-time algorithm [15] using the rescaling introduced by Betke in [3]. The same running time O( √ log n/ρA) was achieved by Wei Yu et al [21] by adapting the MirrorProx algorithm of Nemirovski [12].…”

Section: Denote By Supp(l+) ⊆ [N] the Maximum Support Of A Point In L+mentioning

confidence: 80%

“…This is equivalent to the existence of convex combinations λ and µ such that z = λiai = − µjaj . The claim follows by showing that λi or µi can be positive only if i ∈ S. This holds since x = λ + µ is a solution to (15), with S ⊇ supp(x) = supp(λ) ∪ supp(µ).…”

Section: B1 Analyzing the Relative Volumementioning

confidence: 93%

“…That is, αai = − j λj aj for some coefficients λ ≥ 0. Setting xj = λj if j = i and xi = α + λi gives a solution to (15) with xi > 0, a contradiction to i / ∈ S. Let us now take an index i ∈ S. Let x be a maximum support solution to (15) with i xi = 1; in particular, xi > 0. We observe that xiai ∈ PA.…”

Section: B1 Analyzing the Relative Volumementioning

confidence: 99%

“…The algorithms we propose fit into a line of research developed over the past 10 years [3,8,4,6,5,15,2,20,16], where simple iterative updates, such as variants of perceptron [17] or of the relaxation method [1,11], are combined with some form of rescaling in order to get polynomial time algorithms for linear programming.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Rescaled Coordinate Descent Methods for Linear Programming

Dadush

Végh

Zambelli

2016

Integer Programming and Combinatorial Optimization

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Abstract. We propose two simple polynomial-time algorithms to find a positive solution to Ax = 0. Both algorithms iterate between coordinate descent steps similar to von Neumann's algorithm, and rescaling steps. In both cases, either the updating step leads to a substantial decrease in the norm, or we can infer that the condition measure is small and rescale in order to improve the geometry. We also show how the algorithms can be extended to find a solution of maximum support for the system Ax = 0, x ≥ 0.

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Section: Denote By Supp(l+) ⊆ [N] the Maximum Support Of A Point In L+mentioning

confidence: 80%

Section: B1 Analyzing the Relative Volumementioning

confidence: 93%

Section: B1 Analyzing the Relative Volumementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Rescaled Coordinate Descent Methods for Linear Programming

Dadush

Végh

Zambelli

2016

Integer Programming and Combinatorial Optimization

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“…(1) Improved rescaling: We design a rescaling method that applies for a parameter of ∆ = Θ( 1 n ), which improves over the threshold ∆ = Θ( 1 m n ) required by [PS16]. This results in a smaller number of iterations that are needed per phase until one can rescale the system.…”

Section: Algorithmmentioning

confidence: 99%

An Improved Deterministic Rescaling for Linear Programming Algorithms

Hoberg

Rothvoß

2017

Integer Programming and Combinatorial Optimization

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The perceptron algorithm for linear programming, arising from machine learning, has been around since the 1950s. While not a polynomial-time algorithm, it is useful in practice due to its simplicity and robustness. In 2004, Dunagan and Vempala showed that a randomized rescaling turns the perceptron method into a polynomial time algorithm, and later Peña and Soheili gave a deterministic rescaling. In this paper, we give a deterministic rescaling for the perceptron algorithm that improves upon the previous rescaling methods by making it possible to rescale much earlier. This results in a faster running time for the rescaled perceptron algorithm. We will also demonstrate that the same rescaling methods yield a polynomial time algorithm based on the multiplicative weights update method. This draws a connection to an area that has received a lot of recent attention in theoretical computer science.

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Solving Conic Systems via Projection and Rescaling

Peña

Soheili

2016

Math. Program.

Self Cite

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We propose a simple projection and rescaling algorithm to solve the feasibility problem find x ∈ L ∩ Ω, where L and Ω are respectively a linear subspace and the interior of a symmetric cone in a finite-dimensional vector space V . This projection and rescaling algorithm is inspired by previous work on rescaled versions of the perceptron algorithm and by Chubanov's projectionbased method for linear feasibility problems. As in these predecessors, each main iteration of our algorithm contains two steps: a basic procedure and a rescaling step. When L ∩ Ω = ∅, the projection and rescaling algorithm finds a pointis attained when L ∩ Ω contains the center of the symmetric cone Ω.We describe several possible implementations for the basic procedure including a perceptron scheme and a smooth perceptron scheme. The perceptron scheme requires O(r 4 ) perceptron updates and the smooth perceptron scheme requires O(r 2 ) smooth perceptron updates, where r stands for the Jordan algebra rank of V . IntroductionWe propose a simple algorithm based on projection and rescaling operations to solve the feasibility problem findwhere L and Ω are respectively a linear subspace and the interior of a symmetric cone in a finite-dimensional vector space V . Problem (1) is fundamental in optimization as it encompasses a large class of feasibility problems. For example, for A ∈ R m×n and b ∈ R m , the problem Ax = b, x > 0 can be formulated as (1) by taking L = {(x, t) ∈ R n+1 : Ax − tb = 0} and Ω = R n+1 ++ . For A ∈ R m×n , c ∈ R n , the problem A

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A deterministic rescaled perceptron algorithm

Cited by 14 publications

References 21 publications

Rescaled Coordinate Descent Methods for Linear Programming

Rescaled Coordinate Descent Methods for Linear Programming

An Improved Deterministic Rescaling for Linear Programming Algorithms

Solving Conic Systems via Projection and Rescaling

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