The Backbone Method for Ultra-High Dimensional Sparse Machine Learning

We present the backbone method, a generic framework that enables sparse and interpretable supervised machine learning methods to scale to ultra-high dimensional problems. We solve sparse regression problems with $10^7$ features in minutes and $10^8$ features in hours, as well as decision tree problems with $10^5$ features in minutes. The proposed method operates in two phases; we first determine the backbone set, that consists of potentially relevant features, by solving a number of tractable subproblems; then, we solve a reduced problem, considering only the backbone features. For the sparse regression problem, we show that, under certain assumptions and with high probability, the backbone set consists of the true relevant features. Numerical experiments on both synthetic and real-world datasets demonstrate that our method outperforms or competes with state-of-the-art methods in ultra-high dimensional problems, and competes with optimal solutions in problems where exact methods scale, both in terms of recovering the true relevant features and in its out-of-sample predictive performance.