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Decision trees are widely-used classification and regression models becauseof their interpretability and good accuracy. Classical methods such as CART arebased on greedy approaches but a growing attention has recently been devoted tooptimal decision trees. We investigate the nonlinear continuous optimizationformulation proposed in Blanquero et al. (EJOR, vol. 284, 2020; COR, vol. 132,2021) for (sparse) optimal randomized classification trees. Sparsity isimportant not only for feature selection but also to improve interpretability.We first consider alternative methods to sparsify such trees based on concaveapproximations of the $l_{0}$ ``norm". Promising results are obtained on 24datasets in comparison with $l_1$ and $l_{\infty}$ regularizations. Then, wederive bounds on the VC dimension of multivariate randomized classificationtrees. Finally, since training is computationally challenging for largedatasets, we propose a general decomposition scheme and an efficient version ofit. Experiments on larger datasets show that the proposed decomposition methodis able to significantly reduce the training times without compromising theaccuracy.

On multivariate randomized classification trees: $l_0$-based sparsity, VC~dimension and decomposition methods