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

We propose a novel high-dimensional linear regression estimator: the <italic>Discrete Dantzig Selector</italic>, 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 (<monospace>MILO</monospace>) approach to obtain <italic>certifiably optimal</italic> 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 <italic>quadratic</italic> optimization, recently proposed by Bertsimas <italic>et al.</italic> and the continuous nonconvex quadratic optimization framework of Liu <italic>et al.</italic>. We propose new discrete first-order methods, which when paired with the state-of-the-art <monospace>MILO</monospace> solvers, lead to good solutions for the <italic>Discrete Dantzig Selector</italic> 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 <monospace>MILO</monospace> solvers. We demonstrate both theoretically and empirically that in a wide range of regimes the statistical properties of the <italic>Discrete Dantzig Selector</italic> are superior to those of popular <inline-formula> <tex-math notation="LaTeX">$\ell _{1}$ </tex-math></inline-formula>-based approaches. We illustrate that our approach can handle problem instances with <inline-formula> <tex-math notation="LaTeX">$p =10,\!000$ </tex-math></inline-formula> features with certifiable optimality making it a highly scalable combinatorial variable selection approach in sparse linear modeling.