Best Subset Selection via a Modern Optimization Lens

TL;DR

This paper introduces a mixed integer optimization approach for best subset selection in linear regression, achieving high-quality solutions efficiently and outperforming traditional methods like Lasso in predictive accuracy and sparsity.

Contribution

It develops a novel MIO-based algorithm that guarantees near-optimal solutions with early stopping, handles side constraints, and extends to absolute deviation loss functions.

Findings

Solves subset selection problems with thousands of features in minutes.

Provides solutions with provable suboptimality guarantees.

Outperforms Lasso and similar methods in predictive power and sparsity.

Abstract

In the last twenty-five years (1990-2014), algorithmic advances in integer optimization combined with hardware improvements have resulted in an astonishing 200 billion factor speedup in solving Mixed Integer Optimization (MIO) problems. We present a MIO approach for solving the classical best subset selection problem of choosing $k$ out of $p$ features in linear regression given $n$ observations. We develop a discrete extension of modern first order continuous optimization methods to find high quality feasible solutions that we use as warm starts to a MIO solver that finds provably optimal solutions. The resulting algorithm (a) provides a solution with a guarantee on its suboptimality even if we terminate the algorithm early, (b) can accommodate side constraints on the coefficients of the linear regression and (c) extends to finding best subset solutions for the least absolute deviation loss function. Using a wide variety of synthetic and real datasets, we demonstrate that our approach solves problems with $n$ in the 1000s and $p$ in the 100s in minutes to provable optimality, and finds near optimal solutions for $n$ in the 100s and $p$ in the 1000s in minutes. We also establish via numerical experiments that the MIO approach performs better than {\texttt {Lasso}} and other popularly used sparse learning procedures, in terms of achieving sparse solutions with good predictive power.