Tree ensembles achieve state-of-the-art performance on numerous prediction tasks. We propose Joint Optimization of Piecewise Linear ENsembles (JOPLEN), which jointly fits piecewise linear models at all leaf nodes of an existing tree ensemble. In addition to enhancing the expressiveness of an ensemble, JOPLEN allows several common penalties, including sparsity-promoting matrix norms and subspace-norms, to be applied to nonlinear prediction. We demonstrate the performance of JOPLEN on over 100 regression and classification datasets and with a variety of penalties. JOPLEN leads to improved prediction performance relative to not only standard random forest and gradient boosted tree ensembles, but also other methods for enhancing tree ensembles. We demonstrate that JOPLEN with a nuclear norm penalty learns subspace-aligned functions. Additionally, JOPLEN combined with a Dirty LASSO penalty is an effective feature selection method for nonlinear prediction in multitask learning.
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