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We consider the least squares regression problem, penalized with acombination of the $\ell_{0}$ and squared $\ell_{2}$ penalty functions (a.k.a.$\ell_0 \ell_2$ regularization). Recent work shows that the resultingestimators are of key importance in many high-dimensional statistical settings.However, exact computation of these estimators remains a major challenge.Indeed, modern exact methods, based on mixed integer programming (MIP), facedifficulties when the number of features $p \sim 10^4$. In this work, wepresent a new exact MIP framework for $\ell_0\ell_2$-regularized regressionthat can scale to $p \sim 10^7$, achieving speedups of at least $5000$x,compared to state-of-the-art exact methods. Unlike recent work, which relies onmodern commercial MIP solvers, we design a specialized nonlinearbranch-and-bound (BnB) framework, by critically exploiting the problemstructure. A key distinguishing component in our framework lies in efficientlysolving the node relaxations using a specialized first-order method, based oncoordinate descent (CD). Our CD-based method effectively leverages informationacross the BnB nodes, through using warm starts, active sets, and gradientscreening. In addition, we design a novel method for obtaining dual bounds fromprimal CD solutions, which certifiably works in high dimensions. Experiments onsynthetic and real high-dimensional datasets demonstrate that our framework isnot only significantly faster than the state of the art, but can also delivercertifiably optimal solutions to statistically challenging instances thatcannot be handled with existing methods. We open source the implementationthrough our toolkit L0BnB.

Sparse Regression at Scale: Branch-and-Bound rooted in First-Order Optimization