Constraint 2b checks the loss of each sample and the derivation is shown as follows. Constraint 2d to 2i are mainly adopted from Bertsimas and Dunn [2019] on Chapter 8.2. Here we briefly explain the meaning and derivations of these constraints. Constraint 2g and 2h are set to ensure that if there's no split on Constraint 2i enforces the hierarchical structure of the tree. Algorithm 1 depicts the details of the Branch-and-bound scheme for training the optimal decision tree.

Variations in complex traits are influenced by multiple genetic variants, environmental risk factors, and their interactions. Though substantial progress has been made in identifying single genetic variants associated with complex traits, detecting the gene-gene and gene-environment interactions remains a great challenge. When a large number of genetic variants and environmental risk factors are involved, searching for interactions is limited to pair-wise interactions due to the exponentially increased feature space and computational intensity. Alternatively, recursive partitioning approaches, such as random forests, have gained popularity in high-dimensional genetic association studies. In this article, we propose a U-Statistic-based random forest approach, referred to as Forest U-Test, for genetic association studies with quantitative traits. Through simulation studies, we showed that the Forest U-Test outperformed existing methods. The proposed method was also applied to study Cannabis Dependence CD, using three independent datasets from the Study of Addiction: Genetics and Environment. A significant joint association was detected with an empirical p-value less than 0.001. The finding was also replicated in two independent datasets with p-values of 5.93e-19 and 4.70e-17, respectively.

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Decision trees offer this inter-pretability, especially when they are small.

This section summarizes the different algorithms used for the Section 5 numerical studies. For all other benchmarks we use max_depth =3 and num_boost_rounds = 50 . ' and activate the deterministic Default values are used for all other hyperparameters. Figure 1 presents results of benchmark problems with known constraints. Domain bounds without decimals indicate integer-valued variable types.

Tree ensembles can be well-suited for black-box optimization tasks such as algorithm tuning and neural architecture search, as they achieve good predictive performance with little or no manual tuning, naturally handle discrete feature spaces, and are relatively insensitive to outliers in the training data.