4.5 Article

The Discrete Dantzig Selector: Estimating Sparse Linear Models via Mixed Integer Linear Optimization

期刊

IEEE TRANSACTIONS ON INFORMATION THEORY
卷 63, 期 5, 页码 3053-3075

出版社

IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC
DOI: 10.1109/TIT.2017.2658023

关键词

Sparse linear regression; Dantzig selector; l(0)-minimization; mixed integer linear optimization; mathematical optimization; nonconvex optimization

资金

  1. ONR [N000141512342]
  2. Moore Sloan Foundation
  3. NSF [DMS-1209057]
  4. U.S. Department of Defense (DOD) [N000141512342] Funding Source: U.S. Department of Defense (DOD)

向作者/读者索取更多资源

We propose a novel high-dimensional linear regression estimator: the Discrete Dantzig Selector, which minimizes the number of nonzero regression coefficients subject to a budget on the maximal absolute correlation between the features and residuals. Motivated by the significant advances in integer optimization over the past 10-15 years, we present a mixed integer linear optimization (MILO) approach to obtain certifiably optimal global solutions to this nonconvex optimization problem. The current state of algorithmics in integer optimization makes our proposal substantially more computationally attractive than the least squares subset selection framework based on integer quadratic optimization, recently proposed by Bertsimas et al. and the continuous nonconvex quadratic optimization framework of Liu et al.. We propose new discrete first-order methods, which when paired with the state-of-the-art MILO solvers, lead to good solutions for the Discrete Dantzig Selector problem for a given computational budget. We illustrate that our integrated approach provides globally optimal solutions in significantly shorter computation times, when compared to off-the-shelf MILO solvers. We demonstrate both theoretically and empirically that in a wide range of regimes the statistical properties of the Discrete Dantzig Selector are superior to those of popular l(1)-based approaches. We illustrate that our approach can handle problem instances with p = 10,000 features with certifiable optimality making it a highly scalable combinatorial variable selection approach in sparse linear modeling.

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