Fast Best Subset Selection: Coordinate Descent and Local Combinatorial Optimization Algorithms

The $L_0$-regularized least squares problem (a.k.a. best subsets) is central\nto sparse statistical learning and has attracted significant attention across\nthe wider statistics, machine learning, and optimization communities. Recent\nwork has shown that modern mixed integer optimization (MIO) solvers can be used\nto address small to moderate instances of this problem. In spite of the\nusefulness of $L_0$-based estimators and generic MIO solvers, there is a steep\ncomputational price to pay when compared to popular sparse learning algorithms\n(e.g., based on $L_1$ regularization). In this paper, we aim to push the\nfrontiers of computation for a family of $L_0$-regularized problems with\nadditional convex penalties. We propose a new hierarchy of necessary optimality\nconditions for these problems. We develop fast algorithms, based on coordinate\ndescent and local combinatorial optimization, that are guaranteed to converge\nto solutions satisfying these optimality conditions. From a statistical\nviewpoint, an interesting story emerges. When the signal strength is high, our\ncombinatorial optimization algorithms have an edge in challenging statistical\nsettings. When the signal is lower, pure $L_0$ benefits from additional convex\nregularization. We empirically demonstrate that our family of $L_0$-based\nestimators can outperform the state-of-the-art sparse learning algorithms in\nterms of a combination of prediction, estimation, and variable selection\nmetrics under various regimes (e.g., different signal strengths, feature\ncorrelations, number of samples and features). Our new open-source sparse\nlearning toolkit L0Learn (available on CRAN and Github) reaches up to a\nthree-fold speedup (with $p$ up to $10^6$) when compared to competing toolkits\nsuch as glmnet and ncvreg.\n