TY - JOUR

T1 - Faster randomized infeasible interior point methods for tall/wide linear programs

AU - Chowdhury, Agniva

AU - London, Palma

AU - Avron, Haim

AU - Drineas, Petros

N1 - Publisher Copyright:
© 2020 Neural information processing systems foundation. All rights reserved.

PY - 2020

Y1 - 2020

N2 - Linear programming (LP) is used in many machine learning applications, such as l1-regularized SVMs, basis pursuit, nonnegative matrix factorization, etc. Interior Point Methods (IPMs) are one of the most popular methods to solve LPs both in theory and in practice. Their underlying complexity is dominated by the cost of solving a system of linear equations at each iteration. In this paper, we consider infeasible IPMs for the special case where the number of variables is much larger than the number of constraints (i.e., wide), or vice-versa (i.e., tall) by taking the dual. Using tools from Randomized Linear Algebra, we present a preconditioning technique that, when combined with the Conjugate Gradient iterative solver, provably guarantees that infeasible IPM algorithms (suitably modified to account for the error incurred by the approximate solver), converge to a feasible, approximately optimal solution, without increasing their iteration complexity. Our empirical evaluations verify our theoretical results on both real and synthetic data.

AB - Linear programming (LP) is used in many machine learning applications, such as l1-regularized SVMs, basis pursuit, nonnegative matrix factorization, etc. Interior Point Methods (IPMs) are one of the most popular methods to solve LPs both in theory and in practice. Their underlying complexity is dominated by the cost of solving a system of linear equations at each iteration. In this paper, we consider infeasible IPMs for the special case where the number of variables is much larger than the number of constraints (i.e., wide), or vice-versa (i.e., tall) by taking the dual. Using tools from Randomized Linear Algebra, we present a preconditioning technique that, when combined with the Conjugate Gradient iterative solver, provably guarantees that infeasible IPM algorithms (suitably modified to account for the error incurred by the approximate solver), converge to a feasible, approximately optimal solution, without increasing their iteration complexity. Our empirical evaluations verify our theoretical results on both real and synthetic data.

UR - http://www.scopus.com/inward/record.url?scp=85108403763&partnerID=8YFLogxK

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AN - SCOPUS:85108403763

SN - 1049-5258

VL - 2020-December

JO - Advances in Neural Information Processing Systems

JF - Advances in Neural Information Processing Systems

T2 - 34th Conference on Neural Information Processing Systems, NeurIPS 2020

Y2 - 6 December 2020 through 12 December 2020

ER -