Provable Boolean Interaction Recovery from Tree Ensemble obtained via Random Forests

Merle Behr, Yu Wang, Xiao Li, and Bin Yu

Random Forests (RF) are at the cutting edge of supervised machine learning in terms of prediction performance, especially in genomics. Iterative Random Forests (iRF) use a tree ensemble from iteratively modified RF to obtain predictive and stable non-linear high-order Boolean interactions of features. They have shown great promise for high-order biological interaction discovery that is central to advancing functional genomics and precision medicine. However, theoretical studies into how tree-based methods discover high-order feature interactions are missing. In this paper, to enable such theoretical studies, we first introduce a novel discontinuous nonlinear regression model, called Locally Spiky Sparse (LSS) model, which is inspired by the thresholding behavior in many biological processes. Specifically, LSS model assumes that the regression function is a linear combination of piece-wise constant Boolean interaction terms. We define a quantity called depth-weighted prevalence (DWP) for a set of signed features S and a given RF tree ensemble. We prove that, with high probability under the LSS model, DWP of S attains a universal upper bound that does not involve any model coefficients, if and only if S corresponds to a union of Boolean interactions in the LSS model. As a consequence, we show that RF yields consistent interaction discovery under the LSS model. Simulation results show that DWP can recover the interactions under the LSS model even when some assumptions such as the uniformity assumption are violated.   To appear in Proceedings of the National Academy of Science (PNAS).