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

TL;DR

This paper introduces the Discrete Dantzig Selector, a new sparse linear regression estimator optimized via mixed integer linear programming, offering superior statistical properties and scalability for high-dimensional problems.

Contribution

It presents a novel MILO-based approach for globally optimal sparse regression, outperforming existing subset selection and nonconvex quadratic methods in efficiency and scalability.

Findings

Provides certifiably optimal solutions for problems with 10,000 features.

Demonstrates superior statistical performance over $\, ext{l}_1$-based methods.

Achieves shorter computation times with integrated MILO and first-order methods.

Abstract

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 in [8] and the continuous nonconvex quadratic optimization framework of [33]. We propose new discrete first-order methods, which when paired with 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 $\ell_{1}$-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.